Integrand size = 24, antiderivative size = 564 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\frac {b^2 e^2 n^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g)^3 (f+g x)}+\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (e f-d g) (f+g x)^2}-\frac {b e^2 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g)^3 (f+g x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}-\frac {b^3 e^3 n^3 \log (f+g x)}{g (e f-d g)^3}+\frac {2 b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g (e f-d g)^3}+\frac {b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}-\frac {b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}-\frac {b^3 e^3 n^3 \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}+\frac {2 b^2 e^3 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3}+\frac {2 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g (e f-d g)^3}+\frac {2 b^3 e^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e f-d g}{g (d+e x)}\right )}{g (e f-d g)^3} \]
b^2*e^2*n^2*(e*x+d)*(a+b*ln(c*(e*x+d)^n))/(-d*g+e*f)^3/(g*x+f)+1/2*b*e*n*( a+b*ln(c*(e*x+d)^n))^2/g/(-d*g+e*f)/(g*x+f)^2-b*e^2*n*(e*x+d)*(a+b*ln(c*(e *x+d)^n))^2/(-d*g+e*f)^3/(g*x+f)-1/3*(a+b*ln(c*(e*x+d)^n))^3/g/(g*x+f)^3-b ^3*e^3*n^3*ln(g*x+f)/g/(-d*g+e*f)^3+2*b^2*e^3*n^2*(a+b*ln(c*(e*x+d)^n))*ln (e*(g*x+f)/(-d*g+e*f))/g/(-d*g+e*f)^3+b^2*e^3*n^2*(a+b*ln(c*(e*x+d)^n))*ln (1+(-d*g+e*f)/g/(e*x+d))/g/(-d*g+e*f)^3-b*e^3*n*(a+b*ln(c*(e*x+d)^n))^2*ln (1+(-d*g+e*f)/g/(e*x+d))/g/(-d*g+e*f)^3-b^3*e^3*n^3*polylog(2,(d*g-e*f)/g/ (e*x+d))/g/(-d*g+e*f)^3+2*b^2*e^3*n^2*(a+b*ln(c*(e*x+d)^n))*polylog(2,(d*g -e*f)/g/(e*x+d))/g/(-d*g+e*f)^3+2*b^3*e^3*n^3*polylog(2,-g*(e*x+d)/(-d*g+e *f))/g/(-d*g+e*f)^3+2*b^3*e^3*n^3*polylog(3,(d*g-e*f)/g/(e*x+d))/g/(-d*g+e *f)^3
Time = 0.71 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\frac {3 b e (e f-d g)^2 n (f+g x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+6 b e^2 (e f-d g) n (f+g x)^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-6 b (e f-d g)^3 n \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+6 b e^3 n (f+g x)^3 \log (d+e x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-2 (e f-d g)^3 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^3-6 b e^3 n (f+g x)^3 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log (f+g x)+6 b^2 n^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (e^2 g (d+e x) (f+g x)^2+g \left (3 d e^2 f^2-3 d^2 e f g+d^3 g^2+e^3 x \left (3 f^2+3 f g x+g^2 x^2\right )\right ) \log ^2(d+e x)+3 e^3 (f+g x)^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )+e (f+g x) \log (d+e x) \left (g^2 (d+e x)^2-4 e g (d+e x) (f+g x)-2 e^2 (f+g x)^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )-2 e^3 (f+g x)^3 \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+b^3 n^3 \left (2 g \left (3 d e^2 f^2-3 d^2 e f g+d^3 g^2+e^3 x \left (3 f^2+3 f g x+g^2 x^2\right )\right ) \log ^3(d+e x)-6 e^3 (f+g x)^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )+3 e (f+g x) \log ^2(d+e x) \left (g^2 (d+e x)^2-4 e g (d+e x) (f+g x)-2 e^2 (f+g x)^2 \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )+18 e^3 (f+g x)^3 \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )+6 e^2 (f+g x)^2 \log (d+e x) \left (g (d+e x)+3 e (f+g x) \log \left (\frac {e (f+g x)}{e f-d g}\right )-2 e (f+g x) \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )\right )+12 e^3 (f+g x)^3 \operatorname {PolyLog}\left (3,\frac {g (d+e x)}{-e f+d g}\right )\right )}{6 g (e f-d g)^3 (f+g x)^3} \]
(3*b*e*(e*f - d*g)^2*n*(f + g*x)*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x) ^n])^2 + 6*b*e^2*(e*f - d*g)*n*(f + g*x)^2*(a - b*n*Log[d + e*x] + b*Log[c *(d + e*x)^n])^2 - 6*b*(e*f - d*g)^3*n*Log[d + e*x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 6*b*e^3*n*(f + g*x)^3*Log[d + e*x]*(a - b*n*Lo g[d + e*x] + b*Log[c*(d + e*x)^n])^2 - 2*(e*f - d*g)^3*(a - b*n*Log[d + e* x] + b*Log[c*(d + e*x)^n])^3 - 6*b*e^3*n*(f + g*x)^3*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x] + 6*b^2*n^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(e^2*g*(d + e*x)*(f + g*x)^2 + g*(3*d*e^2*f^2 - 3* d^2*e*f*g + d^3*g^2 + e^3*x*(3*f^2 + 3*f*g*x + g^2*x^2))*Log[d + e*x]^2 + 3*e^3*(f + g*x)^3*Log[(e*(f + g*x))/(e*f - d*g)] + e*(f + g*x)*Log[d + e*x ]*(g^2*(d + e*x)^2 - 4*e*g*(d + e*x)*(f + g*x) - 2*e^2*(f + g*x)^2*Log[(e* (f + g*x))/(e*f - d*g)]) - 2*e^3*(f + g*x)^3*PolyLog[2, (g*(d + e*x))/(-(e *f) + d*g)]) + b^3*n^3*(2*g*(3*d*e^2*f^2 - 3*d^2*e*f*g + d^3*g^2 + e^3*x*( 3*f^2 + 3*f*g*x + g^2*x^2))*Log[d + e*x]^3 - 6*e^3*(f + g*x)^3*Log[(e*(f + g*x))/(e*f - d*g)] + 3*e*(f + g*x)*Log[d + e*x]^2*(g^2*(d + e*x)^2 - 4*e* g*(d + e*x)*(f + g*x) - 2*e^2*(f + g*x)^2*Log[(e*(f + g*x))/(e*f - d*g)]) + 18*e^3*(f + g*x)^3*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] + 6*e^2*(f + g*x)^2*Log[d + e*x]*(g*(d + e*x) + 3*e*(f + g*x)*Log[(e*(f + g*x))/(e*f - d*g)] - 2*e*(f + g*x)*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) + 12*e^3* (f + g*x)^3*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)]))/(6*g*(e*f - d*g)...
Time = 2.52 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2845, 2858, 27, 2789, 2756, 2789, 2751, 16, 2755, 2754, 2779, 2821, 2838, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx\) |
\(\Big \downarrow \) 2845 |
\(\displaystyle \frac {b e n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (f+g x)^3}dx}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2858 |
\(\displaystyle \frac {b n \int \frac {e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) \left (e \left (f-\frac {d g}{e}\right )+g (d+e x)\right )^3}d(d+e x)}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b e^3 n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (e f-d g+g (d+e x))^3}d(d+e x)}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2789 |
\(\displaystyle \frac {b e^3 n \left (\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (e f-d g+g (d+e x))^2}d(d+e x)}{e f-d g}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g+g (d+e x))^3}d(d+e x)}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2756 |
\(\displaystyle \frac {b e^3 n \left (\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (e f-d g+g (d+e x))^2}d(d+e x)}{e f-d g}-\frac {g \left (\frac {b n \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (e f-d g+g (d+e x))^2}d(d+e x)}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (g (d+e x)-d g+e f)^2}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2789 |
\(\displaystyle \frac {b e^3 n \left (\frac {\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (e f-d g+g (d+e x))}d(d+e x)}{e f-d g}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g+g (d+e x))^2}d(d+e x)}{e f-d g}}{e f-d g}-\frac {g \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (e f-d g+g (d+e x))}d(d+e x)}{e f-d g}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{(e f-d g+g (d+e x))^2}d(d+e x)}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (g (d+e x)-d g+e f)^2}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2751 |
\(\displaystyle \frac {b e^3 n \left (\frac {\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (e f-d g+g (d+e x))}d(d+e x)}{e f-d g}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g+g (d+e x))^2}d(d+e x)}{e f-d g}}{e f-d g}-\frac {g \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (e f-d g+g (d+e x))}d(d+e x)}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {b n \int \frac {1}{e f-d g+g (d+e x)}d(d+e x)}{e f-d g}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (g (d+e x)-d g+e f)^2}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {b e^3 n \left (\frac {\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (e f-d g+g (d+e x))}d(d+e x)}{e f-d g}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g+g (d+e x))^2}d(d+e x)}{e f-d g}}{e f-d g}-\frac {g \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (e f-d g+g (d+e x))}d(d+e x)}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {b n \log (g (d+e x)-d g+e f)}{g (e f-d g)}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (g (d+e x)-d g+e f)^2}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2755 |
\(\displaystyle \frac {b e^3 n \left (\frac {\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (e f-d g+g (d+e x))}d(d+e x)}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {2 b n \int \frac {a+b \log \left (c (d+e x)^n\right )}{e f-d g+g (d+e x)}d(d+e x)}{e f-d g}\right )}{e f-d g}}{e f-d g}-\frac {g \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (e f-d g+g (d+e x))}d(d+e x)}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {b n \log (g (d+e x)-d g+e f)}{g (e f-d g)}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (g (d+e x)-d g+e f)^2}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2754 |
\(\displaystyle \frac {b e^3 n \left (\frac {\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(d+e x) (e f-d g+g (d+e x))}d(d+e x)}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {2 b n \left (\frac {\log \left (\frac {g (d+e x)}{e f-d g}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b n \int \frac {\log \left (\frac {g (d+e x)}{e f-d g}+1\right )}{d+e x}d(d+e x)}{g}\right )}{e f-d g}\right )}{e f-d g}}{e f-d g}-\frac {g \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (e f-d g+g (d+e x))}d(d+e x)}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {b n \log (g (d+e x)-d g+e f)}{g (e f-d g)}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (g (d+e x)-d g+e f)^2}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2779 |
\(\displaystyle \frac {b e^3 n \left (\frac {\frac {\frac {2 b n \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e f-d g}{g (d+e x)}+1\right )}{d+e x}d(d+e x)}{e f-d g}-\frac {\log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e f-d g}}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {2 b n \left (\frac {\log \left (\frac {g (d+e x)}{e f-d g}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b n \int \frac {\log \left (\frac {g (d+e x)}{e f-d g}+1\right )}{d+e x}d(d+e x)}{g}\right )}{e f-d g}\right )}{e f-d g}}{e f-d g}-\frac {g \left (\frac {b n \left (\frac {\frac {b n \int \frac {\log \left (\frac {e f-d g}{g (d+e x)}+1\right )}{d+e x}d(d+e x)}{e f-d g}-\frac {\log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e f-d g}}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {b n \log (g (d+e x)-d g+e f)}{g (e f-d g)}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (g (d+e x)-d g+e f)^2}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle \frac {b e^3 n \left (\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{d+e x}d(d+e x)\right )}{e f-d g}-\frac {\log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e f-d g}}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {2 b n \left (\frac {\log \left (\frac {g (d+e x)}{e f-d g}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b n \int \frac {\log \left (\frac {g (d+e x)}{e f-d g}+1\right )}{d+e x}d(d+e x)}{g}\right )}{e f-d g}\right )}{e f-d g}}{e f-d g}-\frac {g \left (\frac {b n \left (\frac {\frac {b n \int \frac {\log \left (\frac {e f-d g}{g (d+e x)}+1\right )}{d+e x}d(d+e x)}{e f-d g}-\frac {\log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e f-d g}}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {b n \log (g (d+e x)-d g+e f)}{g (e f-d g)}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (g (d+e x)-d g+e f)^2}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {b e^3 n \left (\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{d+e x}d(d+e x)\right )}{e f-d g}-\frac {\log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e f-d g}}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {2 b n \left (\frac {\log \left (\frac {g (d+e x)}{e f-d g}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g}\right )}{e f-d g}\right )}{e f-d g}}{e f-d g}-\frac {g \left (\frac {b n \left (\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{e f-d g}-\frac {\log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e f-d g}}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {b n \log (g (d+e x)-d g+e f)}{g (e f-d g)}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (g (d+e x)-d g+e f)^2}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {b e^3 n \left (\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (3,-\frac {e f-d g}{g (d+e x)}\right )\right )}{e f-d g}-\frac {\log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e f-d g}}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {2 b n \left (\frac {\log \left (\frac {g (d+e x)}{e f-d g}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g}\right )}{e f-d g}\right )}{e f-d g}}{e f-d g}-\frac {g \left (\frac {b n \left (\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {e f-d g}{g (d+e x)}\right )}{e f-d g}-\frac {\log \left (\frac {e f-d g}{g (d+e x)}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e f-d g}}{e f-d g}-\frac {g \left (\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(e f-d g) (g (d+e x)-d g+e f)}-\frac {b n \log (g (d+e x)-d g+e f)}{g (e f-d g)}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g (g (d+e x)-d g+e f)^2}\right )}{e f-d g}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 g (f+g x)^3}\) |
-1/3*(a + b*Log[c*(d + e*x)^n])^3/(g*(f + g*x)^3) + (b*e^3*n*(-((g*(-1/2*( a + b*Log[c*(d + e*x)^n])^2/(g*(e*f - d*g + g*(d + e*x))^2) + (b*n*(-((g*( ((d + e*x)*(a + b*Log[c*(d + e*x)^n]))/((e*f - d*g)*(e*f - d*g + g*(d + e* x))) - (b*n*Log[e*f - d*g + g*(d + e*x)])/(g*(e*f - d*g))))/(e*f - d*g)) + (-(((a + b*Log[c*(d + e*x)^n])*Log[1 + (e*f - d*g)/(g*(d + e*x))])/(e*f - d*g)) + (b*n*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))])/(e*f - d*g))/(e*f - d*g)))/g))/(e*f - d*g)) + (-((g*(((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2 )/((e*f - d*g)*(e*f - d*g + g*(d + e*x))) - (2*b*n*(((a + b*Log[c*(d + e*x )^n])*Log[1 + (g*(d + e*x))/(e*f - d*g)])/g + (b*n*PolyLog[2, -((g*(d + e* x))/(e*f - d*g))])/g))/(e*f - d*g)))/(e*f - d*g)) + (-(((a + b*Log[c*(d + e*x)^n])^2*Log[1 + (e*f - d*g)/(g*(d + e*x))])/(e*f - d*g)) + (2*b*n*((a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((e*f - d*g)/(g*(d + e*x)))] + b*n*Poly Log[3, -((e*f - d*g)/(g*(d + e*x)))]))/(e*f - d*g))/(e*f - d*g))/(e*f - d* g)))/g
3.1.59.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3}}{\left (g x +f \right )^{4}}d x\]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{4}} \,d x } \]
integral((b^3*log((e*x + d)^n*c)^3 + 3*a*b^2*log((e*x + d)^n*c)^2 + 3*a^2* b*log((e*x + d)^n*c) + a^3)/(g^4*x^4 + 4*f*g^3*x^3 + 6*f^2*g^2*x^2 + 4*f^3 *g*x + f^4), x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}{\left (f + g x\right )^{4}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{4}} \,d x } \]
1/2*(2*e^2*log(e*x + d)/(e^3*f^3*g - 3*d*e^2*f^2*g^2 + 3*d^2*e*f*g^3 - d^3 *g^4) - 2*e^2*log(g*x + f)/(e^3*f^3*g - 3*d*e^2*f^2*g^2 + 3*d^2*e*f*g^3 - d^3*g^4) + (2*e*g*x + 3*e*f - d*g)/(e^2*f^4*g - 2*d*e*f^3*g^2 + d^2*f^2*g^ 3 + (e^2*f^2*g^3 - 2*d*e*f*g^4 + d^2*g^5)*x^2 + 2*(e^2*f^3*g^2 - 2*d*e*f^2 *g^3 + d^2*f*g^4)*x))*a^2*b*e*n - 1/3*b^3*log((e*x + d)^n)^3/(g^4*x^3 + 3* f*g^3*x^2 + 3*f^2*g^2*x + f^3*g) - a^2*b*log((e*x + d)^n*c)/(g^4*x^3 + 3*f *g^3*x^2 + 3*f^2*g^2*x + f^3*g) - 1/3*a^3/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g ^2*x + f^3*g) + integrate((b^3*d*g*log(c)^3 + 3*a*b^2*d*g*log(c)^2 + (3*a* b^2*d*g + (e*f*n + 3*d*g*log(c))*b^3 + (3*a*b^2*e*g + (e*g*n + 3*e*g*log(c ))*b^3)*x)*log((e*x + d)^n)^2 + (b^3*e*g*log(c)^3 + 3*a*b^2*e*g*log(c)^2)* x + 3*(b^3*d*g*log(c)^2 + 2*a*b^2*d*g*log(c) + (b^3*e*g*log(c)^2 + 2*a*b^2 *e*g*log(c))*x)*log((e*x + d)^n))/(e*g^5*x^5 + d*f^4*g + (4*e*f*g^4 + d*g^ 5)*x^4 + 2*(3*e*f^2*g^3 + 2*d*f*g^4)*x^3 + 2*(2*e*f^3*g^2 + 3*d*f^2*g^3)*x ^2 + (e*f^4*g + 4*d*f^3*g^2)*x), x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3}}{{\left (g x + f\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{(f+g x)^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3}{{\left (f+g\,x\right )}^4} \,d x \]