ODE
\[ (5 y(x)+3 x+6) y'(x)=7 y(x)+x+2 \] ODE Classification
[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]
Book solution method
Equation linear in the variables, \(y'(x)=f\left ( \frac {X_1}{X_2} \right ) \)
Mathematica ✓
cpu = 0.355231 (sec), leaf count = 4961
\[\left \{\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,1\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,2\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,3\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,4\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,5\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,6\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,7\right ]}-3 (x+2)\right )\right \},\left \{y(x)\to \frac {1}{5} \left (\frac {1}{\text {Root}\left [\left (16777216 e^{\frac {24 c_1}{25}} x^8+268435456 e^{\frac {24 c_1}{25}} x^7+1879048192 e^{\frac {24 c_1}{25}} x^6+7516192768 e^{\frac {24 c_1}{25}} x^5+18790481920 e^{\frac {24 c_1}{25}} x^4+30064771072 e^{\frac {24 c_1}{25}} x^3+30064771072 e^{\frac {24 c_1}{25}} x^2+16777216 x^2+17179869184 e^{\frac {24 c_1}{25}} x+67108864 x+4294967296 e^{\frac {24 c_1}{25}}+67108864\right ) \text {$\#$1}^8+\left (-16777216 e^{\frac {24 c_1}{25}} x^7-234881024 e^{\frac {24 c_1}{25}} x^6-1409286144 e^{\frac {24 c_1}{25}} x^5-4697620480 e^{\frac {24 c_1}{25}} x^4-9395240960 e^{\frac {24 c_1}{25}} x^3-11274289152 e^{\frac {24 c_1}{25}} x^2-7516192768 e^{\frac {24 c_1}{25}} x-16777216 x-2147483648 e^{\frac {24 c_1}{25}}-33554432\right ) \text {$\#$1}^7+\left (7340032 e^{\frac {24 c_1}{25}} x^6+88080384 e^{\frac {24 c_1}{25}} x^5+440401920 e^{\frac {24 c_1}{25}} x^4+1174405120 e^{\frac {24 c_1}{25}} x^3+1761607680 e^{\frac {24 c_1}{25}} x^2+1409286144 e^{\frac {24 c_1}{25}} x+469762048 e^{\frac {24 c_1}{25}}+4194304\right ) \text {$\#$1}^6+\left (-1835008 e^{\frac {24 c_1}{25}} x^5-18350080 e^{\frac {24 c_1}{25}} x^4-73400320 e^{\frac {24 c_1}{25}} x^3-146800640 e^{\frac {24 c_1}{25}} x^2-146800640 e^{\frac {24 c_1}{25}} x-58720256 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^5+\left (286720 e^{\frac {24 c_1}{25}} x^4+2293760 e^{\frac {24 c_1}{25}} x^3+6881280 e^{\frac {24 c_1}{25}} x^2+9175040 e^{\frac {24 c_1}{25}} x+4587520 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^4+\left (-28672 e^{\frac {24 c_1}{25}} x^3-172032 e^{\frac {24 c_1}{25}} x^2-344064 e^{\frac {24 c_1}{25}} x-229376 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^3+\left (1792 e^{\frac {24 c_1}{25}} x^2+7168 e^{\frac {24 c_1}{25}} x+7168 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}^2+\left (-64 e^{\frac {24 c_1}{25}} x-128 e^{\frac {24 c_1}{25}}\right ) \text {$\#$1}+e^{\frac {24 c_1}{25}}\& ,8\right ]}-3 (x+2)\right )\right \}\right \}\]
Maple ✓
cpu = 0.03 (sec), leaf count = 47
\[ \left \{ -{\frac {4}{3}\ln \left ( {\frac {-y \left ( x \right ) +2+x}{2+x}} \right ) }+{\frac {1}{3}\ln \left ( {\frac {-5\,y \left ( x \right ) -2-x}{2+x}} \right ) }-\ln \left ( 2+x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[(6 + 3*x + 5*y[x])*y'[x] == 2 + x + 7*y[x],y[x],x]
Mathematica raw output
{{y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 1]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 2]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 3]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 4]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 5]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 6]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 7]^(-1))/5}, {y[x] -> (-3*(2 + x) + Root[E^((24*C[1])/25) + (-128*E^((24*C[1])/25) - 64*E^((24*C[1])/25)*x)*#1 + (7168*E^((24*C[1])/25) + 7168*E^((24*C[1])/25)*x + 1792*E^((24*C[1])/25)*x^2)*#1^2 + (-229376*E^((24*C[1])/25) - 344064*E^((24*C[1])/25)*x - 172032*E^((24*C[1])/25)*x^2 - 28672*E^((24*C[1])/25)*x^3)*#1^3 + (4587520*E^((24*C[1])/25) + 9175040*E^((24*C[1])/25)*x + 6881280*E^((24*C[1])/25)*x^2 + 2293760*E^((24*C[1])/25)*x^3 + 286720*E^((24*C[1])/25)*x^4)*#1^4 + (-58720256*E^((24*C[1])/25) - 146800640*E^((24*C[1])/25)*x - 146800640*E^((24*C[1])/25)*x^2 - 73400320*E^((24*C[1])/25)*x^3 - 18350080*E^((24*C[1])/25)*x^4 - 1835008*E^((24*C[1])/25)*x^5)*#1^5 + (4194304 + 469762048*E^((24*C[1])/25) + 1409286144*E^((24*C[1])/25)*x + 1761607680*E^((24*C[1])/25)*x^2 + 1174405120*E^((24*C[1])/25)*x^3 + 440401920*E^((24*C[1])/25)*x^4 + 88080384*E^((24*C[1])/25)*x^5 + 7340032*E^((24*C[1])/25)*x^6)*#1^6 + (-33554432 - 2147483648*E^((24*C[1])/25) - 16777216*x - 7516192768*E^((24*C[1])/25)*x - 11274289152*E^((24*C[1])/25)*x^2 - 9395240960*E^((24*C[1])/25)*x^3 - 4697620480*E^((24*C[1])/25)*x^4 - 1409286144*E^((24*C[1])/25)*x^5 - 234881024*E^((24*C[1])/25)*x^6 - 16777216*E^((24*C[1])/25)*x^7)*#1^7 + (67108864 + 4294967296*E^((24*C[1])/25) + 67108864*x + 17179869184*E^((24*C[1])/25)*x + 16777216*x^2 + 30064771072*E^((24*C[1])/25)*x^2 + 30064771072*E^((24*C[1])/25)*x^3 + 18790481920*E^((24*C[1])/25)*x^4 + 7516192768*E^((24*C[1])/25)*x^5 + 1879048192*E^((24*C[1])/25)*x^6 + 268435456*E^((24*C[1])/25)*x^7 + 16777216*E^((24*C[1])/25)*x^8)*#1^8 & , 8]^(-1))/5}}
Maple raw input
dsolve((6+3*x+5*y(x))*diff(y(x),x) = 2+x+7*y(x), y(x),'implicit')
Maple raw output
-4/3*ln((-y(x)+2+x)/(2+x))+1/3*ln((-5*y(x)-2-x)/(2+x))-ln(2+x)-_C1 = 0