Optimal. Leaf size=302 \[ \frac{6 B^2 n^2 (b c-a d) \text{PolyLog}\left (2,\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(b g-a h) (d g-c h)}-\frac{6 B^3 n^3 (b c-a d) \text{PolyLog}\left (3,\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}+\frac{3 B n (b c-a d) \log \left (1-\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(b g-a h) (d g-c h)}+\frac{(a+b x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(g+h x) (b g-a h)} \]
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Rubi [B] time = 0.807412, antiderivative size = 650, normalized size of antiderivative = 2.15, number of steps used = 14, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6742, 2490, 36, 31, 2503, 2502, 2315, 2506, 6610} \[ \frac{6 A B^2 n^2 (b c-a d) \text{PolyLog}\left (2,1-\frac{(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}+\frac{6 B^3 n^2 (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,1-\frac{(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}-\frac{6 B^3 n^3 (b c-a d) \text{PolyLog}\left (3,1-\frac{(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}+\frac{3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x) (b g-a h)}-\frac{3 A^2 B n (b c-a d) \log (c+d x)}{(b g-a h) (d g-c h)}+\frac{3 A^2 B n (b c-a d) \log (g+h x)}{(b g-a h) (d g-c h)}+\frac{3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x) (b g-a h)}+\frac{6 A B^2 n (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}+\frac{B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x) (b g-a h)}+\frac{3 B^3 n (b c-a d) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}-\frac{A^3}{h (g+h x)} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2490
Rule 36
Rule 31
Rule 2503
Rule 2502
Rule 2315
Rule 2506
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx &=\int \left (\frac{A^3}{(g+h x)^2}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2}\right ) \, dx\\ &=-\frac{A^3}{h (g+h x)}+\left (3 A^2 B\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2} \, dx+\left (3 A B^2\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2} \, dx+B^3 \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2} \, dx\\ &=-\frac{A^3}{h (g+h x)}+\frac{3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}-\frac{\left (3 A^2 B (b c-a d) n\right ) \int \frac{1}{(c+d x) (g+h x)} \, dx}{b g-a h}-\frac{\left (6 A B^2 (b c-a d) n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{b g-a h}-\frac{\left (3 B^3 (b c-a d) n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{b g-a h}\\ &=-\frac{A^3}{h (g+h x)}+\frac{3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{6 A B^2 (b c-a d) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac{3 B^3 (b c-a d) n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}-\frac{\left (3 A^2 B d (b c-a d) n\right ) \int \frac{1}{c+d x} \, dx}{(b g-a h) (d g-c h)}+\frac{\left (3 A^2 B (b c-a d) h n\right ) \int \frac{1}{g+h x} \, dx}{(b g-a h) (d g-c h)}-\frac{\left (6 A B^2 (b c-a d)^2 n^2\right ) \int \frac{\log \left (-\frac{(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h) (d g-c h)}-\frac{\left (6 B^3 (b c-a d)^2 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h) (d g-c h)}\\ &=-\frac{A^3}{h (g+h x)}-\frac{3 A^2 B (b c-a d) n \log (c+d x)}{(b g-a h) (d g-c h)}+\frac{3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{3 A^2 B (b c-a d) n \log (g+h x)}{(b g-a h) (d g-c h)}+\frac{6 A B^2 (b c-a d) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac{3 B^3 (b c-a d) n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac{6 B^3 (b c-a d) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac{\left (6 A B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b g-a h}\right )}{1+\frac{(-b c+a d) x}{b g-a h}} \, dx,x,\frac{g+h x}{c+d x}\right )}{(b g-a h)^2 (d g-c h)}-\frac{\left (6 B^3 (b c-a d)^2 n^3\right ) \int \frac{\text{Li}_2\left (1+\frac{(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h) (d g-c h)}\\ &=-\frac{A^3}{h (g+h x)}-\frac{3 A^2 B (b c-a d) n \log (c+d x)}{(b g-a h) (d g-c h)}+\frac{3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{3 A^2 B (b c-a d) n \log (g+h x)}{(b g-a h) (d g-c h)}+\frac{6 A B^2 (b c-a d) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac{3 B^3 (b c-a d) n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac{6 A B^2 (b c-a d) n^2 \text{Li}_2\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac{6 B^3 (b c-a d) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}-\frac{6 B^3 (b c-a d) n^3 \text{Li}_3\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}\\ \end{align*}
Mathematica [F] time = 3.62669, size = 0, normalized size = 0. \[ \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 2.753, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( hx+g \right ) ^{2}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{3}}\, 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{B^{3} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3} + 3 \, A B^{2} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 3 \, A^{2} B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{3}}{h^{2} x^{2} + 2 \, g h x + g^{2}}, 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 \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (h x + g\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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