Optimal. Leaf size=196 \[ -\frac{B^2 g n^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b d^2}-\frac{B g n (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A+B n\right )}{b d^2}-\frac{B g n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b d}+\frac{g (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b} \]
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Rubi [A] time = 0.442832, antiderivative size = 309, normalized size of antiderivative = 1.58, number of steps used = 15, number of rules used = 12, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{B^2 g n^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b d^2}+\frac{B g n (b c-a d)^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b d^2}+\frac{g (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b}-\frac{A B g n x (b c-a d)}{d}+\frac{B^2 g n^2 (b c-a d)^2 \log ^2(c+d x)}{2 b d^2}+\frac{B^2 g n^2 (b c-a d)^2 \log (c+d x)}{b d^2}-\frac{B^2 g n^2 (b c-a d)^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b d^2}-\frac{B^2 g n (a+b x) (b c-a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 2528
Rule 2486
Rule 31
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int (a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\frac{g (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}-\frac{(B n) \int \frac{(b c-a d) g^2 (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b g}\\ &=\frac{g (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}-\frac{(B (b c-a d) g n) \int \frac{(a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b}\\ &=\frac{g (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}-\frac{(B (b c-a d) g n) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac{(-b c+a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x)}\right ) \, dx}{b}\\ &=\frac{g (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}-\frac{(B (b c-a d) g n) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d}+\frac{\left (B (b c-a d)^2 g 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 d}\\ &=-\frac{A B (b c-a d) g n x}{d}+\frac{g (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac{B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}-\frac{\left (B^2 (b c-a d) g n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{d}-\frac{\left (B^2 (b c-a d)^2 g 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 d^2}\\ &=-\frac{A B (b c-a d) g n x}{d}-\frac{B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b d}+\frac{g (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac{B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}-\frac{\left (B^2 (b c-a d)^2 g 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 d^2}+\frac{\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac{1}{c+d x} \, dx}{b d}\\ &=-\frac{A B (b c-a d) g n x}{d}-\frac{B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b d}+\frac{g (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac{B^2 (b c-a d)^2 g n^2 \log (c+d x)}{b d^2}+\frac{B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}-\frac{\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{d^2}+\frac{\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b d}\\ &=-\frac{A B (b c-a d) g n x}{d}-\frac{B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b d}+\frac{g (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac{B^2 (b c-a d)^2 g n^2 \log (c+d x)}{b d^2}-\frac{B^2 (b c-a d)^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac{B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}+\frac{\left (B^2 (b c-a d)^2 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b d^2}+\frac{\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b d}\\ &=-\frac{A B (b c-a d) g n x}{d}-\frac{B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b d}+\frac{g (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac{B^2 (b c-a d)^2 g n^2 \log (c+d x)}{b d^2}-\frac{B^2 (b c-a d)^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac{B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}+\frac{B^2 (b c-a d)^2 g n^2 \log ^2(c+d x)}{2 b d^2}+\frac{\left (B^2 (b c-a d)^2 g 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 d^2}\\ &=-\frac{A B (b c-a d) g n x}{d}-\frac{B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b d}+\frac{g (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac{B^2 (b c-a d)^2 g n^2 \log (c+d x)}{b d^2}-\frac{B^2 (b c-a d)^2 g n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac{B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}+\frac{B^2 (b c-a d)^2 g n^2 \log ^2(c+d x)}{2 b d^2}-\frac{B^2 (b c-a d)^2 g n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b d^2}\\ \end{align*}
Mathematica [A] time = 0.199556, size = 215, normalized size = 1.1 \[ \frac{g \left ((a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2-\frac{B n (b c-a d) \left (B n (b c-a d) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 (b c-a d) \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 B n (b c-a d) \log (c+d x)+2 A b d x\right )}{d^2}\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.276, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) \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 = 3.80973, size = 1118, normalized size = 5.7 \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 (A^{2} b g x + A^{2} a g +{\left (B^{2} b g x + B^{2} a g\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B b g x + A B a g\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ), 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{\left (b g x + a g\right )}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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