Optimal. Leaf size=177 \[ -\frac{6 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{3 b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac{6 b^3 p^3 q^3 \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{h}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h} \]
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Rubi [A] time = 0.412099, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2396, 2433, 2374, 2383, 6589, 2445} \[ -\frac{6 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{3 b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac{6 b^3 p^3 q^3 \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{h}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h} \]
Antiderivative was successfully verified.
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Rule 2396
Rule 2433
Rule 2374
Rule 2383
Rule 6589
Rule 2445
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{g+h x} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}-\operatorname{Subst}\left (\frac{(3 b f p q) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}-\operatorname{Subst}\left (\frac{(3 b p q) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \log \left (\frac{f \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}+\frac{3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}-\operatorname{Subst}\left (\frac{\left (6 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right ) \text{Li}_2\left (-\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}+\frac{3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}-\frac{6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}+\operatorname{Subst}\left (\frac{\left (6 b^3 p^3 q^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}+\frac{3 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}-\frac{6 b^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}+\frac{6 b^3 p^3 q^3 \text{Li}_4\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}\\ \end{align*}
Mathematica [B] time = 0.264883, size = 646, normalized size = 3.65 \[ \frac{-6 b^2 p^2 q^2 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+3 b p q \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+6 b^3 p^3 q^3 \text{PolyLog}\left (4,\frac{h (e+f x)}{e h-f g}\right )+3 a^2 b \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )-3 a^2 b p q \log (e+f x) \log (g+h x)+3 a^2 b p q \log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )+a^3 \log (g+h x)+3 a b^2 \log (g+h x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-6 a b^2 p q \log (e+f x) \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )+6 a b^2 p q \log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+3 a b^2 p^2 q^2 \log ^2(e+f x) \log (g+h x)-3 a b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )+3 b^3 p^2 q^2 \log ^2(e+f x) \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )-3 b^3 p^2 q^2 \log ^2(e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+b^3 \log (g+h x) \log ^3\left (c \left (d (e+f x)^p\right )^q\right )-3 b^3 p q \log (e+f x) \log (g+h x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+3 b^3 p q \log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-b^3 p^3 q^3 \log ^3(e+f x) \log (g+h x)+b^3 p^3 q^3 \log ^3(e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.715, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{3}}{hx+g}}\, 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*} \frac{a^{3} \log \left (h x + g\right )}{h} + \int \frac{b^{3} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{3} + 3 \,{\left (\log \left (c\right )^{2} + 2 \, \log \left (c\right ) \log \left (d^{q}\right ) + \log \left (d^{q}\right )^{2}\right )} a b^{2} +{\left (\log \left (c\right )^{3} + 3 \, \log \left (c\right )^{2} \log \left (d^{q}\right ) + 3 \, \log \left (c\right ) \log \left (d^{q}\right )^{2} + \log \left (d^{q}\right )^{3}\right )} b^{3} + 3 \, a^{2} b{\left (\log \left (c\right ) + \log \left (d^{q}\right )\right )} + 3 \,{\left (b^{3}{\left (\log \left (c\right ) + \log \left (d^{q}\right )\right )} + a b^{2}\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} + 3 \,{\left ({\left (\log \left (c\right )^{2} + 2 \, \log \left (c\right ) \log \left (d^{q}\right ) + \log \left (d^{q}\right )^{2}\right )} b^{3} + 2 \, a b^{2}{\left (\log \left (c\right ) + \log \left (d^{q}\right )\right )} + a^{2} b\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h x + g}\,{d x} \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 (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 3 \, a b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 3 \, a^{2} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a^{3}}{h x + g}, 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 (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}}{h x + g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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