Optimal. Leaf size=136 \[ -\frac{2 B n (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 (a+b x) (b c-a d)}-\frac{(c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^2 (a+b x) (b c-a d)}-\frac{2 B^2 n^2 (c+d x)}{g^2 (a+b x) (b c-a d)} \]
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Rubi [C] time = 0.840441, antiderivative size = 512, normalized size of antiderivative = 3.76, 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, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{2 B^2 d n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{2 B^2 d n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{2 B d n \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b g^2 (b c-a d)}-\frac{2 B n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b g^2 (a+b x)}+\frac{2 B d n \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b g^2 (b c-a d)}-\frac{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b g^2 (a+b x)}+\frac{B^2 d n^2 \log ^2(a+b x)}{b g^2 (b c-a d)}+\frac{B^2 d n^2 \log ^2(c+d x)}{b g^2 (b c-a d)}-\frac{2 B^2 d n^2 \log (a+b x)}{b g^2 (b c-a d)}-\frac{2 B^2 d n^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}+\frac{2 B^2 d n^2 \log (c+d x)}{b g^2 (b c-a d)}-\frac{2 B^2 d n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{2 B^2 n^2}{b g^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 2528
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
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}{(a g+b g x)^2} \, dx &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g^2 (a+b 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)^2 (c+d x)} \, dx}{b g}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g^2 (a+b 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)^2 (c+d x)} \, dx}{b g^2}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g^2 (a+b x)}+\frac{(2 B (b c-a d) n) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b 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 (a+b x)}+\frac{d^2 \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}{b g^2}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g^2 (a+b x)}+\frac{(2 B n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{g^2}-\frac{(2 B d n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{(b c-a d) g^2}+\frac{\left (2 B d^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b (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 )}{b g^2 (a+b x)}-\frac{2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b (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}{b g^2 (a+b x)}+\frac{2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (2 B^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b g^2}+\frac{\left (2 B^2 d 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}{b (b c-a d) g^2}-\frac{\left (2 B^2 d 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}{b (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 )}{b g^2 (a+b x)}-\frac{2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b (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}{b g^2 (a+b x)}+\frac{2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (2 B^2 d 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}{b (b c-a d) g^2}-\frac{\left (2 B^2 d 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}{b (b c-a d) g^2}+\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b g^2}\\ &=-\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b g^2 (a+b x)}-\frac{2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b (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}{b g^2 (a+b x)}+\frac{2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (2 B^2 d n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{(b c-a d) g^2}-\frac{\left (2 B^2 d n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{(b c-a d) g^2}-\frac{\left (2 B^2 d^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b (b c-a d) g^2}+\frac{\left (2 B^2 d^2 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b (b c-a d) g^2}+\frac{\left (2 B^2 (b c-a d) n^2\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 g^2}\\ &=-\frac{2 B^2 n^2}{b g^2 (a+b x)}-\frac{2 B^2 d n^2 \log (a+b x)}{b (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 )}{b g^2 (a+b x)}-\frac{2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b (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}{b g^2 (a+b x)}+\frac{2 B^2 d n^2 \log (c+d x)}{b (b c-a d) g^2}-\frac{2 B^2 d n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d) g^2}-\frac{2 B^2 d n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}+\frac{\left (2 B^2 d n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{(b c-a d) g^2}+\frac{\left (2 B^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b (b c-a d) g^2}+\frac{\left (2 B^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b (b c-a d) g^2}+\frac{\left (2 B^2 d^2 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac{2 B^2 n^2}{b g^2 (a+b x)}-\frac{2 B^2 d n^2 \log (a+b x)}{b (b c-a d) g^2}+\frac{B^2 d n^2 \log ^2(a+b x)}{b (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 )}{b g^2 (a+b x)}-\frac{2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b (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}{b g^2 (a+b x)}+\frac{2 B^2 d n^2 \log (c+d x)}{b (b c-a d) g^2}-\frac{2 B^2 d n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{B^2 d n^2 \log ^2(c+d x)}{b (b c-a d) g^2}-\frac{2 B^2 d n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}+\frac{\left (2 B^2 d 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 )}{b (b c-a d) g^2}+\frac{\left (2 B^2 d 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 )}{b (b c-a d) g^2}\\ &=-\frac{2 B^2 n^2}{b g^2 (a+b x)}-\frac{2 B^2 d n^2 \log (a+b x)}{b (b c-a d) g^2}+\frac{B^2 d n^2 \log ^2(a+b x)}{b (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 )}{b g^2 (a+b x)}-\frac{2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b (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}{b g^2 (a+b x)}+\frac{2 B^2 d n^2 \log (c+d x)}{b (b c-a d) g^2}-\frac{2 B^2 d n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{2 B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{B^2 d n^2 \log ^2(c+d x)}{b (b c-a d) g^2}-\frac{2 B^2 d n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}-\frac{2 B^2 d n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d) g^2}-\frac{2 B^2 d n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}\\ \end{align*}
Mathematica [C] time = 0.598637, size = 330, normalized size = 2.43 \[ -\frac{\frac{B n \left (-B d n (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+B d n (a+b x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 d (a+b x) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 d (a+b x) \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 B n (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )}{b c-a d}+\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b g^2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.436, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \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 = 1.23068, size = 581, normalized size = 4.27 \begin{align*} -2 \, A B n{\left (\frac{1}{b^{2} g^{2} x + a b g^{2}} + \frac{d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac{d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} -{\left (2 \, n{\left (\frac{1}{b^{2} g^{2} x + a b g^{2}} + \frac{d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac{d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\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 + a d\right )} \log \left (b x + a\right )^{2} +{\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \,{\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \,{\left (b d x + a d -{\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n^{2}}{a b^{2} c g^{2} - a^{2} b d g^{2} +{\left (b^{3} c g^{2} - a b^{2} d 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}}{b^{2} g^{2} x + a b g^{2}} - \frac{2 \, A B \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{b^{2} g^{2} x + a b g^{2}} - \frac{A^{2}}{b^{2} g^{2} x + a b g^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.928436, size = 555, normalized size = 4.08 \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} b c 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} b c n\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 2 \,{\left (B^{2} b c n^{2} + A B b c 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^{3} c - a b^{2} d\right )} g^{2} x +{\left (a b^{2} c - a^{2} b d\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 (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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