Optimal. Leaf size=123 \[ \frac{2 b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac{2 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{h}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h} \]
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Rubi [A] time = 0.275149, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2396, 2433, 2374, 6589, 2445} \[ \frac{2 b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac{2 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{h}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h} \]
Antiderivative was successfully verified.
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Rule 2396
Rule 2433
Rule 2374
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 )^2}{g+h x} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{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 )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}-\operatorname{Subst}\left (\frac{(2 b f p q) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \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 )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}-\operatorname{Subst}\left (\frac{(2 b p q) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q x^{p q}\right )\right ) \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 )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}+\frac{2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}-\operatorname{Subst}\left (\frac{\left (2 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\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 )^2 \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h}+\frac{2 b p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}-\frac{2 b^2 p^2 q^2 \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{h}\\ \end{align*}
Mathematica [B] time = 0.11753, size = 324, normalized size = 2.63 \[ \frac{2 b p q \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )-2 b^2 p^2 q^2 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right )+a^2 \log (g+h x)+2 a b \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )-2 a b p q \log (e+f x) \log (g+h x)+2 a b p q \log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )+b^2 \log (g+h x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-2 b^2 p q \log (e+f x) \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )+2 b^2 p q \log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+b^2 p^2 q^2 \log ^2(e+f x) \log (g+h x)-b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, 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 ) ^{2}}{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^{2} \log \left (h x + g\right )}{h} + \int \frac{b^{2} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} +{\left (\log \left (c\right )^{2} + 2 \, \log \left (c\right ) \log \left (d^{q}\right ) + \log \left (d^{q}\right )^{2}\right )} b^{2} + 2 \, a b{\left (\log \left (c\right ) + \log \left (d^{q}\right )\right )} + 2 \,{\left (b^{2}{\left (\log \left (c\right ) + \log \left (d^{q}\right )\right )} + a 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^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, a b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a^{2}}{h x + g}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{g + h x}\, dx \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 )}^{2}}{h x + g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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