Optimal. Leaf size=390 \[ \frac{3 B d i^3 n (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac{2 d^2 i^3 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^2}+\frac{d i^3 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g^2}-\frac{i^3 (c+d x) (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2 (a+b x)}-\frac{3 d i^3 (b c-a d)^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^2}-\frac{B d^2 i^3 n x (b c-a d)}{2 b^3 g^2}-\frac{B i^3 n (c+d x) (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac{B d i^3 n (b c-a d)^2 \log \left (\frac{a+b x}{c+d x}\right )}{2 b^4 g^2}-\frac{5 B d i^3 n (b c-a d)^2 \log (c+d x)}{2 b^4 g^2} \]
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Rubi [A] time = 0.6914, antiderivative size = 543, normalized size of antiderivative = 1.39, number of steps used = 21, number of rules used = 14, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.326, Rules used = {2528, 2486, 31, 2525, 12, 72, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{3 B d i^3 n (b c-a d)^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^2}-\frac{a^2 B d^3 i^3 n \log (a+b x)}{2 b^4 g^2}+\frac{d^3 i^3 x^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g^2}+\frac{3 d i^3 (b c-a d)^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^2}-\frac{i^3 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^2 (a+b x)}+\frac{A d^2 i^3 x (3 b c-2 a d)}{b^3 g^2}+\frac{B d^2 i^3 (a+b x) (3 b c-2 a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}-\frac{B d^2 i^3 n x (b c-a d)}{2 b^3 g^2}-\frac{B i^3 n (b c-a d)^3}{b^4 g^2 (a+b x)}-\frac{3 B d i^3 n (b c-a d)^2 \log ^2(a+b x)}{2 b^4 g^2}-\frac{B d i^3 n (b c-a d)^2 \log (a+b x)}{b^4 g^2}+\frac{B d i^3 n (b c-a d)^2 \log (c+d x)}{b^4 g^2}-\frac{B d i^3 n (3 b c-2 a d) (b c-a d) \log (c+d x)}{b^4 g^2}+\frac{3 B d i^3 n (b c-a d)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{B c^2 d i^3 n \log (c+d x)}{2 b^2 g^2} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2486
Rule 31
Rule 2525
Rule 12
Rule 72
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{(132 c+132 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^2} \, dx &=\int \left (\frac{2299968 d^2 (3 b c-2 a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac{2299968 d^3 x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}+\frac{2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)^2}+\frac{6899904 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2 (a+b x)}\right ) \, dx\\ &=\frac{\left (2299968 d^3\right ) \int x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g^2}+\frac{\left (2299968 d^2 (3 b c-2 a d)\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^3 g^2}+\frac{\left (6899904 d (b c-a d)^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^3 g^2}+\frac{\left (2299968 (b c-a d)^3\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^3 g^2}\\ &=\frac{2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}+\frac{1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac{2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac{6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{\left (2299968 B d^2 (3 b c-2 a d)\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 g^2}-\frac{\left (1149984 B d^3 n\right ) \int \frac{(b c-a d) x^2}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac{\left (6899904 B d (b c-a d)^2 n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 g^2}+\frac{\left (2299968 B (b c-a d)^3 n\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}\\ &=\frac{2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}+\frac{2299968 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}+\frac{1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac{2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac{6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac{\left (1149984 B d^3 (b c-a d) n\right ) \int \frac{x^2}{(a+b x) (c+d x)} \, dx}{b^2 g^2}-\frac{\left (2299968 B d^2 (3 b c-2 a d) (b c-a d) n\right ) \int \frac{1}{c+d x} \, dx}{b^4 g^2}-\frac{\left (6899904 B d (b c-a d)^2 n\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^4 g^2}+\frac{\left (2299968 B (b c-a d)^4 n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^2}\\ &=\frac{2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}+\frac{2299968 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}+\frac{1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac{2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac{6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}-\frac{2299968 B d (3 b c-2 a d) (b c-a d) n \log (c+d x)}{b^4 g^2}-\frac{\left (1149984 B d^3 (b c-a d) n\right ) \int \left (\frac{1}{b d}+\frac{a^2}{b (b c-a d) (a+b x)}+\frac{c^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{b^2 g^2}-\frac{\left (6899904 B d (b c-a d)^2 n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 g^2}+\frac{\left (6899904 B d^2 (b c-a d)^2 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 g^2}+\frac{\left (2299968 B (b c-a d)^4 n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^4 g^2}\\ &=\frac{2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}-\frac{1149984 B d^2 (b c-a d) n x}{b^3 g^2}-\frac{2299968 B (b c-a d)^3 n}{b^4 g^2 (a+b x)}-\frac{1149984 a^2 B d^3 n \log (a+b x)}{b^4 g^2}-\frac{2299968 B d (b c-a d)^2 n \log (a+b x)}{b^4 g^2}+\frac{2299968 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}+\frac{1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac{2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac{6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{1149984 B c^2 d n \log (c+d x)}{b^2 g^2}-\frac{2299968 B d (3 b c-2 a d) (b c-a d) n \log (c+d x)}{b^4 g^2}+\frac{2299968 B d (b c-a d)^2 n \log (c+d x)}{b^4 g^2}+\frac{6899904 B d (b c-a d)^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac{\left (6899904 B d (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^2}-\frac{\left (6899904 B d (b c-a d)^2 n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^2}\\ &=\frac{2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}-\frac{1149984 B d^2 (b c-a d) n x}{b^3 g^2}-\frac{2299968 B (b c-a d)^3 n}{b^4 g^2 (a+b x)}-\frac{1149984 a^2 B d^3 n \log (a+b x)}{b^4 g^2}-\frac{2299968 B d (b c-a d)^2 n \log (a+b x)}{b^4 g^2}-\frac{3449952 B d (b c-a d)^2 n \log ^2(a+b x)}{b^4 g^2}+\frac{2299968 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}+\frac{1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac{2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac{6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{1149984 B c^2 d n \log (c+d x)}{b^2 g^2}-\frac{2299968 B d (3 b c-2 a d) (b c-a d) n \log (c+d x)}{b^4 g^2}+\frac{2299968 B d (b c-a d)^2 n \log (c+d x)}{b^4 g^2}+\frac{6899904 B d (b c-a d)^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}-\frac{\left (6899904 B d (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^2}\\ &=\frac{2299968 A d^2 (3 b c-2 a d) x}{b^3 g^2}-\frac{1149984 B d^2 (b c-a d) n x}{b^3 g^2}-\frac{2299968 B (b c-a d)^3 n}{b^4 g^2 (a+b x)}-\frac{1149984 a^2 B d^3 n \log (a+b x)}{b^4 g^2}-\frac{2299968 B d (b c-a d)^2 n \log (a+b x)}{b^4 g^2}-\frac{3449952 B d (b c-a d)^2 n \log ^2(a+b x)}{b^4 g^2}+\frac{2299968 B d^2 (3 b c-2 a d) (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^2}+\frac{1149984 d^3 x^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac{2299968 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2 (a+b x)}+\frac{6899904 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^2}+\frac{1149984 B c^2 d n \log (c+d x)}{b^2 g^2}-\frac{2299968 B d (3 b c-2 a d) (b c-a d) n \log (c+d x)}{b^4 g^2}+\frac{2299968 B d (b c-a d)^2 n \log (c+d x)}{b^4 g^2}+\frac{6899904 B d (b c-a d)^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^2}+\frac{6899904 B d (b c-a d)^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^2}\\ \end{align*}
Mathematica [A] time = 0.41922, size = 394, normalized size = 1.01 \[ \frac{i^3 \left (-3 B d n (b c-a d)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-a^2 B d^3 n \log (a+b x)+b^2 d^3 x^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+6 d (b c-a d)^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+2 A b d^2 x (3 b c-2 a d)+2 B d^2 (a+b x) (3 b c-2 a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-b B d^2 n x (b c-a d)-\frac{2 B n (b c-a d)^3}{a+b x}-2 B d n (b c-a d)^2 \log (a+b x)+2 B d n (b c-a d)^2 \log (c+d x)-2 B d n (a d-b c) (2 a d-3 b c) \log (c+d x)+b^2 B c^2 d n \log (c+d x)\right )}{2 b^4 g^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.699, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{3}}{ \left ( bgx+ag \right ) ^{2}} \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.
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Maxima [B] time = 2.76488, size = 2410, normalized size = 6.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A d^{3} i^{3} x^{3} + 3 \, A c d^{2} i^{3} x^{2} + 3 \, A c^{2} d i^{3} x + A c^{3} i^{3} +{\left (B d^{3} i^{3} x^{3} + 3 \, B c d^{2} i^{3} x^{2} + 3 \, B c^{2} d i^{3} x + B c^{3} i^{3}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}^{3}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{{\left (b g x + a g\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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