Optimal. Leaf size=274 \[ \frac{24 b^3 f p^3 q^3 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)}-\frac{12 b^2 f p^2 q^2 \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h (f g-e h)}-\frac{24 b^4 f p^4 q^4 \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{h (f g-e h)}-\frac{4 b f p q \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h (f g-e h)}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x) (f g-e h)} \]
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Rubi [A] time = 0.52508, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2397, 2396, 2433, 2374, 2383, 6589, 2445} \[ \frac{24 b^3 f p^3 q^3 \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (f g-e h)}-\frac{12 b^2 f p^2 q^2 \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h (f g-e h)}-\frac{24 b^4 f p^4 q^4 \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{h (f g-e h)}-\frac{4 b f p q \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h (f g-e h)}+\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(g+h x) (f g-e h)} \]
Antiderivative was successfully verified.
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Rule 2397
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 )^4}{(g+h x)^2} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^4}{(g+h x)^2} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(f g-e h) (g+h x)}-\operatorname{Subst}\left (\frac{(4 b f p q) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx}{f g-e h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(f g-e h) (g+h x)}-\frac{4 b f p q \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 (f g-e h)}+\operatorname{Subst}\left (\frac{\left (12 b^2 f^2 p^2 q^2\right ) \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 (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(f g-e h) (g+h x)}-\frac{4 b f p q \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 (f g-e h)}+\operatorname{Subst}\left (\frac{\left (12 b^2 f p^2 q^2\right ) \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 (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(f g-e h) (g+h x)}-\frac{4 b f p q \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 (f g-e h)}-\frac{12 b^2 f p^2 q^2 \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 (f g-e h)}+\operatorname{Subst}\left (\frac{\left (24 b^3 f p^3 q^3\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 (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(f g-e h) (g+h x)}-\frac{4 b f p q \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 (f g-e h)}-\frac{12 b^2 f p^2 q^2 \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 (f g-e h)}+\frac{24 b^3 f p^3 q^3 \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 (f g-e h)}-\operatorname{Subst}\left (\frac{\left (24 b^4 f p^4 q^4\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 (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^4}{(f g-e h) (g+h x)}-\frac{4 b f p q \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 (f g-e h)}-\frac{12 b^2 f p^2 q^2 \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 (f g-e h)}+\frac{24 b^3 f p^3 q^3 \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 (f g-e h)}-\frac{24 b^4 f p^4 q^4 \text{Li}_4\left (-\frac{h (e+f x)}{f g-e h}\right )}{h (f g-e h)}\\ \end{align*}
Mathematica [B] time = 0.57738, size = 1301, normalized size = 4.75 \[ \frac{f g a^4-e h a^4-4 b f g p q \log (e+f x) a^3-4 b f h p q x \log (e+f x) a^3+4 b f g \log \left (c \left (d (e+f x)^p\right )^q\right ) a^3-4 b e h \log \left (c \left (d (e+f x)^p\right )^q\right ) a^3+4 b f g p q \log \left (\frac{f (g+h x)}{f g-e h}\right ) a^3+4 b f h p q x \log \left (\frac{f (g+h x)}{f g-e h}\right ) a^3+6 b^2 f g p^2 q^2 \log ^2(e+f x) a^2+6 b^2 f h p^2 q^2 x \log ^2(e+f x) a^2+6 b^2 f g \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) a^2-6 b^2 e h \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) a^2-12 b^2 f g p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) a^2-12 b^2 f h p q x \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) a^2+12 b^2 f g p q \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right ) a^2+12 b^2 f h p q x \log \left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right ) a^2-4 b^3 f g p^3 q^3 \log ^3(e+f x) a-4 b^3 f h p^3 q^3 x \log ^3(e+f x) a+4 b^3 f g \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) a-4 b^3 e h \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) a-12 b^3 f g p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) a-12 b^3 f h p q x \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) a+12 b^3 f g p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) a+12 b^3 f h p^2 q^2 x \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) a+12 b^3 f g p q \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right ) a+12 b^3 f h p q x \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right ) a+b^4 f g p^4 q^4 \log ^4(e+f x)+b^4 f h p^4 q^4 x \log ^4(e+f x)+b^4 f g \log ^4\left (c \left (d (e+f x)^p\right )^q\right )-b^4 e h \log ^4\left (c \left (d (e+f x)^p\right )^q\right )-4 b^4 f g p q \log (e+f x) \log ^3\left (c \left (d (e+f x)^p\right )^q\right )-4 b^4 f h p q x \log (e+f x) \log ^3\left (c \left (d (e+f x)^p\right )^q\right )+6 b^4 f g p^2 q^2 \log ^2(e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+6 b^4 f h p^2 q^2 x \log ^2(e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-4 b^4 f g p^3 q^3 \log ^3(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )-4 b^4 f h p^3 q^3 x \log ^3(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )+4 b^4 f g p q \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right )+4 b^4 f h p q x \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right )+12 b^2 f p^2 q^2 (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )-24 b^3 f p^3 q^3 (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text{PolyLog}\left (3,\frac{h (e+f x)}{e h-f g}\right )+24 b^4 f g p^4 q^4 \text{PolyLog}\left (4,\frac{h (e+f x)}{e h-f g}\right )+24 b^4 f h p^4 q^4 x \text{PolyLog}\left (4,\frac{h (e+f x)}{e h-f g}\right )}{h (e h-f g) (g+h x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.686, 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 ) ^{4}}{ \left ( hx+g \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*} \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^{4} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{4} + 4 \, a b^{3} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 6 \, a^{2} b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 4 \, a^{3} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a^{4}}{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 (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{4}}{{\left (h x + g\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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