Optimal. Leaf size=373 \[ \frac{B i^3 n (b c-a d)^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^4 g}+\frac{i^3 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g}+\frac{d i^3 (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g}-\frac{i^3 (b c-a d)^3 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g}+\frac{i^3 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b g}-\frac{B i^3 n (c+d x)^2 (b c-a d)}{6 b^2 g}-\frac{5 B d i^3 n x (b c-a d)^2}{6 b^3 g}-\frac{5 B i^3 n (b c-a d)^3 \log \left (\frac{a+b x}{c+d x}\right )}{6 b^4 g}-\frac{11 B i^3 n (b c-a d)^3 \log (c+d x)}{6 b^4 g} \]
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Rubi [A] time = 0.600166, antiderivative size = 455, normalized size of antiderivative = 1.22, number of steps used = 22, number of rules used = 13, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.302, Rules used = {2528, 2486, 31, 2524, 2418, 2390, 12, 2301, 2394, 2393, 2391, 2525, 43} \[ \frac{B i^3 n (b c-a d)^3 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g}+\frac{i^3 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g}+\frac{i^3 (b c-a d)^3 \log (a g+b g x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g}+\frac{A d i^3 x (b c-a d)^2}{b^3 g}+\frac{i^3 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b g}+\frac{B d i^3 (a+b x) (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g}-\frac{B i^3 n (c+d x)^2 (b c-a d)}{6 b^2 g}-\frac{5 B d i^3 n x (b c-a d)^2}{6 b^3 g}-\frac{B i^3 n (b c-a d)^3 \log ^2(g (a+b x))}{2 b^4 g}-\frac{5 B i^3 n (b c-a d)^3 \log (a+b x)}{6 b^4 g}-\frac{B i^3 n (b c-a d)^3 \log (c+d x)}{b^4 g}+\frac{B i^3 n (b c-a d)^3 \log (a g+b g x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2486
Rule 31
Rule 2524
Rule 2418
Rule 2390
Rule 12
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 2525
Rule 43
Rubi steps
\begin{align*} \int \frac{(131 c+131 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} \, dx &=\int \left (\frac{2248091 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}+\frac{17161 d (b c-a d) (131 c+131 d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac{131 d (131 c+131 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b g}+\frac{2248091 (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 (a g+b g x)}\right ) \, dx\\ &=\frac{\left (2248091 (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 g+b g x} \, dx}{b^3}+\frac{(131 d) \int (131 c+131 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b g}+\frac{(17161 d (b c-a d)) \int (131 c+131 d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g}+\frac{\left (2248091 d (b c-a d)^2\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^3 g}\\ &=\frac{2248091 A d (b c-a d)^2 x}{b^3 g}+\frac{2248091 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g}+\frac{2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b g}+\frac{2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}+\frac{\left (2248091 B d (b c-a d)^2\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 g}-\frac{(B n) \int \frac{2248091 (b c-a d) (c+d x)^2}{a+b x} \, dx}{3 b g}-\frac{(131 B (b c-a d) n) \int \frac{17161 (b c-a d) (c+d x)}{a+b x} \, dx}{2 b^2 g}-\frac{\left (2248091 B (b c-a d)^3 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 g+b g x)}{a+b x} \, dx}{b^4 g}\\ &=\frac{2248091 A d (b c-a d)^2 x}{b^3 g}+\frac{2248091 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g}+\frac{2248091 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g}+\frac{2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b g}+\frac{2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}-\frac{(2248091 B (b c-a d) n) \int \frac{(c+d x)^2}{a+b x} \, dx}{3 b g}-\frac{\left (2248091 B (b c-a d)^2 n\right ) \int \frac{c+d x}{a+b x} \, dx}{2 b^2 g}-\frac{\left (2248091 B (b c-a d)^3 n\right ) \int \left (\frac{b \log (a g+b g x)}{a+b x}-\frac{d \log (a g+b g x)}{c+d x}\right ) \, dx}{b^4 g}-\frac{\left (2248091 B d (b c-a d)^3 n\right ) \int \frac{1}{c+d x} \, dx}{b^4 g}\\ &=\frac{2248091 A d (b c-a d)^2 x}{b^3 g}+\frac{2248091 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g}+\frac{2248091 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g}+\frac{2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b g}-\frac{2248091 B (b c-a d)^3 n \log (c+d x)}{b^4 g}+\frac{2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}-\frac{(2248091 B (b c-a d) n) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{3 b g}-\frac{\left (2248091 B (b c-a d)^2 n\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{2 b^2 g}-\frac{\left (2248091 B (b c-a d)^3 n\right ) \int \frac{\log (a g+b g x)}{a+b x} \, dx}{b^3 g}+\frac{\left (2248091 B d (b c-a d)^3 n\right ) \int \frac{\log (a g+b g x)}{c+d x} \, dx}{b^4 g}\\ &=\frac{2248091 A d (b c-a d)^2 x}{b^3 g}-\frac{11240455 B d (b c-a d)^2 n x}{6 b^3 g}-\frac{2248091 B (b c-a d) n (c+d x)^2}{6 b^2 g}-\frac{11240455 B (b c-a d)^3 n \log (a+b x)}{6 b^4 g}+\frac{2248091 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g}+\frac{2248091 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g}+\frac{2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b g}-\frac{2248091 B (b c-a d)^3 n \log (c+d x)}{b^4 g}+\frac{2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}+\frac{2248091 B (b c-a d)^3 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^4 g}-\frac{\left (2248091 B (b c-a d)^3 n\right ) \int \frac{\log \left (\frac{b g (c+d x)}{b c g-a d g}\right )}{a g+b g x} \, dx}{b^3}-\frac{\left (2248091 B (b c-a d)^3 n\right ) \operatorname{Subst}\left (\int \frac{g \log (x)}{x} \, dx,x,a g+b g x\right )}{b^4 g^2}\\ &=\frac{2248091 A d (b c-a d)^2 x}{b^3 g}-\frac{11240455 B d (b c-a d)^2 n x}{6 b^3 g}-\frac{2248091 B (b c-a d) n (c+d x)^2}{6 b^2 g}-\frac{11240455 B (b c-a d)^3 n \log (a+b x)}{6 b^4 g}+\frac{2248091 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g}+\frac{2248091 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g}+\frac{2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b g}-\frac{2248091 B (b c-a d)^3 n \log (c+d x)}{b^4 g}+\frac{2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}+\frac{2248091 B (b c-a d)^3 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^4 g}-\frac{\left (2248091 B (b c-a d)^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a g+b g x\right )}{b^4 g}-\frac{\left (2248091 B (b c-a d)^3 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^4 g}\\ &=\frac{2248091 A d (b c-a d)^2 x}{b^3 g}-\frac{11240455 B d (b c-a d)^2 n x}{6 b^3 g}-\frac{2248091 B (b c-a d) n (c+d x)^2}{6 b^2 g}-\frac{11240455 B (b c-a d)^3 n \log (a+b x)}{6 b^4 g}-\frac{2248091 B (b c-a d)^3 n \log ^2(g (a+b x))}{2 b^4 g}+\frac{2248091 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g}+\frac{2248091 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b^2 g}+\frac{2248091 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b g}-\frac{2248091 B (b c-a d)^3 n \log (c+d x)}{b^4 g}+\frac{2248091 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{b^4 g}+\frac{2248091 B (b c-a d)^3 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^4 g}+\frac{2248091 B (b c-a d)^3 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g}\\ \end{align*}
Mathematica [A] time = 0.272942, size = 368, normalized size = 0.99 \[ \frac{i^3 \left (-3 B n (b c-a d)^3 \left (\log (g (a+b x)) \left (\log (g (a+b x))-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+3 b^2 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 b^3 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+6 (b c-a d)^3 \log (g (a+b x)) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+6 A b d x (b c-a d)^2-B n (b c-a d) \left (2 b d x (b c-a d)+2 (b c-a d)^2 \log (a+b x)+b^2 (c+d x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-6 B n (b c-a d)^3 \log (c+d x)-3 B n (b c-a d)^2 ((b c-a d) \log (a+b x)+b d x)\right )}{6 b^4 g} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.678, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{3}}{bgx+ag} \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 = 2.64474, size = 1262, normalized size = 3.38 \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 g x + a g}, 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 )}}{b g x + a g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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