Optimal. Leaf size=263 \[ -\frac{b^2 B g^2 n \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^3 i^3}-\frac{b^2 g^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i^3}-\frac{g^2 (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d i^3 (c+d x)^2}-\frac{A b g^2 (a+b x)}{d^2 i^3 (c+d x)}-\frac{b B g^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{d^2 i^3 (c+d x)}+\frac{b B g^2 n (a+b x)}{d^2 i^3 (c+d x)}+\frac{B g^2 n (a+b x)^2}{4 d i^3 (c+d x)^2} \]
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Rubi [A] time = 0.597234, antiderivative size = 356, normalized size of antiderivative = 1.35, number of steps used = 18, number of rules used = 11, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac{b^2 B g^2 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^3 i^3}+\frac{b^2 g^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i^3}+\frac{2 b g^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i^3 (c+d x)}-\frac{g^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3 i^3 (c+d x)^2}-\frac{b^2 B g^2 n \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^3 i^3}-\frac{3 b^2 B g^2 n \log (a+b x)}{2 d^3 i^3}-\frac{3 b B g^2 n (b c-a d)}{2 d^3 i^3 (c+d x)}+\frac{B g^2 n (b c-a d)^2}{4 d^3 i^3 (c+d x)^2}+\frac{b^2 B g^2 n \log ^2(c+d x)}{2 d^3 i^3}+\frac{3 b^2 B g^2 n \log (c+d x)}{2 d^3 i^3} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2525
Rule 12
Rule 44
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int \frac{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(152 c+152 d x)^3} \, dx &=\int \left (\frac{(-b c+a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3511808 d^2 (c+d x)^3}-\frac{b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1755904 d^2 (c+d x)^2}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3511808 d^2 (c+d x)}\right ) \, dx\\ &=\frac{\left (b^2 g^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3511808 d^2}-\frac{\left (b (b c-a d) g^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{1755904 d^2}+\frac{\left ((b c-a d)^2 g^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3511808 d^2}\\ &=-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7023616 d^3 (c+d x)^2}+\frac{b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1755904 d^3 (c+d x)}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}-\frac{\left (b^2 B g^2 n\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}{3511808 d^3}-\frac{\left (b B (b c-a d) g^2 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{1755904 d^3}+\frac{\left (B (b c-a d)^2 g^2 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{7023616 d^3}\\ &=-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7023616 d^3 (c+d x)^2}+\frac{b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1755904 d^3 (c+d x)}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}-\frac{\left (b^2 B g^2 n\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{3511808 d^3}-\frac{\left (b B (b c-a d)^2 g^2 n\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{1755904 d^3}+\frac{\left (B (b c-a d)^3 g^2 n\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{7023616 d^3}\\ &=-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7023616 d^3 (c+d x)^2}+\frac{b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1755904 d^3 (c+d x)}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}-\frac{\left (b^3 B g^2 n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3511808 d^3}+\frac{\left (b^2 B g^2 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3511808 d^2}-\frac{\left (b B (b c-a d)^2 g^2 n\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}{1755904 d^3}+\frac{\left (B (b c-a d)^3 g^2 n\right ) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7023616 d^3}\\ &=\frac{B (b c-a d)^2 g^2 n}{14047232 d^3 (c+d x)^2}-\frac{3 b B (b c-a d) g^2 n}{7023616 d^3 (c+d x)}-\frac{3 b^2 B g^2 n \log (a+b x)}{7023616 d^3}-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7023616 d^3 (c+d x)^2}+\frac{b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1755904 d^3 (c+d x)}+\frac{3 b^2 B g^2 n \log (c+d x)}{7023616 d^3}-\frac{b^2 B g^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3511808 d^3}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}+\frac{\left (b^2 B g^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3511808 d^3}+\frac{\left (b^2 B g^2 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3511808 d^2}\\ &=\frac{B (b c-a d)^2 g^2 n}{14047232 d^3 (c+d x)^2}-\frac{3 b B (b c-a d) g^2 n}{7023616 d^3 (c+d x)}-\frac{3 b^2 B g^2 n \log (a+b x)}{7023616 d^3}-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7023616 d^3 (c+d x)^2}+\frac{b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1755904 d^3 (c+d x)}+\frac{3 b^2 B g^2 n \log (c+d x)}{7023616 d^3}-\frac{b^2 B g^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3511808 d^3}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}+\frac{b^2 B g^2 n \log ^2(c+d x)}{7023616 d^3}+\frac{\left (b^2 B g^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3511808 d^3}\\ &=\frac{B (b c-a d)^2 g^2 n}{14047232 d^3 (c+d x)^2}-\frac{3 b B (b c-a d) g^2 n}{7023616 d^3 (c+d x)}-\frac{3 b^2 B g^2 n \log (a+b x)}{7023616 d^3}-\frac{(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7023616 d^3 (c+d x)^2}+\frac{b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1755904 d^3 (c+d x)}+\frac{3 b^2 B g^2 n \log (c+d x)}{7023616 d^3}-\frac{b^2 B g^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3511808 d^3}+\frac{b^2 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3511808 d^3}+\frac{b^2 B g^2 n \log ^2(c+d x)}{7023616 d^3}-\frac{b^2 B g^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3511808 d^3}\\ \end{align*}
Mathematica [A] time = 0.355202, size = 259, normalized size = 0.98 \[ \frac{g^2 \left (-2 b^2 B n \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+4 b^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{8 b (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-\frac{2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(c+d x)^2}-6 b^2 B n \log (a+b x)-\frac{6 b B n (b c-a d)}{c+d x}+\frac{B n (b c-a d)^2}{(c+d x)^2}+6 b^2 B n \log (c+d x)\right )}{4 d^3 i^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.68, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bgx+ag \right ) ^{2}}{ \left ( dix+ci \right ) ^{3}} \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 [F] time = 0., size = 0, normalized size = 0. \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 b^{2} g^{2} x^{2} + 2 \, A a b g^{2} x + A a^{2} g^{2} +{\left (B b^{2} g^{2} x^{2} + 2 \, B a b g^{2} x + B a^{2} g^{2}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{d^{3} i^{3} x^{3} + 3 \, c d^{2} i^{3} x^{2} + 3 \, c^{2} d i^{3} x + c^{3} i^{3}}, 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 (b g x + a g\right )}^{2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{{\left (d i x + c i\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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