Optimal. Leaf size=294 \[ \frac{B^2 n^2 (b c-a d) (-a d h-b c h+2 b d g) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}+\frac{B n (b c-a d) (-a d h-b c h+2 b d g) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b^2 d^2}-\frac{(b g-a h)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 b^2 h}-\frac{B h n (a+b x) (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b^2 d}+\frac{(g+h x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 h}+\frac{B^2 h n^2 (b c-a d)^2 \log (c+d x)}{b^2 d^2} \]
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Rubi [A] time = 0.955147, antiderivative size = 449, normalized size of antiderivative = 1.53, number of steps used = 20, number of rules used = 11, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.355, Rules used = {6742, 2492, 72, 2514, 2486, 31, 2488, 2411, 2343, 2333, 2315} \[ -\frac{B^2 n^2 (b g-a h)^2 \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right )}{b^2 h}-\frac{B^2 n^2 (d g-c h)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d^2 h}-\frac{A B n (b g-a h)^2 \log (a+b x)}{b^2 h}+\frac{A B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}-\frac{A B h n x (b c-a d)}{b d}+\frac{B^2 h n^2 (b c-a d)^2 \log (c+d x)}{b^2 d^2}+\frac{B^2 n (b g-a h)^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 h}-\frac{B^2 h n (a+b x) (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 d}-\frac{B^2 n (d g-c h)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d^2 h}+\frac{B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac{A^2 (g+h x)^2}{2 h}+\frac{A B n (d g-c h)^2 \log (c+d x)}{d^2 h} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2492
Rule 72
Rule 2514
Rule 2486
Rule 31
Rule 2488
Rule 2411
Rule 2343
Rule 2333
Rule 2315
Rubi steps
\begin{align*} \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx &=\int \left (A^2 (g+h x)+2 A B (g+h x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 (g+h x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac{A^2 (g+h x)^2}{2 h}+(2 A B) \int (g+h x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx+B^2 \int (g+h x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac{A^2 (g+h x)^2}{2 h}+\frac{A B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac{(A B (b c-a d) n) \int \frac{(g+h x)^2}{(a+b x) (c+d x)} \, dx}{h}-\frac{\left (B^2 (b c-a d) n\right ) \int \frac{(g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{h}\\ &=\frac{A^2 (g+h x)^2}{2 h}+\frac{A B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac{(A B (b c-a d) n) \int \left (\frac{h^2}{b d}+\frac{(b g-a h)^2}{b (b c-a d) (a+b x)}+\frac{(d g-c h)^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{h}-\frac{\left (B^2 (b c-a d) n\right ) \int \left (\frac{h^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac{(b g-a h)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)}+\frac{(d g-c h)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{d (-b c+a d) (c+d x)}\right ) \, dx}{h}\\ &=-\frac{A B (b c-a d) h n x}{b d}+\frac{A^2 (g+h x)^2}{2 h}-\frac{A B (b g-a h)^2 n \log (a+b x)}{b^2 h}+\frac{A B (d g-c h)^2 n \log (c+d x)}{d^2 h}+\frac{A B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac{\left (B^2 (b c-a d) h n\right ) \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx}{b d}-\frac{\left (B^2 (b g-a h)^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{b h}+\frac{\left (B^2 (d g-c h)^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{d h}\\ &=-\frac{A B (b c-a d) h n x}{b d}+\frac{A^2 (g+h x)^2}{2 h}-\frac{A B (b g-a h)^2 n \log (a+b x)}{b^2 h}+\frac{A B (d g-c h)^2 n \log (c+d x)}{d^2 h}-\frac{B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 d}+\frac{A B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^2 (b g-a h)^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 h}-\frac{B^2 (d g-c h)^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 )}{d^2 h}+\frac{B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac{\left (B^2 (b c-a d)^2 h n^2\right ) \int \frac{1}{c+d x} \, dx}{b^2 d}-\frac{\left (B^2 (b c-a d) (b g-a h)^2 n^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^2 h}+\frac{\left (B^2 (b c-a d) (d g-c h)^2 n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{d^2 h}\\ &=-\frac{A B (b c-a d) h n x}{b d}+\frac{A^2 (g+h x)^2}{2 h}-\frac{A B (b g-a h)^2 n \log (a+b x)}{b^2 h}+\frac{A B (d g-c h)^2 n \log (c+d x)}{d^2 h}+\frac{B^2 (b c-a d)^2 h n^2 \log (c+d x)}{b^2 d^2}-\frac{B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 d}+\frac{A B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^2 (b g-a h)^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 h}-\frac{B^2 (d g-c h)^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 )}{d^2 h}+\frac{B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac{\left (B^2 (b c-a d) (b g-a h)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b c-a d}{d x}\right )}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{b^3 h}+\frac{\left (B^2 (b c-a d) (d g-c h)^2 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 )}{d^3 h}\\ &=-\frac{A B (b c-a d) h n x}{b d}+\frac{A^2 (g+h x)^2}{2 h}-\frac{A B (b g-a h)^2 n \log (a+b x)}{b^2 h}+\frac{A B (d g-c h)^2 n \log (c+d x)}{d^2 h}+\frac{B^2 (b c-a d)^2 h n^2 \log (c+d x)}{b^2 d^2}-\frac{B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 d}+\frac{A B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^2 (b g-a h)^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 h}-\frac{B^2 (d g-c h)^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 )}{d^2 h}+\frac{B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac{\left (B^2 (b c-a d) (b g-a h)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\left (\frac{b c-a d}{b}+\frac{d}{b x}\right ) x} \, dx,x,\frac{1}{a+b x}\right )}{b^3 h}-\frac{\left (B^2 (b c-a d) (d g-c h)^2 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 )}{d^3 h}\\ &=-\frac{A B (b c-a d) h n x}{b d}+\frac{A^2 (g+h x)^2}{2 h}-\frac{A B (b g-a h)^2 n \log (a+b x)}{b^2 h}+\frac{A B (d g-c h)^2 n \log (c+d x)}{d^2 h}+\frac{B^2 (b c-a d)^2 h n^2 \log (c+d x)}{b^2 d^2}-\frac{B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 d}+\frac{A B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^2 (b g-a h)^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 h}-\frac{B^2 (d g-c h)^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 )}{d^2 h}+\frac{B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}+\frac{\left (B^2 (b c-a d) (b g-a h)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\frac{d}{b}+\frac{(b c-a d) x}{b}} \, dx,x,\frac{1}{a+b x}\right )}{b^3 h}-\frac{\left (B^2 (b c-a d) (d g-c h)^2 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 )}{d^3 h}\\ &=-\frac{A B (b c-a d) h n x}{b d}+\frac{A^2 (g+h x)^2}{2 h}-\frac{A B (b g-a h)^2 n \log (a+b x)}{b^2 h}+\frac{A B (d g-c h)^2 n \log (c+d x)}{d^2 h}+\frac{B^2 (b c-a d)^2 h n^2 \log (c+d x)}{b^2 d^2}-\frac{B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 d}+\frac{A B (g+h x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^2 (b g-a h)^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b^2 h}-\frac{B^2 (d g-c h)^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 )}{d^2 h}+\frac{B^2 (g+h x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h}-\frac{B^2 (d g-c h)^2 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{d^2 h}-\frac{B^2 (b g-a h)^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b^2 h}\\ \end{align*}
Mathematica [A] time = 0.985289, size = 472, normalized size = 1.61 \[ \frac{2 B^2 n^2 (b c-a d) (a d h+b c h-2 b d g) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )-2 B n \log (a+b x) \left (a d \left (A (a d h-2 b d g)+B d (a h-2 b g) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B n (-a d h+b c h-2 b d g)\right )-B n (b c-a d) (a d h+b c h-2 b d g) \log \left (\frac{b (c+d x)}{b c-a d}\right )+b^2 B c n (c h-2 d g) \log (c+d x)\right )+b \left (d \left (2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) (a B d n (h x-2 g)+b x (2 A d g+A d h x-B c h n))+2 a B n (-2 A d g+A d h x+B c h n-2 B d g n)+b B^2 d x (2 g+h x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+A b x (2 A d g+A d h x-2 B c h n)\right )+2 B n \log (c+d x) \left (B n \left (b c^2 h-a d (c h+2 d g)\right )+b B c (c h-2 d g) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A b c (c h-2 d g)\right )+b B^2 c n^2 (c h-2 d g) \log ^2(c+d x)\right )+a B^2 d^2 n^2 (a h-2 b g) \log ^2(a+b x)}{2 b^2 d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 2.088, size = 11007, normalized size = 37.4 \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 = 3.61767, size = 1219, normalized size = 4.15 \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} h x + A^{2} g +{\left (B^{2} h x + B^{2} g\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \,{\left (A B h x + A B g\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 (h x + g\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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