Optimal. Leaf size=404 \[ -3 B f^2 g n \text{PolyLog}\left (2,-\frac{b x}{a}\right )+3 B f^2 g n \text{PolyLog}\left (2,-\frac{d x}{c}\right )-\frac{1}{2} B g^3 n \log (x) \left (\frac{b^2}{a^2}-\frac{d^2}{c^2}\right )+\frac{b^2 B g^3 n \log (a+b x)}{2 a^2}+3 f^2 g \log (x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{3 f g^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{a (c+d x) \left (a-\frac{c (a+b x)}{c+d x}\right )}-\frac{g^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 x^2}+\frac{B f^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B f^3 n (b c-a d) \log (c+d x)}{b d}+\frac{3 B f g^2 n (b c-a d) \log \left (a-\frac{c (a+b x)}{c+d x}\right )}{a c}-\frac{B g^3 n (b c-a d)}{2 a c x}-3 B f^2 g n \log (x) \log \left (\frac{b x}{a}+1\right )+A f^3 x-\frac{B d^2 g^3 n \log (c+d x)}{2 c^2}+3 B f^2 g n \log (x) \log \left (\frac{d x}{c}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.493055, antiderivative size = 385, normalized size of antiderivative = 0.95, number of steps used = 20, number of rules used = 10, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {2528, 2486, 31, 2525, 12, 72, 2524, 2357, 2317, 2391} \[ -3 B f^2 g n \text{PolyLog}\left (2,-\frac{b x}{a}\right )+3 B f^2 g n \text{PolyLog}\left (2,-\frac{d x}{c}\right )-\frac{1}{2} B g^3 n \log (x) \left (\frac{b^2}{a^2}-\frac{d^2}{c^2}\right )+\frac{b^2 B g^3 n \log (a+b x)}{2 a^2}+3 f^2 g \log (x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{3 f g^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{x}-\frac{g^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 x^2}+\frac{B f^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B f^3 n (b c-a d) \log (c+d x)}{b d}+\frac{3 B f g^2 n \log (x) (b c-a d)}{a c}-\frac{B g^3 n (b c-a d)}{2 a c x}-3 B f^2 g n \log (x) \log \left (\frac{b x}{a}+1\right )-\frac{3 b B f g^2 n \log (a+b x)}{a}+A f^3 x-\frac{B d^2 g^3 n \log (c+d x)}{2 c^2}+3 B f^2 g n \log (x) \log \left (\frac{d x}{c}+1\right )+\frac{3 B d f g^2 n \log (c+d x)}{c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2528
Rule 2486
Rule 31
Rule 2525
Rule 12
Rule 72
Rule 2524
Rule 2357
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \left (f+\frac{g}{x}\right )^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (f^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x^3}+\frac{3 f g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x^2}+\frac{3 f^2 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}\right ) \, dx\\ &=f^3 \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx+\left (3 f^2 g\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{x} \, dx+\left (3 f g^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{x^2} \, dx+g^3 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{x^3} \, dx\\ &=A f^3 x-\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 x^2}-\frac{3 f g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}+3 f^2 g \log (x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+\left (B f^3\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx-\left (3 B f^2 g n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (x)}{a+b x} \, dx+\left (3 B f g^2 n\right ) \int \frac{b c-a d}{x (a+b x) (c+d x)} \, dx+\frac{1}{2} \left (B g^3 n\right ) \int \frac{b c-a d}{x^2 (a+b x) (c+d x)} \, dx\\ &=A f^3 x+\frac{B f^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 x^2}-\frac{3 f g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}+3 f^2 g \log (x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-\frac{\left (B (b c-a d) f^3 n\right ) \int \frac{1}{c+d x} \, dx}{b}-\left (3 B f^2 g n\right ) \int \left (\frac{b \log (x)}{a+b x}-\frac{d \log (x)}{c+d x}\right ) \, dx+\left (3 B (b c-a d) f g^2 n\right ) \int \frac{1}{x (a+b x) (c+d x)} \, dx+\frac{1}{2} \left (B (b c-a d) g^3 n\right ) \int \frac{1}{x^2 (a+b x) (c+d x)} \, dx\\ &=A f^3 x+\frac{B f^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 x^2}-\frac{3 f g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}+3 f^2 g \log (x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-\frac{B (b c-a d) f^3 n \log (c+d x)}{b d}-\left (3 b B f^2 g n\right ) \int \frac{\log (x)}{a+b x} \, dx+\left (3 B d f^2 g n\right ) \int \frac{\log (x)}{c+d x} \, dx+\left (3 B (b c-a d) f g^2 n\right ) \int \left (\frac{1}{a c x}+\frac{b^2}{a (-b c+a d) (a+b x)}+\frac{d^2}{c (b c-a d) (c+d x)}\right ) \, dx+\frac{1}{2} \left (B (b c-a d) g^3 n\right ) \int \left (\frac{1}{a c x^2}+\frac{-b c-a d}{a^2 c^2 x}-\frac{b^3}{a^2 (-b c+a d) (a+b x)}-\frac{d^3}{c^2 (b c-a d) (c+d x)}\right ) \, dx\\ &=-\frac{B (b c-a d) g^3 n}{2 a c x}+A f^3 x+\frac{3 B (b c-a d) f g^2 n \log (x)}{a c}-\frac{B (b c-a d) (b c+a d) g^3 n \log (x)}{2 a^2 c^2}-\frac{3 b B f g^2 n \log (a+b x)}{a}+\frac{b^2 B g^3 n \log (a+b x)}{2 a^2}-3 B f^2 g n \log (x) \log \left (1+\frac{b x}{a}\right )+\frac{B f^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 x^2}-\frac{3 f g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}+3 f^2 g \log (x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-\frac{B (b c-a d) f^3 n \log (c+d x)}{b d}+\frac{3 B d f g^2 n \log (c+d x)}{c}-\frac{B d^2 g^3 n \log (c+d x)}{2 c^2}+3 B f^2 g n \log (x) \log \left (1+\frac{d x}{c}\right )+\left (3 B f^2 g n\right ) \int \frac{\log \left (1+\frac{b x}{a}\right )}{x} \, dx-\left (3 B f^2 g n\right ) \int \frac{\log \left (1+\frac{d x}{c}\right )}{x} \, dx\\ &=-\frac{B (b c-a d) g^3 n}{2 a c x}+A f^3 x+\frac{3 B (b c-a d) f g^2 n \log (x)}{a c}-\frac{B (b c-a d) (b c+a d) g^3 n \log (x)}{2 a^2 c^2}-\frac{3 b B f g^2 n \log (a+b x)}{a}+\frac{b^2 B g^3 n \log (a+b x)}{2 a^2}-3 B f^2 g n \log (x) \log \left (1+\frac{b x}{a}\right )+\frac{B f^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 x^2}-\frac{3 f g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}+3 f^2 g \log (x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-\frac{B (b c-a d) f^3 n \log (c+d x)}{b d}+\frac{3 B d f g^2 n \log (c+d x)}{c}-\frac{B d^2 g^3 n \log (c+d x)}{2 c^2}+3 B f^2 g n \log (x) \log \left (1+\frac{d x}{c}\right )-3 B f^2 g n \text{Li}_2\left (-\frac{b x}{a}\right )+3 B f^2 g n \text{Li}_2\left (-\frac{d x}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.413278, size = 336, normalized size = 0.83 \[ -3 B f^2 g n \left (\text{PolyLog}\left (2,-\frac{b x}{a}\right )-\text{PolyLog}\left (2,-\frac{d x}{c}\right )+\log (x) \left (\log \left (\frac{b x}{a}+1\right )-\log \left (\frac{d x}{c}+1\right )\right )\right )+\frac{B g^3 n \left (\log (x) \left (a^2 d^2 x-b^2 c^2 x\right )+b^2 c^2 x \log (a+b x)+a \left (-a d^2 x \log (c+d x)+a c d-b c^2\right )\right )}{2 a^2 c^2 x}+3 f^2 g \log (x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{3 f g^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{x}-\frac{g^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 x^2}+\frac{B f^3 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B f^3 n (b c-a d) \log (c+d x)}{b d}+\frac{3 B f g^2 n (\log (x) (b c-a d)-b c \log (a+b x)+a d \log (c+d x))}{a c}+A f^3 x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.154, size = 0, normalized size = 0. \begin{align*} \int \left ( f+{\frac{g}{x}} \right ) ^{3} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \, 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*} B f^{3} n{\left (\frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} - 3 \, B f g^{2} n{\left (\frac{b \log \left (b x + a\right )}{a} - \frac{d \log \left (d x + c\right )}{c} - \frac{{\left (b c - a d\right )} \log \left (x\right )}{a c}\right )} + \frac{1}{2} \, B g^{3} n{\left (\frac{b^{2} \log \left (b x + a\right )}{a^{2}} - \frac{d^{2} \log \left (d x + c\right )}{c^{2}} - \frac{b c - a d}{a c x} - \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (x\right )}{a^{2} c^{2}}\right )} + B f^{3} x \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A f^{3} x - 3 \, B f^{2} g \int -\frac{\log \left ({\left (b x + a\right )}^{n}\right ) - \log \left ({\left (d x + c\right )}^{n}\right ) + \log \left (e\right )}{x}\,{d x} + 3 \, A f^{2} g \log \left (x\right ) - \frac{3 \, B f g^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{x} - \frac{3 \, A f g^{2}}{x} - \frac{B g^{3} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{2 \, x^{2}} - \frac{A g^{3}}{2 \, x^{2}} \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{A f^{3} x^{3} + 3 \, A f^{2} g x^{2} + 3 \, A f g^{2} x + A g^{3} +{\left (B f^{3} x^{3} + 3 \, B f^{2} g x^{2} + 3 \, B f g^{2} x + B g^{3}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{x^{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{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}{\left (f + \frac{g}{x}\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]