Optimal. Leaf size=163 \[ \frac{(a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^2 (c+d x) (b c-a d)}-\frac{2 A B n (a+b x)}{g^2 (c+d x) (b c-a d)}-\frac{2 B^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{g^2 (c+d x) (b c-a d)}+\frac{2 B^2 n^2 (a+b x)}{g^2 (c+d x) (b c-a d)} \]
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Rubi [C] time = 0.773967, antiderivative size = 514, normalized size of antiderivative = 3.15, number of steps used = 24, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac{2 b B^2 n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d g^2 (b c-a d)}+\frac{2 b B^2 n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d g^2 (b c-a d)}+\frac{2 b B n \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d g^2 (b c-a d)}+\frac{2 B n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d g^2 (c+d x)}-\frac{2 b B n \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d g^2 (b c-a d)}-\frac{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d g^2 (c+d x)}-\frac{b B^2 n^2 \log ^2(a+b x)}{d g^2 (b c-a d)}-\frac{b B^2 n^2 \log ^2(c+d x)}{d g^2 (b c-a d)}-\frac{2 b B^2 n^2 \log (a+b x)}{d g^2 (b c-a d)}+\frac{2 b B^2 n^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d g^2 (b c-a d)}+\frac{2 b B^2 n^2 \log (c+d x)}{d g^2 (b c-a d)}+\frac{2 b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d g^2 (b c-a d)}-\frac{2 B^2 n^2}{d g^2 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 2528
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 44
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^2} \, dx &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d g^2 (c+d x)}+\frac{(2 B n) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{g (a+b x) (c+d x)^2} \, dx}{d g}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d g^2 (c+d x)}+\frac{(2 B (b c-a d) n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^2} \, dx}{d g^2}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d g^2 (c+d x)}+\frac{(2 B (b c-a d) n) \int \left (\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{d g^2}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d g^2 (c+d x)}-\frac{(2 B n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{g^2}-\frac{(2 b B n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{(b c-a d) g^2}+\frac{\left (2 b^2 B n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{d (b c-a d) g^2}\\ &=\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d g^2 (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^2}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d g^2 (c+d x)}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d) g^2}-\frac{\left (2 B^2 n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{d g^2}-\frac{\left (2 b B^2 n^2\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}{d (b c-a d) g^2}+\frac{\left (2 b B^2 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{d (b c-a d) g^2}\\ &=\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d g^2 (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^2}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d g^2 (c+d x)}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d) g^2}-\frac{\left (2 b B^2 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{d (b c-a d) g^2}+\frac{\left (2 b B^2 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{d (b c-a d) g^2}-\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{d g^2}\\ &=\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d g^2 (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^2}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d g^2 (c+d x)}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d) g^2}+\frac{\left (2 b B^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{(b c-a d) g^2}-\frac{\left (2 b B^2 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{(b c-a d) g^2}-\frac{\left (2 b^2 B^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{d (b c-a d) g^2}+\frac{\left (2 b^2 B^2 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{d (b c-a d) g^2}-\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{d g^2}\\ &=-\frac{2 B^2 n^2}{d g^2 (c+d x)}-\frac{2 b B^2 n^2 \log (a+b x)}{d (b c-a d) g^2}+\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d g^2 (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^2}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d g^2 (c+d x)}+\frac{2 b B^2 n^2 \log (c+d x)}{d (b c-a d) g^2}+\frac{2 b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d (b c-a d) g^2}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d) g^2}+\frac{2 b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d (b c-a d) g^2}-\frac{\left (2 b B^2 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{(b c-a d) g^2}-\frac{\left (2 b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{d (b c-a d) g^2}-\frac{\left (2 b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{d (b c-a d) g^2}-\frac{\left (2 b^2 B^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{d (b c-a d) g^2}\\ &=-\frac{2 B^2 n^2}{d g^2 (c+d x)}-\frac{2 b B^2 n^2 \log (a+b x)}{d (b c-a d) g^2}-\frac{b B^2 n^2 \log ^2(a+b x)}{d (b c-a d) g^2}+\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d g^2 (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^2}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d g^2 (c+d x)}+\frac{2 b B^2 n^2 \log (c+d x)}{d (b c-a d) g^2}+\frac{2 b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d (b c-a d) g^2}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d) g^2}-\frac{b B^2 n^2 \log ^2(c+d x)}{d (b c-a d) g^2}+\frac{2 b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d (b c-a d) g^2}-\frac{\left (2 b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{d (b c-a d) g^2}-\frac{\left (2 b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d (b c-a d) g^2}\\ &=-\frac{2 B^2 n^2}{d g^2 (c+d x)}-\frac{2 b B^2 n^2 \log (a+b x)}{d (b c-a d) g^2}-\frac{b B^2 n^2 \log ^2(a+b x)}{d (b c-a d) g^2}+\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d g^2 (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^2}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d g^2 (c+d x)}+\frac{2 b B^2 n^2 \log (c+d x)}{d (b c-a d) g^2}+\frac{2 b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d (b c-a d) g^2}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d) g^2}-\frac{b B^2 n^2 \log ^2(c+d x)}{d (b c-a d) g^2}+\frac{2 b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d (b c-a d) g^2}+\frac{2 b B^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{d (b c-a d) g^2}+\frac{2 b B^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d (b c-a d) g^2}\\ \end{align*}
Mathematica [C] time = 0.446407, size = 331, normalized size = 2.03 \[ \frac{\frac{B n \left (-b B n (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+b B n (c+d x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 B n (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )}{b c-a d}-\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d g^2 (c+d x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.441, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dgx+cg \right ) ^{2}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.26026, size = 578, normalized size = 3.55 \begin{align*} 2 \, A B n{\left (\frac{1}{d^{2} g^{2} x + c d g^{2}} + \frac{b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} g^{2}} - \frac{b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} g^{2}}\right )} +{\left (2 \, n{\left (\frac{1}{d^{2} g^{2} x + c d g^{2}} + \frac{b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} g^{2}} - \frac{b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} g^{2}}\right )} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) - \frac{{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} +{\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \,{\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \,{\left (b d x + b c +{\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n^{2}}{b c^{2} d g^{2} - a c d^{2} g^{2} +{\left (b c d^{2} g^{2} - a d^{3} g^{2}\right )} x}\right )} B^{2} - \frac{B^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )^{2}}{d^{2} g^{2} x + c d g^{2}} - \frac{2 \, A B \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{d^{2} g^{2} x + c d g^{2}} - \frac{A^{2}}{d^{2} g^{2} x + c d g^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.879876, size = 555, normalized size = 3.4 \begin{align*} -\frac{A^{2} b c - A^{2} a d + 2 \,{\left (B^{2} b c - B^{2} a d\right )} n^{2} +{\left (B^{2} b c - B^{2} a d\right )} \log \left (e\right )^{2} -{\left (B^{2} b d n^{2} x + B^{2} a d n^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )^{2} - 2 \,{\left (A B b c - A B a d\right )} n + 2 \,{\left (A B b c - A B a d -{\left (B^{2} b c - B^{2} a d\right )} n -{\left (B^{2} b d n x + B^{2} a d n\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 2 \,{\left (B^{2} a d n^{2} - A B a d n +{\left (B^{2} b d n^{2} - A B b d n\right )} x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} g^{2} x +{\left (b c^{2} d - a c d^{2}\right )} g^{2}} \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 (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (d g x + c g\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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