3.2 Test file Number [57] 3-Logarithms/3.1.4-f-x-^m-d+e-x^r-^q-a+b-log-c-x^n-^p

3.2.1 Mathematica

Integral number [166] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx \]

[B]   time = 0.104714 (sec), size = 72 ,normalized size = 2.77 \[ \frac {x (f x)^m \left ((m+1) \, _2F_1\left (1,m+1;m+2;-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (1,m+1,m+1;m+2,m+2;-\frac {e x}{d}\right )\right )}{d (m+1)^2} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x),x]

[Out]

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{1, 1 + m, 1 + m}, {2 + m, 2 + m}, -((e*x)/d)]) + (1 + m)*Hypergeometric2F
1[1, 1 + m, 2 + m, -((e*x)/d)]*(a + b*Log[c*x^n])))/(d*(1 + m)^2)

Integral number [167] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx \]

[B]   time = 0.107026 (sec), size = 72 ,normalized size = 2.77 \[ \frac {x (f x)^m \left ((m+1) \, _2F_1\left (2,m+1;m+2;-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (2,m+1,m+1;m+2,m+2;-\frac {e x}{d}\right )\right )}{d^2 (m+1)^2} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x)^2,x]

[Out]

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{2, 1 + m, 1 + m}, {2 + m, 2 + m}, -((e*x)/d)]) + (1 + m)*Hypergeometric2F
1[2, 1 + m, 2 + m, -((e*x)/d)]*(a + b*Log[c*x^n])))/(d^2*(1 + m)^2)

Integral number [168] \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx \]

[B]   time = 0.250616 (sec), size = 173 ,normalized size = 9.61 \[ \frac {(a+b x)^m \left (\frac {b x}{a}+1\right )^{-m} \left (a b (m+2) n x \, _3F_2\left (1,1,-m-1;2,2;-\frac {b x}{a}\right )+\left (-a^2 \left (\left (\frac {b x}{a}+1\right )^m-1\right )+b^2 (m+1) x^2 \left (\frac {b x}{a}+1\right )^m+a b m x \left (\frac {b x}{a}+1\right )^m\right ) \log \left (c x^n\right )-n \left (a^2 \left (\left (\frac {b x}{a}+1\right )^m-1\right )+b^2 x^2 \left (\frac {b x}{a}+1\right )^m+2 a b x \left (\frac {b x}{a}+1\right )^m\right )\right )}{b^2 (m+1) (m+2)} \]

[In]

Integrate[x*(a + b*x)^m*Log[c*x^n],x]

[Out]

((a + b*x)^m*(-(n*(2*a*b*x*(1 + (b*x)/a)^m + b^2*x^2*(1 + (b*x)/a)^m + a^2*(-1 + (1 + (b*x)/a)^m))) + a*b*(2 +
m)*n*x*HypergeometricPFQ[{1, 1, -1 - m}, {2, 2}, -((b*x)/a)] + (a*b*m*x*(1 + (b*x)/a)^m + b^2*(1 + m)*x^2*(1
+ (b*x)/a)^m - a^2*(-1 + (1 + (b*x)/a)^m))*Log[c*x^n]))/(b^2*(1 + m)*(2 + m)*(1 + (b*x)/a)^m)

Integral number [170] \[ \int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx \]

[B]   time = 0.0644198 (sec), size = 89 ,normalized size = 4.45 \[ \frac {\left (\frac {a}{b x}+1\right )^{-m} (a+b x)^m \left (m \log \left (c x^n\right ) \, _2F_1\left (-m,-m;1-m;-\frac {a}{b x}\right )-n \, _3F_2\left (-m,-m,-m;1-m,1-m;-\frac {a}{b x}\right )\right )}{m^2} \]

[In]

Integrate[((a + b*x)^m*Log[c*x^n])/x,x]

[Out]

((a + b*x)^m*(-(n*HypergeometricPFQ[{-m, -m, -m}, {1 - m, 1 - m}, -(a/(b*x))]) + m*Hypergeometric2F1[-m, -m, 1
- m, -(a/(b*x))]*Log[c*x^n]))/(m^2*(1 + a/(b*x))^m)

Integral number [322] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx \]

[B]   time = 0.208145 (sec), size = 108 ,normalized size = 3.86 \[ \frac {x (f x)^m \left ((m+1) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {e x^2}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (1,\frac {m}{2}+\frac {1}{2},\frac {m}{2}+\frac {1}{2};\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};-\frac {e x^2}{d}\right )\right )}{d (m+1)^2} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^2),x]

[Out]

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{1, 1/2 + m/2, 1/2 + m/2}, {3/2 + m/2, 3/2 + m/2}, -((e*x^2)/d)]) + (1 + m
)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)]*(a + b*Log[c*x^n])))/(d*(1 + m)^2)

Integral number [323] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2} \, dx \]

[B]   time = 0.128838 (sec), size = 108 ,normalized size = 3.86 \[ \frac {x (f x)^m \left ((m+1) \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {e x^2}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (2,\frac {m}{2}+\frac {1}{2},\frac {m}{2}+\frac {1}{2};\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};-\frac {e x^2}{d}\right )\right )}{d^2 (m+1)^2} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^2)^2,x]

[Out]

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{2, 1/2 + m/2, 1/2 + m/2}, {3/2 + m/2, 3/2 + m/2}, -((e*x^2)/d)]) + (1 + m
)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)]*(a + b*Log[c*x^n])))/(d^2*(1 + m)^2)

Integral number [406] \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[B]   time = 0.124222 (sec), size = 87 ,normalized size = 3.35 \[ \frac {x^4 \left (4 \, _2F_1\left (1,\frac {4}{r};\frac {r+4}{r};-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (1,\frac {4}{r},\frac {4}{r};1+\frac {4}{r},1+\frac {4}{r};-\frac {e x^r}{d}\right )\right )}{16 d} \]

[In]

Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e*x^r),x]

[Out]

(x^4*(-(b*n*HypergeometricPFQ[{1, 4/r, 4/r}, {1 + 4/r, 1 + 4/r}, -((e*x^r)/d)]) + 4*Hypergeometric2F1[1, 4/r,
(4 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/(16*d)

Integral number [407] \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[B]   time = 0.106702 (sec), size = 87 ,normalized size = 3.62 \[ \frac {x^2 \left (2 \, _2F_1\left (1,\frac {2}{r};\frac {r+2}{r};-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (1,\frac {2}{r},\frac {2}{r};1+\frac {2}{r},1+\frac {2}{r};-\frac {e x^r}{d}\right )\right )}{4 d} \]

[In]

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x^r),x]

[Out]

(x^2*(-(b*n*HypergeometricPFQ[{1, 2/r, 2/r}, {1 + 2/r, 1 + 2/r}, -((e*x^r)/d)]) + 2*Hypergeometric2F1[1, 2/r,
(2 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/(4*d)

Integral number [409] \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )} \, dx \]

[B]   time = 0.117141 (sec), size = 86 ,normalized size = 3.31 \[ -\frac {b n \, _3F_2\left (1,-\frac {2}{r},-\frac {2}{r};1-\frac {2}{r},1-\frac {2}{r};-\frac {e x^r}{d}\right )+2 \, _2F_1\left (1,-\frac {2}{r};\frac {r-2}{r};-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d x^2} \]

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)),x]

[Out]

-1/4*(b*n*HypergeometricPFQ[{1, -2/r, -2/r}, {1 - 2/r, 1 - 2/r}, -((e*x^r)/d)] + 2*Hypergeometric2F1[1, -2/r,
(-2 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n]))/(d*x^2)

Integral number [410] \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[B]   time = 0.114966 (sec), size = 87 ,normalized size = 3.35 \[ \frac {x^3 \left (3 \, _2F_1\left (1,\frac {3}{r};\frac {r+3}{r};-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (1,\frac {3}{r},\frac {3}{r};1+\frac {3}{r},1+\frac {3}{r};-\frac {e x^r}{d}\right )\right )}{9 d} \]

[In]

Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x^r),x]

[Out]

(x^3*(-(b*n*HypergeometricPFQ[{1, 3/r, 3/r}, {1 + 3/r, 1 + 3/r}, -((e*x^r)/d)]) + 3*Hypergeometric2F1[1, 3/r,
(3 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/(9*d)

Integral number [411] \[ \int \frac {a+b \log \left (c x^n\right )}{d+e x^r} \, dx \]

[B]   time = 0.085968 (sec), size = 69 ,normalized size = 3. \[ \frac {x \left (\, _2F_1\left (1,\frac {1}{r};1+\frac {1}{r};-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (1,\frac {1}{r},\frac {1}{r};1+\frac {1}{r},1+\frac {1}{r};-\frac {e x^r}{d}\right )\right )}{d} \]

[In]

Integrate[(a + b*Log[c*x^n])/(d + e*x^r),x]

[Out]

(x*(-(b*n*HypergeometricPFQ[{1, r^(-1), r^(-1)}, {1 + r^(-1), 1 + r^(-1)}, -((e*x^r)/d)]) + Hypergeometric2F1[
1, r^(-1), 1 + r^(-1), -((e*x^r)/d)]*(a + b*Log[c*x^n])))/d

Integral number [412] \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )} \, dx \]

[B]   time = 0.103256 (sec), size = 83 ,normalized size = 3.19 \[ -\frac {b n \, _3F_2\left (1,-\frac {1}{r},-\frac {1}{r};1-\frac {1}{r},1-\frac {1}{r};-\frac {e x^r}{d}\right )+\, _2F_1\left (1,-\frac {1}{r};\frac {r-1}{r};-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d x} \]

[In]

Integrate[(a + b*Log[c*x^n])/(x^2*(d + e*x^r)),x]

[Out]

-((b*n*HypergeometricPFQ[{1, -r^(-1), -r^(-1)}, {1 - r^(-1), 1 - r^(-1)}, -((e*x^r)/d)] + Hypergeometric2F1[1,
-r^(-1), (-1 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n]))/(d*x))

Integral number [413] \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.261741 (sec), size = 140 ,normalized size = 5.38 \[ \frac {x^4 \left (-b n (r-4) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {4}{r},\frac {4}{r};1+\frac {4}{r},1+\frac {4}{r};-\frac {e x^r}{d}\right )+4 \left (d+e x^r\right ) \, _2F_1\left (1,\frac {4}{r};\frac {r+4}{r};-\frac {e x^r}{d}\right ) \left (a (r-4)+b (r-4) \log \left (c x^n\right )-b n\right )+16 d \left (a+b \log \left (c x^n\right )\right )\right )}{16 d^2 r \left (d+e x^r\right )} \]

[In]

Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

[Out]

(x^4*(-(b*n*(-4 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 4/r, 4/r}, {1 + 4/r, 1 + 4/r}, -((e*x^r)/d)]) + 16*d*(a
+ b*Log[c*x^n]) + 4*(d + e*x^r)*Hypergeometric2F1[1, 4/r, (4 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-4 + r) + b*(
-4 + r)*Log[c*x^n])))/(16*d^2*r*(d + e*x^r))

Integral number [414] \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.239464 (sec), size = 140 ,normalized size = 5.83 \[ \frac {x^2 \left (-b n (r-2) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {2}{r},\frac {2}{r};1+\frac {2}{r},1+\frac {2}{r};-\frac {e x^r}{d}\right )+2 \left (d+e x^r\right ) \, _2F_1\left (1,\frac {2}{r};\frac {r+2}{r};-\frac {e x^r}{d}\right ) \left (a (r-2)+b (r-2) \log \left (c x^n\right )-b n\right )+4 d \left (a+b \log \left (c x^n\right )\right )\right )}{4 d^2 r \left (d+e x^r\right )} \]

[In]

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

[Out]

(x^2*(-(b*n*(-2 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 2/r, 2/r}, {1 + 2/r, 1 + 2/r}, -((e*x^r)/d)]) + 4*d*(a
+ b*Log[c*x^n]) + 2*(d + e*x^r)*Hypergeometric2F1[1, 2/r, (2 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-2 + r) + b*(-
2 + r)*Log[c*x^n])))/(4*d^2*r*(d + e*x^r))

Integral number [416] \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx \]

[B]   time = 3.16006 (sec), size = 205 ,normalized size = 7.88 \[ \frac {4 b e n (r+2) x^r \left (d+e x^r\right ) \, _3F_2\left (1,1-\frac {2}{r},1-\frac {2}{r};2-\frac {2}{r},2-\frac {2}{r};-\frac {e x^r}{d}\right )-(r-2) \left (4 e x^r \left (d+e x^r\right ) \, _2F_1\left (1,\frac {r-2}{r};2-\frac {2}{r};-\frac {e x^r}{d}\right ) \left (a (r+2)+b (r+2) \log \left (c x^n\right )-b n\right )+d (r-2) \left (2 a \left (d r+e (r+2) x^r\right )+2 b \log \left (c x^n\right ) \left (d r+e (r+2) x^r\right )+b n r \left (d+e x^r\right )\right )\right )}{4 d^3 (r-2)^2 r x^2 \left (d+e x^r\right )} \]

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2),x]

[Out]

(4*b*e*n*(2 + r)*x^r*(d + e*x^r)*HypergeometricPFQ[{1, 1 - 2/r, 1 - 2/r}, {2 - 2/r, 2 - 2/r}, -((e*x^r)/d)] -
(-2 + r)*(4*e*x^r*(d + e*x^r)*Hypergeometric2F1[1, (-2 + r)/r, 2 - 2/r, -((e*x^r)/d)]*(-(b*n) + a*(2 + r) + b*
(2 + r)*Log[c*x^n]) + d*(-2 + r)*(b*n*r*(d + e*x^r) + 2*a*(d*r + e*(2 + r)*x^r) + 2*b*(d*r + e*(2 + r)*x^r)*Lo
g[c*x^n])))/(4*d^3*(-2 + r)^2*r*x^2*(d + e*x^r))

Integral number [417] \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.249699 (sec), size = 140 ,normalized size = 5.38 \[ \frac {x^3 \left (-b n (r-3) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {3}{r},\frac {3}{r};1+\frac {3}{r},1+\frac {3}{r};-\frac {e x^r}{d}\right )+3 \left (d+e x^r\right ) \, _2F_1\left (1,\frac {3}{r};\frac {r+3}{r};-\frac {e x^r}{d}\right ) \left (a (r-3)+b (r-3) \log \left (c x^n\right )-b n\right )+9 d \left (a+b \log \left (c x^n\right )\right )\right )}{9 d^2 r \left (d+e x^r\right )} \]

[In]

Integrate[(x^2*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

[Out]

(x^3*(-(b*n*(-3 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 3/r, 3/r}, {1 + 3/r, 1 + 3/r}, -((e*x^r)/d)]) + 9*d*(a
+ b*Log[c*x^n]) + 3*(d + e*x^r)*Hypergeometric2F1[1, 3/r, (3 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-3 + r) + b*(-
3 + r)*Log[c*x^n])))/(9*d^2*r*(d + e*x^r))

Integral number [418] \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 2.61975 (sec), size = 161 ,normalized size = 7. \[ \frac {x \left (-b n (r-1) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {1}{r},\frac {1}{r};1+\frac {1}{r},1+\frac {1}{r};-\frac {e x^r}{d}\right )+a e r x^r \, _2F_1\left (2,\frac {1}{r};1+\frac {1}{r};-\frac {e x^r}{d}\right )+a d r \, _2F_1\left (2,\frac {1}{r};1+\frac {1}{r};-\frac {e x^r}{d}\right )-b \left (d+e x^r\right ) \left (n-(r-1) \log \left (c x^n\right )\right ) \, _2F_1\left (1,\frac {1}{r};1+\frac {1}{r};-\frac {e x^r}{d}\right )+b d \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )} \]

[In]

Integrate[(a + b*Log[c*x^n])/(d + e*x^r)^2,x]

[Out]

(x*(a*d*r*Hypergeometric2F1[2, r^(-1), 1 + r^(-1), -((e*x^r)/d)] + a*e*r*x^r*Hypergeometric2F1[2, r^(-1), 1 +
r^(-1), -((e*x^r)/d)] - b*n*(-1 + r)*(d + e*x^r)*HypergeometricPFQ[{1, r^(-1), r^(-1)}, {1 + r^(-1), 1 + r^(-1
)}, -((e*x^r)/d)] + b*d*Log[c*x^n] - b*(d + e*x^r)*Hypergeometric2F1[1, r^(-1), 1 + r^(-1), -((e*x^r)/d)]*(n -
(-1 + r)*Log[c*x^n])))/(d^2*r*(d + e*x^r))

Integral number [419] \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.204882 (sec), size = 135 ,normalized size = 5.19 \[ \frac {-b n (r+1) \left (d+e x^r\right ) \, _3F_2\left (1,-\frac {1}{r},-\frac {1}{r};1-\frac {1}{r},1-\frac {1}{r};-\frac {e x^r}{d}\right )-\left (d+e x^r\right ) \, _2F_1\left (1,-\frac {1}{r};\frac {r-1}{r};-\frac {e x^r}{d}\right ) \left (a r+a+b (r+1) \log \left (c x^n\right )-b n\right )+d \left (a+b \log \left (c x^n\right )\right )}{d^2 r x \left (d+e x^r\right )} \]

[In]

Integrate[(a + b*Log[c*x^n])/(x^2*(d + e*x^r)^2),x]

[Out]

(-(b*n*(1 + r)*(d + e*x^r)*HypergeometricPFQ[{1, -r^(-1), -r^(-1)}, {1 - r^(-1), 1 - r^(-1)}, -((e*x^r)/d)]) +
d*(a + b*Log[c*x^n]) - (d + e*x^r)*Hypergeometric2F1[1, -r^(-1), (-1 + r)/r, -((e*x^r)/d)]*(a - b*n + a*r + b
*(1 + r)*Log[c*x^n]))/(d^2*r*x*(d + e*x^r))

Integral number [444] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[B]   time = 0.156347 (sec), size = 111 ,normalized size = 3.96 \[ \frac {x (f x)^m \left ((m+1) \left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (1,\frac {m+1}{r};\frac {m+r+1}{r};-\frac {e x^r}{d}\right )-b n \, _3F_2\left (1,\frac {m}{r}+\frac {1}{r},\frac {m}{r}+\frac {1}{r};\frac {m}{r}+\frac {1}{r}+1,\frac {m}{r}+\frac {1}{r}+1;-\frac {e x^r}{d}\right )\right )}{d (m+1)^2} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^r),x]

[Out]

(x*(f*x)^m*(-(b*n*HypergeometricPFQ[{1, r^(-1) + m/r, r^(-1) + m/r}, {1 + r^(-1) + m/r, 1 + r^(-1) + m/r}, -((
e*x^r)/d)]) + (1 + m)*Hypergeometric2F1[1, (1 + m)/r, (1 + m + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/(d*(1
+ m)^2)

Integral number [445] \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

[B]   time = 0.386001 (sec), size = 177 ,normalized size = 6.32 \[ \frac {x (f x)^m \left (b n (m-r+1) \left (d+e x^r\right ) \, _3F_2\left (1,\frac {m}{r}+\frac {1}{r},\frac {m}{r}+\frac {1}{r};\frac {m}{r}+\frac {1}{r}+1,\frac {m}{r}+\frac {1}{r}+1;-\frac {e x^r}{d}\right )-(m+1) \left (\left (d+e x^r\right ) \, _2F_1\left (1,\frac {m+1}{r};\frac {m+r+1}{r};-\frac {e x^r}{d}\right ) \left (a (m-r+1)+b (m-r+1) \log \left (c x^n\right )+b n\right )-d (m+1) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{d^2 (m+1)^2 r \left (d+e x^r\right )} \]

[In]

Integrate[((f*x)^m*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

[Out]

(x*(f*x)^m*(b*n*(1 + m - r)*(d + e*x^r)*HypergeometricPFQ[{1, r^(-1) + m/r, r^(-1) + m/r}, {1 + r^(-1) + m/r,
1 + r^(-1) + m/r}, -((e*x^r)/d)] - (1 + m)*(-(d*(1 + m)*(a + b*Log[c*x^n])) + (d + e*x^r)*Hypergeometric2F1[1,
(1 + m)/r, (1 + m + r)/r, -((e*x^r)/d)]*(b*n + a*(1 + m - r) + b*(1 + m - r)*Log[c*x^n]))))/(d^2*(1 + m)^2*r*
(d + e*x^r))