Optimal. Leaf size=427 \[ \frac{b p q (h i-g j)^3 \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{h^4}+\frac{(i+j x)^2 (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h^2}+\frac{(h i-g j)^3 \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^4}+\frac{(i+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac{a j x (h i-g j)^2}{h^3}+\frac{b j (e+f x) (h i-g j)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^3}-\frac{b p q (f i-e j)^2 \log (e+f x) (h i-g j)}{2 f^2 h^2}-\frac{b j p q x (f i-e j)^2}{3 f^2 h}-\frac{b p q (f i-e j)^3 \log (e+f x)}{3 f^3 h}-\frac{b j p q x (f i-e j) (h i-g j)}{2 f h^2}-\frac{b p q (i+j x)^2 (f i-e j)}{6 f h}-\frac{b p q (i+j x)^2 (h i-g j)}{4 h^2}-\frac{b j p q x (h i-g j)^2}{h^3}-\frac{b p q (i+j x)^3}{9 h} \]
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Rubi [A] time = 0.816707, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2418, 2389, 2295, 2394, 2393, 2391, 2395, 43, 2445} \[ \frac{b p q (h i-g j)^3 \text{PolyLog}\left (2,-\frac{h (e+f x)}{f g-e h}\right )}{h^4}+\frac{(i+j x)^2 (h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h^2}+\frac{(h i-g j)^3 \log \left (\frac{f (g+h x)}{f g-e h}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h^4}+\frac{(i+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac{a j x (h i-g j)^2}{h^3}+\frac{b j (e+f x) (h i-g j)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^3}-\frac{b p q (f i-e j)^2 \log (e+f x) (h i-g j)}{2 f^2 h^2}-\frac{b j p q x (f i-e j)^2}{3 f^2 h}-\frac{b p q (f i-e j)^3 \log (e+f x)}{3 f^3 h}-\frac{b j p q x (f i-e j) (h i-g j)}{2 f h^2}-\frac{b p q (i+j x)^2 (f i-e j)}{6 f h}-\frac{b p q (i+j x)^2 (h i-g j)}{4 h^2}-\frac{b j p q x (h i-g j)^2}{h^3}-\frac{b p q (i+j x)^3}{9 h} \]
Antiderivative was successfully verified.
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Rule 2418
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rule 2395
Rule 43
Rule 2445
Rubi steps
\begin{align*} \int \frac{(523+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{g+h x} \, dx &=\operatorname{Subst}\left (\int \frac{(523+j x)^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{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 (523 h-g j)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h^3}+\frac{(523 h-g j)^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h^3 (g+h x)}+\frac{j (523 h-g j) (523+j x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h^2}+\frac{j (523+j x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{h}\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 (523+j x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(j (523 h-g j)) \int (523+j x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (j (523 h-g j)^2\right ) \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx}{h^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(523 h-g j)^3 \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{g+h x} \, dx}{h^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{a j (523 h-g j)^2 x}{h^3}+\frac{(523 h-g j) (523+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h^2}+\frac{(523+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac{(523 h-g j)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h^4}+\operatorname{Subst}\left (\frac{\left (b j (523 h-g j)^2\right ) \int \log \left (c d^q (e+f x)^{p q}\right ) \, dx}{h^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(b f p q) \int \frac{(523+j x)^3}{e+f x} \, dx}{3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(b f (523 h-g j) p q) \int \frac{(523+j x)^2}{e+f x} \, dx}{2 h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (b f (523 h-g j)^3 p q\right ) \int \frac{\log \left (\frac{f (g+h x)}{f g-e h}\right )}{e+f x} \, dx}{h^4},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{a j (523 h-g j)^2 x}{h^3}+\frac{(523 h-g j) (523+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h^2}+\frac{(523+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac{(523 h-g j)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h^4}+\operatorname{Subst}\left (\frac{\left (b j (523 h-g j)^2\right ) \operatorname{Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f h^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(b f p q) \int \left (\frac{j (523 f-e j)^2}{f^3}+\frac{(523 f-e j)^3}{f^3 (e+f x)}+\frac{j (523 f-e j) (523+j x)}{f^2}+\frac{j (523+j x)^2}{f}\right ) \, dx}{3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(b f (523 h-g j) p q) \int \left (\frac{j (523 f-e j)}{f^2}+\frac{(523 f-e j)^2}{f^2 (e+f x)}+\frac{j (523+j x)}{f}\right ) \, dx}{2 h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (b (523 h-g j)^3 p q\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{h x}{f g-e h}\right )}{x} \, dx,x,e+f x\right )}{h^4},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{a j (523 h-g j)^2 x}{h^3}-\frac{b j (523 f-e j)^2 p q x}{3 f^2 h}-\frac{b j (523 f-e j) (523 h-g j) p q x}{2 f h^2}-\frac{b j (523 h-g j)^2 p q x}{h^3}-\frac{b (523 f-e j) p q (523+j x)^2}{6 f h}-\frac{b (523 h-g j) p q (523+j x)^2}{4 h^2}-\frac{b p q (523+j x)^3}{9 h}-\frac{b (523 f-e j)^3 p q \log (e+f x)}{3 f^3 h}-\frac{b (523 f-e j)^2 (523 h-g j) p q \log (e+f x)}{2 f^2 h^2}+\frac{b j (523 h-g j)^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h^3}+\frac{(523 h-g j) (523+j x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 h^2}+\frac{(523+j x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h}+\frac{(523 h-g j)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac{f (g+h x)}{f g-e h}\right )}{h^4}+\frac{b (523 h-g j)^3 p q \text{Li}_2\left (-\frac{h (e+f x)}{f g-e h}\right )}{h^4}\\ \end{align*}
Mathematica [A] time = 0.686944, size = 386, normalized size = 0.9 \[ \frac{36 b f^3 p q (h i-g j)^3 \text{PolyLog}\left (2,\frac{h (e+f x)}{e h-f g}\right )+f \left (h j x \left (6 a f^2 \left (6 g^2 j^2-3 g h j (6 i+j x)+h^2 \left (18 i^2+9 i j x+2 j^2 x^2\right )\right )-b p q \left (12 e^2 h^2 j^2-6 e f h j (-3 g j+9 h i+h j x)+f^2 \left (36 g^2 j^2-9 g h j (12 i+j x)+h^2 \left (108 i^2+27 i j x+4 j^2 x^2\right )\right )\right )\right )+36 a f^2 (h i-g j)^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )+6 b f \log \left (c \left (d (e+f x)^p\right )^q\right ) \left (h j \left (6 e \left (g^2 j^2-3 g h i j+3 h^2 i^2\right )+f x \left (6 g^2 j^2-3 g h j (6 i+j x)+h^2 \left (18 i^2+9 i j x+2 j^2 x^2\right )\right )\right )+6 f (h i-g j)^3 \log \left (\frac{f (g+h x)}{f g-e h}\right )\right )\right )+6 b e^2 h^2 j^2 p q \log (e+f x) (2 e h j+3 f g j-9 f h i)}{36 f^3 h^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.856, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( jx+i \right ) ^{3} \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) }{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*} 3 \, a i^{2} j{\left (\frac{x}{h} - \frac{g \log \left (h x + g\right )}{h^{2}}\right )} - \frac{1}{6} \, a j^{3}{\left (\frac{6 \, g^{3} \log \left (h x + g\right )}{h^{4}} - \frac{2 \, h^{2} x^{3} - 3 \, g h x^{2} + 6 \, g^{2} x}{h^{3}}\right )} + \frac{3}{2} \, a i j^{2}{\left (\frac{2 \, g^{2} \log \left (h x + g\right )}{h^{3}} + \frac{h x^{2} - 2 \, g x}{h^{2}}\right )} + \frac{a i^{3} \log \left (h x + g\right )}{h} + \int \frac{{\left (j^{3} \log \left (c\right ) + j^{3} \log \left (d^{q}\right )\right )} b x^{3} + 3 \,{\left (i j^{2} \log \left (c\right ) + i j^{2} \log \left (d^{q}\right )\right )} b x^{2} + 3 \,{\left (i^{2} j \log \left (c\right ) + i^{2} j \log \left (d^{q}\right )\right )} b x +{\left (i^{3} \log \left (c\right ) + i^{3} \log \left (d^{q}\right )\right )} b +{\left (b j^{3} x^{3} + 3 \, b i j^{2} x^{2} + 3 \, b i^{2} j x + b i^{3}\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 j^{3} x^{3} + 3 \, a i j^{2} x^{2} + 3 \, a i^{2} j x + a i^{3} +{\left (b j^{3} x^{3} + 3 \, b i j^{2} x^{2} + 3 \, b i^{2} j x + b i^{3}\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 )}^{3}{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}}{h x + g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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