Optimal. Leaf size=195 \[ -\frac{B^2 n^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b d^2}-\frac{B n (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A+B n\right )}{b d^2}-\frac{B n (a+b x) (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b d}+\frac{(a+b x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 b} \]
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Rubi [A] time = 0.485903, antiderivative size = 308, normalized size of antiderivative = 1.58, number of steps used = 15, number of rules used = 11, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.355, Rules used = {6742, 2492, 43, 2514, 2486, 31, 2488, 2411, 2343, 2333, 2315} \[ -\frac{B^2 n^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b d^2}+\frac{A^2 (a+b x)^2}{2 b}+\frac{A B n (b c-a d)^2 \log (c+d x)}{b d^2}+\frac{A B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{A B n x (b c-a d)}{d}-\frac{B^2 n (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^2}+\frac{B^2 n^2 (b c-a d)^2 \log (c+d x)}{b d^2}+\frac{B^2 (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{B^2 n (a+b x) (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2492
Rule 43
Rule 2514
Rule 2486
Rule 31
Rule 2488
Rule 2411
Rule 2343
Rule 2333
Rule 2315
Rubi steps
\begin{align*} \int (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx &=\int \left (A^2 (a+b x)+2 A B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac{A^2 (a+b x)^2}{2 b}+(2 A B) \int (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+B^2 \int (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac{A^2 (a+b x)^2}{2 b}+\frac{A B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{B^2 (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{(A B (b c-a d) n) \int \frac{a+b x}{c+d x} \, dx}{b}-\frac{\left (B^2 (b c-a d) n\right ) \int \frac{(a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b}\\ &=\frac{A^2 (a+b x)^2}{2 b}+\frac{A B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{B^2 (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{(A B (b c-a d) n) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{b}-\frac{\left (B^2 (b c-a d) n\right ) \int \left (\frac{b \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d}+\frac{(-b c+a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d (c+d x)}\right ) \, dx}{b}\\ &=-\frac{A B (b c-a d) n x}{d}+\frac{A^2 (a+b x)^2}{2 b}+\frac{A B (b c-a d)^2 n \log (c+d x)}{b d^2}+\frac{A B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{B^2 (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{\left (B^2 (b c-a d) n\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{d}+\frac{\left (B^2 (b c-a d)^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b d}\\ &=-\frac{A B (b c-a d) n x}{d}+\frac{A^2 (a+b x)^2}{2 b}+\frac{A B (b c-a d)^2 n \log (c+d x)}{b d^2}-\frac{B^2 (b c-a d) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac{A B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B^2 (b c-a d)^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^2}+\frac{B^2 (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}+\frac{\left (B^2 (b c-a d)^2 n^2\right ) \int \frac{1}{c+d x} \, dx}{b d}+\frac{\left (B^2 (b c-a d)^3 n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{b d^2}\\ &=-\frac{A B (b c-a d) n x}{d}+\frac{A^2 (a+b x)^2}{2 b}+\frac{A B (b c-a d)^2 n \log (c+d x)}{b d^2}+\frac{B^2 (b c-a d)^2 n^2 \log (c+d x)}{b d^2}-\frac{B^2 (b c-a d) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac{A B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B^2 (b c-a d)^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^2}+\frac{B^2 (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}+\frac{\left (B^2 (b c-a d)^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b c+a d}{b x}\right )}{x \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )} \, dx,x,c+d x\right )}{b d^3}\\ &=-\frac{A B (b c-a d) n x}{d}+\frac{A^2 (a+b x)^2}{2 b}+\frac{A B (b c-a d)^2 n \log (c+d x)}{b d^2}+\frac{B^2 (b c-a d)^2 n^2 \log (c+d x)}{b d^2}-\frac{B^2 (b c-a d) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac{A B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B^2 (b c-a d)^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^2}+\frac{B^2 (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{\left (B^2 (b c-a d)^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\left (\frac{-b c+a d}{d}+\frac{b}{d x}\right ) x} \, dx,x,\frac{1}{c+d x}\right )}{b d^3}\\ &=-\frac{A B (b c-a d) n x}{d}+\frac{A^2 (a+b x)^2}{2 b}+\frac{A B (b c-a d)^2 n \log (c+d x)}{b d^2}+\frac{B^2 (b c-a d)^2 n^2 \log (c+d x)}{b d^2}-\frac{B^2 (b c-a d) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac{A B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B^2 (b c-a d)^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^2}+\frac{B^2 (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{\left (B^2 (b c-a d)^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\frac{b}{d}+\frac{(-b c+a d) x}{d}} \, dx,x,\frac{1}{c+d x}\right )}{b d^3}\\ &=-\frac{A B (b c-a d) n x}{d}+\frac{A^2 (a+b x)^2}{2 b}+\frac{A B (b c-a d)^2 n \log (c+d x)}{b d^2}+\frac{B^2 (b c-a d)^2 n^2 \log (c+d x)}{b d^2}-\frac{B^2 (b c-a d) n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac{A B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B^2 (b c-a d)^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d^2}+\frac{B^2 (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b}-\frac{B^2 (b c-a d)^2 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{b d^2}\\ \end{align*}
Mathematica [B] time = 0.708813, size = 656, normalized size = 3.36 \[ \frac{B^2 n^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )}{b d^2}-\frac{2 a^2 A B n}{b}-\frac{2 a^2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{2 a^2 B^2 n^2 \log (c+d x)}{b}-\frac{a^2 B^2 n^2 \log ^2(a+b x)}{2 b}-\frac{2 a^2 B^2 n^2}{b}+a A^2 x+\frac{B n \log (a+b x) \left (a d \left (a d (A+3 B n)+a B d \log \left (e (a+b x)^n (c+d x)^{-n}\right )-b B c n\right )+B n (b c-a d)^2 \log \left (\frac{b (c+d x)}{b c-a d}\right )+b B c n (2 a d-b c) \log (c+d x)\right )}{b d^2}+A b B x^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 a A B x \log \left (e (a+b x)^n (c+d x)^{-n}\right )-\frac{2 a A B c n \log (c+d x)}{d}+a A B n x+\frac{b B^2 c^2 n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2}+\frac{1}{2} b B^2 x^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+a B^2 x \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+a B^2 n x \log \left (e (a+b x)^n (c+d x)^{-n}\right )-\frac{2 a B^2 c n \log (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d}-\frac{b B^2 c n x \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d}-\frac{a B^2 c n^2 \log ^2(c+d x)}{d}-\frac{a B^2 c n^2 \log (c+d x)}{d}+\frac{a B^2 c n^2}{d}+\frac{1}{2} A^2 b x^2+\frac{A b B c^2 n \log (c+d x)}{d^2}-\frac{A b B c n x}{d}+\frac{b B^2 c^2 n^2 \log ^2(c+d x)}{2 d^2}+\frac{b B^2 c^2 n^2 \log (c+d x)}{d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.651, size = 10210, normalized size = 52.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 3.611, size = 1052, normalized size = 5.39 \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 x + A^{2} a +{\left (B^{2} b x + B^{2} a\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \,{\left (A B b x + A B a\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (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 x + a\right )}{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (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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