Optimal. Leaf size=288 \[ \frac{2 b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac{2 b p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac{2 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{h i-g j}+\frac{2 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{j (e+f x)}{f i-e j}\right )}{h i-g j}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}-\frac{\log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j} \]
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Rubi [A] time = 0.898805, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {2418, 2396, 2433, 2374, 6589, 2445} \[ \frac{2 b p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac{2 b p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h i-g j}-\frac{2 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{h i-g j}+\frac{2 b^2 p^2 q^2 \text{PolyLog}\left (3,-\frac{j (e+f x)}{f i-e j}\right )}{h i-g j}+\frac{\log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j}-\frac{\log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h i-g j} \]
Antiderivative was successfully verified.
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Rule 2418
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) (533+j 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) (533+j x)} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{h \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(533 h-g j) (g+h x)}-\frac{j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(533 h-g j) (533+j x)}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{h \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{g+h x} \, dx}{533 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{j \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{533+j x} \, dx}{533 h-g j},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 )}{533 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (533+j x)}{533 f-e j}\right )}{533 h-g j}-\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}{533 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\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 (533+j x)}{533 f-e j}\right )}{e+f x} \, dx}{533 h-g j},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 )}{533 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (533+j x)}{533 f-e j}\right )}{533 h-g j}-\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 )}{533 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\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{533 f-e j}{f}+\frac{j x}{f}\right )}{533 f-e j}\right )}{x} \, dx,x,e+f x\right )}{533 h-g j},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 )}{533 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (533+j x)}{533 f-e j}\right )}{533 h-g j}+\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 )}{533 h-g j}-\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{j (e+f x)}{533 f-e j}\right )}{533 h-g j}-\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 )}{533 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (2 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{533 f-e j}\right )}{x} \, dx,x,e+f x\right )}{533 h-g j},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 )}{533 h-g j}-\frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (533+j x)}{533 f-e j}\right )}{533 h-g j}+\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 )}{533 h-g j}-\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{j (e+f x)}{533 f-e j}\right )}{533 h-g j}-\frac{2 b^2 p^2 q^2 \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{533 h-g j}+\frac{2 b^2 p^2 q^2 \text{Li}_3\left (-\frac{j (e+f x)}{533 f-e j}\right )}{533 h-g j}\\ \end{align*}
Mathematica [B] time = 0.322164, size = 652, normalized size = 2.26 \[ \frac{2 b p q \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )-2 b p q \text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )-2 b^2 p^2 q^2 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right )+2 b^2 p^2 q^2 \text{PolyLog}\left (3,\frac{j (e+f x)}{e j-f i}\right )+a^2 \log (g+h x)-a^2 \log (i+j x)+2 a b \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )-2 a b \log (i+j x) \log \left (c \left (d (e+f x)^p\right )^q\right )-2 a b p q \log (e+f x) \log (g+h x)+2 a b p q \log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )+2 a b p q \log (e+f x) \log (i+j x)-2 a b p q \log (e+f x) \log \left (\frac{f (i+j x)}{f i-e j}\right )+b^2 \log (g+h x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-2 b^2 p q \log (e+f x) \log (g+h x) \log \left (c \left (d (e+f x)^p\right )^q\right )+2 b^2 p q \log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )-b^2 \log (i+j x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+2 b^2 p q \log (e+f x) \log (i+j x) \log \left (c \left (d (e+f x)^p\right )^q\right )-2 b^2 p q \log (e+f x) \log \left (\frac{f (i+j x)}{f i-e j}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+b^2 p^2 q^2 \log ^2(e+f x) \log (g+h x)-b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )-b^2 p^2 q^2 \log ^2(e+f x) \log (i+j x)+b^2 p^2 q^2 \log ^2(e+f x) \log \left (\frac{f (i+j x)}{f i-e j}\right )}{h i-g j} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.031, 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}}{ \left ( hx+g \right ) \left ( jx+i \right ) }}\, 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*} a^{2}{\left (\frac{\log \left (h x + g\right )}{h i - g j} - \frac{\log \left (j x + i\right )}{h i - g j}\right )} + \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 j x^{2} + g i +{\left (h i + g j\right )} x}\,{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 j x^{2} + g i +{\left (h i + g j\right )} x}, 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 )}^{2}}{{\left (h x + g\right )}{\left (j x + i\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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