Optimal. Leaf size=531 \[ \frac{3 B g n \text{PolyLog}\left (2,-\frac{b (f x+g)}{a f-b g}\right )}{f^4}-\frac{3 B g n \text{PolyLog}\left (2,-\frac{d (f x+g)}{c f-d g}\right )}{f^4}+\frac{g^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 f^4 (f x+g)^2}-\frac{3 g^2 (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^3 (f x+g) (a f-b g)}-\frac{3 g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^4}-\frac{b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{B g^3 n (b c-a d)}{2 f^3 (f x+g) (a f-b g) (c f-d g)}+\frac{B g^3 n (b c-a d) \log (f x+g) (a d f+b c f-2 b d g)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac{3 B g^2 n (b c-a d) \log \left (\frac{f x+g}{c+d x}\right )}{f^3 (a f-b g) (c f-d g)}-\frac{B n (b c-a d) \log (c+d x)}{b d f^3}+\frac{3 B g n \log (f x+g) \log \left (\frac{f (a+b x)}{a f-b g}\right )}{f^4}+\frac{A x}{f^3}+\frac{B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}-\frac{3 B g n \log (f x+g) \log \left (\frac{f (c+d x)}{c f-d g}\right )}{f^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.796941, antiderivative size = 562, normalized size of antiderivative = 1.06, number of steps used = 22, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2528, 2486, 31, 2525, 12, 72, 2524, 2418, 2394, 2393, 2391} \[ \frac{3 B g n \text{PolyLog}\left (2,-\frac{b (f x+g)}{a f-b g}\right )}{f^4}-\frac{3 B g n \text{PolyLog}\left (2,-\frac{d (f x+g)}{c f-d g}\right )}{f^4}+\frac{g^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 f^4 (f x+g)^2}-\frac{3 g^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^4 (f x+g)}-\frac{3 g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^4}-\frac{b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{B g^3 n (b c-a d)}{2 f^3 (f x+g) (a f-b g) (c f-d g)}+\frac{B g^3 n (b c-a d) \log (f x+g) (a d f+b c f-2 b d g)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac{3 B g^2 n (b c-a d) \log (f x+g)}{f^3 (a f-b g) (c f-d g)}-\frac{B n (b c-a d) \log (c+d x)}{b d f^3}-\frac{3 b B g^2 n \log (a+b x)}{f^4 (a f-b g)}+\frac{3 B g n \log (f x+g) \log \left (\frac{f (a+b x)}{a f-b g}\right )}{f^4}+\frac{A x}{f^3}+\frac{B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}+\frac{3 B d g^2 n \log (c+d x)}{f^4 (c f-d g)}-\frac{3 B g n \log (f x+g) \log \left (\frac{f (c+d x)}{c f-d g}\right )}{f^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2486
Rule 31
Rule 2525
Rule 12
Rule 72
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{\left (f+\frac{g}{x}\right )^3} \, dx &=\int \left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f^3}-\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)^3}+\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)^2}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)}\right ) \, dx\\ &=\frac{\int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{f^3}-\frac{(3 g) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{g+f x} \, dx}{f^3}+\frac{\left (3 g^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(g+f x)^2} \, dx}{f^3}-\frac{g^3 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(g+f x)^3} \, dx}{f^3}\\ &=\frac{A x}{f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}+\frac{B \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{f^3}+\frac{(3 B g n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (g+f x)}{a+b x} \, dx}{f^4}+\frac{\left (3 B g^2 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x) (g+f x)} \, dx}{f^4}-\frac{\left (B g^3 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x) (g+f x)^2} \, dx}{2 f^4}\\ &=\frac{A x}{f^3}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}-\frac{(B (b c-a d) n) \int \frac{1}{c+d x} \, dx}{b f^3}+\frac{(3 B g n) \int \left (\frac{b \log (g+f x)}{a+b x}-\frac{d \log (g+f x)}{c+d x}\right ) \, dx}{f^4}+\frac{\left (3 B (b c-a d) g^2 n\right ) \int \frac{1}{(a+b x) (c+d x) (g+f x)} \, dx}{f^4}-\frac{\left (B (b c-a d) g^3 n\right ) \int \frac{1}{(a+b x) (c+d x) (g+f x)^2} \, dx}{2 f^4}\\ &=\frac{A x}{f^3}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^3}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}+\frac{(3 b B g n) \int \frac{\log (g+f x)}{a+b x} \, dx}{f^4}-\frac{(3 B d g n) \int \frac{\log (g+f x)}{c+d x} \, dx}{f^4}+\frac{\left (3 B (b c-a d) g^2 n\right ) \int \left (\frac{b^2}{(b c-a d) (-a f+b g) (a+b x)}+\frac{d^2}{(b c-a d) (c f-d g) (c+d x)}+\frac{f^2}{(a f-b g) (c f-d g) (g+f x)}\right ) \, dx}{f^4}-\frac{\left (B (b c-a d) g^3 n\right ) \int \left (\frac{b^3}{(b c-a d) (-a f+b g)^2 (a+b x)}-\frac{d^3}{(b c-a d) (c f-d g)^2 (c+d x)}+\frac{f^2}{(a f-b g) (c f-d g) (g+f x)^2}-\frac{f^2 (b c f+a d f-2 b d g)}{(a f-b g)^2 (c f-d g)^2 (g+f x)}\right ) \, dx}{2 f^4}\\ &=\frac{A x}{f^3}+\frac{B (b c-a d) g^3 n}{2 f^3 (a f-b g) (c f-d g) (g+f x)}-\frac{b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}-\frac{3 b B g^2 n \log (a+b x)}{f^4 (a f-b g)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^3}+\frac{B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}+\frac{3 B d g^2 n \log (c+d x)}{f^4 (c f-d g)}+\frac{3 B (b c-a d) g^2 n \log (g+f x)}{f^3 (a f-b g) (c f-d g)}+\frac{B (b c-a d) g^3 (b c f+a d f-2 b d g) n \log (g+f x)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac{3 B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^4}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}-\frac{3 B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^4}-\frac{(3 B g n) \int \frac{\log \left (\frac{f (a+b x)}{a f-b g}\right )}{g+f x} \, dx}{f^3}+\frac{(3 B g n) \int \frac{\log \left (\frac{f (c+d x)}{c f-d g}\right )}{g+f x} \, dx}{f^3}\\ &=\frac{A x}{f^3}+\frac{B (b c-a d) g^3 n}{2 f^3 (a f-b g) (c f-d g) (g+f x)}-\frac{b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}-\frac{3 b B g^2 n \log (a+b x)}{f^4 (a f-b g)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^3}+\frac{B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}+\frac{3 B d g^2 n \log (c+d x)}{f^4 (c f-d g)}+\frac{3 B (b c-a d) g^2 n \log (g+f x)}{f^3 (a f-b g) (c f-d g)}+\frac{B (b c-a d) g^3 (b c f+a d f-2 b d g) n \log (g+f x)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac{3 B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^4}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}-\frac{3 B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^4}-\frac{(3 B g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a f-b g}\right )}{x} \, dx,x,g+f x\right )}{f^4}+\frac{(3 B g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{c f-d g}\right )}{x} \, dx,x,g+f x\right )}{f^4}\\ &=\frac{A x}{f^3}+\frac{B (b c-a d) g^3 n}{2 f^3 (a f-b g) (c f-d g) (g+f x)}-\frac{b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}-\frac{3 b B g^2 n \log (a+b x)}{f^4 (a f-b g)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^3}+\frac{B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}+\frac{3 B d g^2 n \log (c+d x)}{f^4 (c f-d g)}+\frac{3 B (b c-a d) g^2 n \log (g+f x)}{f^3 (a f-b g) (c f-d g)}+\frac{B (b c-a d) g^3 (b c f+a d f-2 b d g) n \log (g+f x)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac{3 B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^4}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}-\frac{3 B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^4}+\frac{3 B g n \text{Li}_2\left (-\frac{b (g+f x)}{a f-b g}\right )}{f^4}-\frac{3 B g n \text{Li}_2\left (-\frac{d (g+f x)}{c f-d g}\right )}{f^4}\\ \end{align*}
Mathematica [A] time = 1.65344, size = 470, normalized size = 0.89 \[ \frac{6 B g n \left (\text{PolyLog}\left (2,\frac{b (f x+g)}{b g-a f}\right )-\text{PolyLog}\left (2,\frac{d (f x+g)}{d g-c f}\right )+\log (f x+g) \left (\log \left (\frac{f (a+b x)}{a f-b g}\right )-\log \left (\frac{f (c+d x)}{c f-d g}\right )\right )\right )+\frac{g^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(f x+g)^2}-\frac{6 g^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f x+g}-6 g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+B g^3 n (b c-a d) \left (\frac{\frac{d^2 \log (c+d x)}{b c-a d}+\frac{f \left (\frac{(a f-b g) (c f-d g)}{f x+g}+\log (f x+g) (a d f+b c f-2 b d g)\right )}{(a f-b g)^2}}{(c f-d g)^2}-\frac{b^2 \log (a+b x)}{(b c-a d) (a f-b g)^2}\right )+\frac{2 B f (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}+\frac{6 B g^2 n (b \log (a+b x) (d g-c f)+d (a f-b g) \log (c+d x)+f (b c-a d) \log (f x+g))}{(a f-b g) (c f-d g)}-\frac{2 B f n (b c-a d) \log (c+d x)}{b d}+2 A f x}{2 f^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.084, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \left ( f+{\frac{g}{x}} \right ) ^{-3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, A{\left (\frac{6 \, f g^{2} x + 5 \, g^{3}}{f^{6} x^{2} + 2 \, f^{5} g x + f^{4} g^{2}} - \frac{2 \, x}{f^{3}} + \frac{6 \, g \log \left (f x + g\right )}{f^{4}}\right )} - B \int -\frac{x^{3} \log \left ({\left (b x + a\right )}^{n}\right ) - x^{3} \log \left ({\left (d x + c\right )}^{n}\right ) + x^{3} \log \left (e\right )}{f^{3} x^{3} + 3 \, f^{2} g x^{2} + 3 \, f g^{2} x + g^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B x^{3} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A x^{3}}{f^{3} x^{3} + 3 \, f^{2} g x^{2} + 3 \, f g^{2} x + g^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}{{\left (f + \frac{g}{x}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]