Optimal. Leaf size=349 \[ \frac {6 a b^2 j p^2 q^2 x}{h}-\frac {6 b^3 j p^3 q^3 x}{h}+\frac {6 b^3 j p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f h}-\frac {3 b j p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f h}+\frac {j (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f h}+\frac {(h i-g j) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \log \left (\frac {f (g+h x)}{f g-e h}\right )}{h^2}+\frac {3 b (h i-g j) p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text {Li}_2\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^2}-\frac {6 b^2 (h i-g j) p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Li}_3\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^2}+\frac {6 b^3 (h i-g j) p^3 q^3 \text {Li}_4\left (-\frac {h (e+f x)}{f g-e h}\right )}{h^2} \]
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Rubi [A]
time = 0.59, antiderivative size = 349, normalized size of antiderivative = 1.00, 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 = {2465, 2436,
2333, 2332, 2443, 2481, 2421, 2430, 6724, 2495} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rule 2421
Rule 2430
Rule 2436
Rule 2443
Rule 2465
Rule 2481
Rule 2495
Rule 6724
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 &=\text {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 )\\ &=\text {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 )\\ &=\text {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 )+\text {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}+\text {Subst}\left (\frac {j \text {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 )-\text {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}-\text {Subst}\left (\frac {(3 b j p q) \text {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 )-\text {Subst}\left (\frac {(3 b (536 h-g j) p q) \text {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}+\text {Subst}\left (\frac {\left (6 b^2 j p^2 q^2\right ) \text {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 )-\text {Subst}\left (\frac {\left (6 b^2 (536 h-g j) p^2 q^2\right ) \text {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}+\text {Subst}\left (\frac {\left (6 b^3 j p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (6 b^3 (536 h-g j) p^3 q^3\right ) \text {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*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1769\) vs. \(2(349)=698\).
time = 0.42, size = 1769, normalized size = 5.07 \begin {gather*} \frac {-3 a^2 b e h j p q+a^3 f h j x-3 a^2 b f h j p q x+6 a b^2 f h j p^2 q^2 x-6 b^3 f h j p^3 q^3 x+3 a^2 b e h j p q \log (e+f x)+6 b^3 e h j p^3 q^3 \log (e+f x)-3 a b^2 e h j p^2 q^2 \log ^2(e+f x)+b^3 e h j p^3 q^3 \log ^3(e+f x)-6 a b^2 e h j p q \log \left (c \left (d (e+f x)^p\right )^q\right )+3 a^2 b f h j x \log \left (c \left (d (e+f x)^p\right )^q\right )-6 a b^2 f h j p q x \log \left (c \left (d (e+f x)^p\right )^q\right )+6 b^3 f h j p^2 q^2 x \log \left (c \left (d (e+f x)^p\right )^q\right )+6 a b^2 e h j p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )-3 b^3 e h j p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )-3 b^3 e h j p q \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+3 a b^2 f h j x \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-3 b^3 f h j p q x \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+3 b^3 e h j p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+b^3 f h j x \log ^3\left (c \left (d (e+f x)^p\right )^q\right )+a^3 f h i \log (g+h x)-a^3 f g j \log (g+h x)-3 a^2 b f h i p q \log (e+f x) \log (g+h x)+3 a^2 b f g j p q \log (e+f x) \log (g+h x)+3 a b^2 f h i p^2 q^2 \log ^2(e+f x) \log (g+h x)-3 a b^2 f g j p^2 q^2 \log ^2(e+f x) \log (g+h x)-b^3 f h i p^3 q^3 \log ^3(e+f x) \log (g+h x)+b^3 f g j p^3 q^3 \log ^3(e+f x) \log (g+h x)+3 a^2 b f h i \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-3 a^2 b f g j \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-6 a b^2 f h i p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+6 a b^2 f g j p q \log (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 b^3 f h i p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-3 b^3 f g j p^2 q^2 \log ^2(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 a b^2 f h i \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-3 a b^2 f g j \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-3 b^3 f h i p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 b^3 f g j p q \log (e+f x) \log ^2\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+b^3 f h i \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)-b^3 f g j \log ^3\left (c \left (d (e+f x)^p\right )^q\right ) \log (g+h x)+3 a^2 b f h i p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-3 a^2 b f g j p q \log (e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-3 a b^2 f h i p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+3 a b^2 f g j p^2 q^2 \log ^2(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+b^3 f h i p^3 q^3 \log ^3(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )-b^3 f g j p^3 q^3 \log ^3(e+f x) \log \left (\frac {f (g+h x)}{f g-e h}\right )+6 a b^2 f h i p q \log (e+f 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 )-6 a b^2 f g j p q \log (e+f 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 )-3 b^3 f h i p^2 q^2 \log ^2(e+f 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 )+3 b^3 f g j p^2 q^2 \log ^2(e+f 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 )+3 b^3 f h i p q \log (e+f 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 )-3 b^3 f g j p q \log (e+f 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 )+3 b f (h i-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 )-6 b^2 f (h i-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 )+6 b^3 f h i p^3 q^3 \text {Li}_4\left (\frac {h (e+f x)}{-f g+e h}\right )-6 b^3 f g j p^3 q^3 \text {Li}_4\left (\frac {h (e+f x)}{-f g+e h}\right )}{f h^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\left (j x +i \right ) \left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{3}}{h x +g}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3} \left (i + j x\right )}{g + h x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (i+j\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3}{g+h\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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