Optimal. Leaf size=463 \[ \frac{2 b h p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}-\frac{2 b h p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac{2 b^2 f p^2 q^2 \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}-\frac{2 b^2 h p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}+\frac{2 b^2 h p^2 q^2 \text{PolyLog}\left (3,-\frac{j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}+\frac{2 b f p q \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f i-e j) (h i-g j)}-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(i+j x) (f i-e j) (h i-g j)}+\frac{h \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}-\frac{h \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2} \]
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Rubi [A] time = 1.18287, antiderivative size = 463, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2418, 2396, 2433, 2374, 6589, 2397, 2394, 2393, 2391, 2445} \[ \frac{2 b h p q \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}-\frac{2 b h p q \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(h i-g j)^2}+\frac{2 b^2 f p^2 q^2 \text{PolyLog}\left (2,-\frac{j (e+f x)}{f i-e j}\right )}{(f i-e j) (h i-g j)}-\frac{2 b^2 h p^2 q^2 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right )}{(h i-g j)^2}+\frac{2 b^2 h p^2 q^2 \text{PolyLog}\left (3,-\frac{j (e+f x)}{f i-e j}\right )}{(h i-g j)^2}+\frac{2 b f p q \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{(f i-e j) (h i-g j)}-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(i+j x) (f i-e j) (h i-g j)}+\frac{h \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2}-\frac{h \log \left (\frac{f (i+j x)}{f i-e j}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(h i-g j)^2} \]
Antiderivative was successfully verified.
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Rule 2418
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 2397
Rule 2394
Rule 2393
Rule 2391
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) (534+j x)^2} \, 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) (534+j x)^2} \, 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^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(534 h-g j)^2 (g+h x)}-\frac{j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(534 h-g j) (534+j x)^2}-\frac{h j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(534 h-g j)^2 (534+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^2 \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{g+h x} \, dx}{(534 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(h j) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{534+j x} \, dx}{(534 h-g j)^2},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}{(534+j x)^2} \, dx}{534 h-g j},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(534 f-e j) (534 h-g j) (534+j x)}+\frac{h \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 )}{(534 h-g j)^2}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (534+j x)}{534 f-e j}\right )}{(534 h-g j)^2}-\operatorname{Subst}\left (\frac{(2 b f h 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}{(534 h-g j)^2},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 h p q) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \log \left (\frac{f (534+j x)}{534 f-e j}\right )}{e+f x} \, dx}{(534 h-g j)^2},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 j p q) \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{534+j x} \, dx}{(534 f-e j) (534 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(534 f-e j) (534 h-g j) (534+j x)}+\frac{h \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 )}{(534 h-g j)^2}+\frac{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (534+j x)}{534 f-e j}\right )}{(534 f-e j) (534 h-g j)}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (534+j x)}{534 f-e j}\right )}{(534 h-g j)^2}-\operatorname{Subst}\left (\frac{(2 b h 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 )}{(534 h-g j)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(2 b h 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{534 f-e j}{f}+\frac{j x}{f}\right )}{534 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(534 h-g j)^2},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 f^2 p^2 q^2\right ) \int \frac{\log \left (\frac{f (534+j x)}{534 f-e j}\right )}{e+f x} \, dx}{(534 f-e j) (534 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(534 f-e j) (534 h-g j) (534+j x)}+\frac{h \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 )}{(534 h-g j)^2}+\frac{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (534+j x)}{534 f-e j}\right )}{(534 f-e j) (534 h-g j)}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (534+j x)}{534 f-e j}\right )}{(534 h-g j)^2}+\frac{2 b h 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 )}{(534 h-g j)^2}-\frac{2 b h 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)}{534 f-e j}\right )}{(534 h-g j)^2}-\operatorname{Subst}\left (\frac{\left (2 b^2 h 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 )}{(534 h-g j)^2},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 h p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{j x}{534 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(534 h-g j)^2},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 f p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{j x}{534 f-e j}\right )}{x} \, dx,x,e+f x\right )}{(534 f-e j) (534 h-g j)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(534 f-e j) (534 h-g j) (534+j x)}+\frac{h \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 )}{(534 h-g j)^2}+\frac{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (534+j x)}{534 f-e j}\right )}{(534 f-e j) (534 h-g j)}-\frac{h \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \log \left (\frac{f (534+j x)}{534 f-e j}\right )}{(534 h-g j)^2}+\frac{2 b h 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 )}{(534 h-g j)^2}+\frac{2 b^2 f p^2 q^2 \text{Li}_2\left (-\frac{j (e+f x)}{534 f-e j}\right )}{(534 f-e j) (534 h-g j)}-\frac{2 b h 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)}{534 f-e j}\right )}{(534 h-g j)^2}-\frac{2 b^2 h p^2 q^2 \text{Li}_3\left (-\frac{h (e+f x)}{f g-e h}\right )}{(534 h-g j)^2}+\frac{2 b^2 h p^2 q^2 \text{Li}_3\left (-\frac{j (e+f x)}{534 f-e j}\right )}{(534 h-g j)^2}\\ \end{align*}
Mathematica [A] time = 0.945503, size = 654, normalized size = 1.41 \[ \frac{-2 b p q \left (-h (i+j x) (f i-e j) \left (\text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )+\log (e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )\right )+h (i+j x) (f i-e j) \left (\text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right )+\log (e+f x) \log \left (\frac{f (i+j x)}{f i-e j}\right )\right )+(h i-g j) (j (e+f x) \log (e+f x)-f (i+j x) \log (i+j x))\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )-b^2 p^2 q^2 \left (-h (i+j x) (f i-e j) \left (-2 \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right )+2 \log (e+f x) \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )+\log ^2(e+f x) \log \left (\frac{f (g+h x)}{f g-e h}\right )\right )+(h i-g j) \left (\log (e+f x) \left (j (e+f x) \log (e+f x)-2 f (i+j x) \log \left (\frac{f (i+j x)}{f i-e j}\right )\right )-2 f (i+j x) \text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right )\right )+h (i+j x) (f i-e j) \left (-2 \text{PolyLog}\left (3,\frac{j (e+f x)}{e j-f i}\right )+2 \log (e+f x) \text{PolyLog}\left (2,\frac{j (e+f x)}{e j-f i}\right )+\log ^2(e+f x) \log \left (\frac{f (i+j x)}{f i-e j}\right )\right )\right )+(f i-e j) (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2+h (i+j x) (f i-e j) \log (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2-h (i+j x) (f i-e j) \log (i+j x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2}{(i+j x) (f i-e j) (h i-g j)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1., 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 ) ^{2}}}\, 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{h \log \left (h x + g\right )}{h^{2} i^{2} - 2 \, g h i j + g^{2} j^{2}} - \frac{h \log \left (j x + i\right )}{h^{2} i^{2} - 2 \, g h i j + g^{2} j^{2}} + \frac{1}{h i^{2} - g i j +{\left (h i j - g j^{2}\right )} x}\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^{2} x^{3} + g i^{2} +{\left (2 \, h i j + g j^{2}\right )} x^{2} +{\left (h i^{2} + 2 \, g i 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^{2} x^{3} + g i^{2} +{\left (2 \, h i j + g j^{2}\right )} x^{2} +{\left (h i^{2} + 2 \, g i 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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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