Optimal. Leaf size=349 \[ -\frac{6 b^2 p^2 q^2 (h i-g j) \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^2}+\frac{3 b p q (h i-g j) \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^2}+\frac{6 b^3 p^3 q^3 (h i-g j) \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{h^2}+\frac{6 a b^2 j p^2 q^2 x}{h}+\frac{(h i-g j) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h^2}-\frac{3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{6 b^3 j p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac{6 b^3 j p^3 q^3 x}{h} \]
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Rubi [A] time = 0.893341, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.303, Rules used = {2418, 2389, 2296, 2295, 2396, 2433, 2374, 2383, 6589, 2445} \[ -\frac{6 b^2 p^2 q^2 (h i-g j) \text{PolyLog}\left (3,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^2}+\frac{3 b p q (h i-g j) \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h^2}+\frac{6 b^3 p^3 q^3 (h i-g j) \text{PolyLog}\left (4,-\frac{h (e+f x)}{f g-e h}\right )}{h^2}+\frac{6 a b^2 j p^2 q^2 x}{h}+\frac{(h i-g j) \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{h^2}-\frac{3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{6 b^3 j p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac{6 b^3 j p^3 q^3 x}{h} \]
Antiderivative was successfully verified.
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Rule 2418
Rule 2389
Rule 2296
Rule 2295
Rule 2396
Rule 2433
Rule 2374
Rule 2383
Rule 6589
Rule 2445
Rubi steps
\begin{align*} \int \frac{(536+j x) \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{(536+j x) \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 )\\ &=\operatorname{Subst}\left (\int \left (\frac{j \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{h}+\frac{(536 h-g j) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{h (g+h 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{j \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3 \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(536 h-g j) \int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{g+h x} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(536 h-g j) \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^2}+\operatorname{Subst}\left (\frac{j \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^3 \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(3 b f (536 h-g j) 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^2},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 )^3}{f h}+\frac{(536 h-g j) \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^2}-\operatorname{Subst}\left (\frac{(3 b j p q) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(3 b (536 h-g j) 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^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{(536 h-g j) \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^2}+\frac{3 b (536 h-g j) 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^2}+\operatorname{Subst}\left (\frac{\left (6 b^2 j p^2 q^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (6 b^2 (536 h-g j) 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^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{6 a b^2 j p^2 q^2 x}{h}-\frac{3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{(536 h-g j) \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^2}+\frac{3 b (536 h-g j) 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^2}-\frac{6 b^2 (536 h-g j) 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^2}+\operatorname{Subst}\left (\frac{\left (6 b^3 j p^2 q^2\right ) \operatorname{Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (6 b^3 (536 h-g j) 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^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{6 a b^2 j p^2 q^2 x}{h}-\frac{6 b^3 j p^3 q^3 x}{h}+\frac{6 b^3 j p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac{3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac{j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac{(536 h-g j) \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^2}+\frac{3 b (536 h-g j) 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^2}-\frac{6 b^2 (536 h-g j) 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^2}+\frac{6 b^3 (536 h-g j) p^3 q^3 \text{Li}_4\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^2}\\ \end{align*}
Mathematica [B] time = 0.677263, size = 1769, normalized size = 5.07 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.628, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( jx+i \right ) \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*} a^{3} j{\left (\frac{x}{h} - \frac{g \log \left (h x + g\right )}{h^{2}}\right )} + \frac{a^{3} i \log \left (h x + g\right )}{h} + \int \frac{3 \,{\left (i \log \left (c\right ) + i \log \left (d^{q}\right )\right )} a^{2} b + 3 \,{\left (i \log \left (c\right )^{2} + 2 \, i \log \left (c\right ) \log \left (d^{q}\right ) + i \log \left (d^{q}\right )^{2}\right )} a b^{2} +{\left (i \log \left (c\right )^{3} + 3 \, i \log \left (c\right )^{2} \log \left (d^{q}\right ) + 3 \, i \log \left (c\right ) \log \left (d^{q}\right )^{2} + i \log \left (d^{q}\right )^{3}\right )} b^{3} +{\left (b^{3} j x + b^{3} i\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{3} + 3 \,{\left ({\left (i \log \left (c\right ) + i \log \left (d^{q}\right )\right )} b^{3} + a b^{2} i +{\left ({\left (j \log \left (c\right ) + j \log \left (d^{q}\right )\right )} b^{3} + a b^{2} j\right )} x\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} +{\left (3 \,{\left (j \log \left (c\right ) + j \log \left (d^{q}\right )\right )} a^{2} b + 3 \,{\left (j \log \left (c\right )^{2} + 2 \, j \log \left (c\right ) \log \left (d^{q}\right ) + j \log \left (d^{q}\right )^{2}\right )} a b^{2} +{\left (j \log \left (c\right )^{3} + 3 \, j \log \left (c\right )^{2} \log \left (d^{q}\right ) + 3 \, j \log \left (c\right ) \log \left (d^{q}\right )^{2} + j \log \left (d^{q}\right )^{3}\right )} b^{3}\right )} x + 3 \,{\left (2 \,{\left (i \log \left (c\right ) + i \log \left (d^{q}\right )\right )} a b^{2} +{\left (i \log \left (c\right )^{2} + 2 \, i \log \left (c\right ) \log \left (d^{q}\right ) + i \log \left (d^{q}\right )^{2}\right )} b^{3} + a^{2} b i +{\left (2 \,{\left (j \log \left (c\right ) + j \log \left (d^{q}\right )\right )} a b^{2} +{\left (j \log \left (c\right )^{2} + 2 \, j \log \left (c\right ) \log \left (d^{q}\right ) + j \log \left (d^{q}\right )^{2}\right )} b^{3} + a^{2} b j\right )} x\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{a^{3} j x + a^{3} i +{\left (b^{3} j x + b^{3} i\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{3} + 3 \,{\left (a b^{2} j x + a b^{2} i\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 3 \,{\left (a^{2} b j x + a^{2} b i\right )} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{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 (j x + i\right )}{\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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