Optimal. Leaf size=259 \[ \frac{2 B d i^2 n (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac{d^2 i^2 (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2}-\frac{i^2 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2 (a+b x)}-\frac{2 d i^2 (b c-a d) \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2}-\frac{B i^2 n (c+d x) (b c-a d)}{b^2 g^2 (a+b x)}-\frac{B d i^2 n (b c-a d) \log (c+d x)}{b^3 g^2} \]
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Rubi [A] time = 0.522551, antiderivative size = 327, normalized size of antiderivative = 1.26, number of steps used = 17, number of rules used = 13, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.302, Rules used = {2528, 2486, 31, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{2 B d i^2 n (b c-a d) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac{2 d i^2 (b c-a d) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2}-\frac{i^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^2 (a+b x)}+\frac{B d^2 i^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{B i^2 n (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac{B d i^2 n (b c-a d) \log ^2(a+b x)}{b^3 g^2}-\frac{B d i^2 n (b c-a d) \log (a+b x)}{b^3 g^2}+\frac{2 B d i^2 n (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{A d^2 i^2 x}{b^2 g^2} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2486
Rule 31
Rule 2525
Rule 12
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{(122 c+122 d x)^2 \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{14884 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}+\frac{14884 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)^2}+\frac{29768 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}\right ) \, dx\\ &=\frac{\left (14884 d^2\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g^2}+\frac{(29768 d (b c-a d)) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g^2}+\frac{\left (14884 (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)^2} \, dx}{b^2 g^2}\\ &=\frac{14884 A d^2 x}{b^2 g^2}-\frac{14884 (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)}+\frac{29768 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac{\left (14884 B d^2\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 g^2}-\frac{(29768 B d (b c-a d) n) \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^3 g^2}+\frac{\left (14884 B (b c-a d)^2 n\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=\frac{14884 A d^2 x}{b^2 g^2}+\frac{14884 B d^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{14884 (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)}+\frac{29768 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}-\frac{(29768 B d (b c-a d) n) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^3 g^2}-\frac{\left (14884 B d^2 (b c-a d) n\right ) \int \frac{1}{c+d x} \, dx}{b^3 g^2}+\frac{\left (14884 B (b c-a d)^3 n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=\frac{14884 A d^2 x}{b^2 g^2}+\frac{14884 B d^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{14884 (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)}+\frac{29768 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}-\frac{14884 B d (b c-a d) n \log (c+d x)}{b^3 g^2}-\frac{(29768 B d (b c-a d) n) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 g^2}+\frac{\left (29768 B d^2 (b c-a d) n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 g^2}+\frac{\left (14884 B (b c-a d)^3 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^3 g^2}\\ &=\frac{14884 A d^2 x}{b^2 g^2}-\frac{14884 B (b c-a d)^2 n}{b^3 g^2 (a+b x)}-\frac{14884 B d (b c-a d) n \log (a+b x)}{b^3 g^2}+\frac{14884 B d^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{14884 (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)}+\frac{29768 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac{29768 B d (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac{(29768 B d (b c-a d) n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}-\frac{(29768 B d (b c-a d) n) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^2}\\ &=\frac{14884 A d^2 x}{b^2 g^2}-\frac{14884 B (b c-a d)^2 n}{b^3 g^2 (a+b x)}-\frac{14884 B d (b c-a d) n \log (a+b x)}{b^3 g^2}-\frac{14884 B d (b c-a d) n \log ^2(a+b x)}{b^3 g^2}+\frac{14884 B d^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{14884 (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)}+\frac{29768 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac{29768 B d (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac{(29768 B d (b c-a d) n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^2}\\ &=\frac{14884 A d^2 x}{b^2 g^2}-\frac{14884 B (b c-a d)^2 n}{b^3 g^2 (a+b x)}-\frac{14884 B d (b c-a d) n \log (a+b x)}{b^3 g^2}-\frac{14884 B d (b c-a d) n \log ^2(a+b x)}{b^3 g^2}+\frac{14884 B d^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g^2}-\frac{14884 (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)}+\frac{29768 d (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^2}+\frac{29768 B d (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac{29768 B d (b c-a d) n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g^2}\\ \end{align*}
Mathematica [A] time = 0.236218, size = 233, normalized size = 0.9 \[ \frac{i^2 \left (B d n (a d-b c) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+2 d (b c-a d) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{(b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+B d^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\frac{B n (b c-a d)^2}{a+b x}+B d n (a d-b c) \log (a+b x)+A b d^2 x\right )}{b^3 g^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.695, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{ \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 = 2.80036, size = 1607, normalized size = 6.2 \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^{2} i^{2} x^{2} + 2 \, A c d i^{2} x + A c^{2} i^{2} +{\left (B d^{2} i^{2} x^{2} + 2 \, B c d i^{2} x + B c^{2} i^{2}\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 )}^{2}{\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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