Optimal. Leaf size=692 \[ \frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}-\frac{\sqrt [3]{a} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^3}-\frac{a^{2/3} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e^2}+\frac{a^{2/3} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e^2}+\frac{\sqrt{3} a^{2/3} d p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 b^{2/3} e^2}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac{d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^4}+\frac{\sqrt [3]{a} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^3}-\frac{\sqrt{3} \sqrt [3]{a} d^2 p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{b} e^3}-\frac{3 d^2 p x}{e^3}+\frac{3 d p x^2}{4 e^2}-\frac{p x^3}{3 e} \]
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Rubi [A] time = 0.891271, antiderivative size = 692, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 20, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.87, Rules used = {2466, 2448, 321, 200, 31, 634, 617, 204, 628, 2455, 292, 2454, 2389, 2295, 2462, 260, 2416, 2394, 2393, 2391} \[ \frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}-\frac{\sqrt [3]{a} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^3}-\frac{a^{2/3} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e^2}+\frac{a^{2/3} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e^2}+\frac{\sqrt{3} a^{2/3} d p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 b^{2/3} e^2}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac{d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^4}+\frac{\sqrt [3]{a} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^3}-\frac{\sqrt{3} \sqrt [3]{a} d^2 p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{b} e^3}-\frac{3 d^2 p x}{e^3}+\frac{3 d p x^2}{4 e^2}-\frac{p x^3}{3 e} \]
Antiderivative was successfully verified.
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Rule 2466
Rule 2448
Rule 321
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 2455
Rule 292
Rule 2454
Rule 2389
Rule 2295
Rule 2462
Rule 260
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3 \log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx &=\int \left (\frac{d^2 \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x \log \left (c \left (a+b x^3\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+b x^3\right )^p\right )}{e}-\frac{d^3 \log \left (c \left (a+b x^3\right )^p\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac{d^2 \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx}{e^3}-\frac{d^3 \int \frac{\log \left (c \left (a+b x^3\right )^p\right )}{d+e x} \, dx}{e^3}-\frac{d \int x \log \left (c \left (a+b x^3\right )^p\right ) \, dx}{e^2}+\frac{\int x^2 \log \left (c \left (a+b x^3\right )^p\right ) \, dx}{e}\\ &=\frac{d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac{\operatorname{Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,x^3\right )}{3 e}+\frac{\left (3 b d^3 p\right ) \int \frac{x^2 \log (d+e x)}{a+b x^3} \, dx}{e^4}-\frac{\left (3 b d^2 p\right ) \int \frac{x^3}{a+b x^3} \, dx}{e^3}+\frac{(3 b d p) \int \frac{x^4}{a+b x^3} \, dx}{2 e^2}\\ &=-\frac{3 d^2 p x}{e^3}+\frac{3 d p x^2}{4 e^2}+\frac{d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac{\operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^3\right )}{3 b e}+\frac{\left (3 b d^3 p\right ) \int \left (\frac{\log (d+e x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\log (d+e x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\log (d+e x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{e^4}+\frac{\left (3 a d^2 p\right ) \int \frac{1}{a+b x^3} \, dx}{e^3}-\frac{(3 a d p) \int \frac{x}{a+b x^3} \, dx}{2 e^2}\\ &=-\frac{3 d^2 p x}{e^3}+\frac{3 d p x^2}{4 e^2}-\frac{p x^3}{3 e}+\frac{d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac{\left (\sqrt [3]{b} d^3 p\right ) \int \frac{\log (d+e x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{e^4}+\frac{\left (\sqrt [3]{b} d^3 p\right ) \int \frac{\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{e^4}+\frac{\left (\sqrt [3]{b} d^3 p\right ) \int \frac{\log (d+e x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{e^4}+\frac{\left (\sqrt [3]{a} d^2 p\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{e^3}+\frac{\left (\sqrt [3]{a} d^2 p\right ) \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{e^3}+\frac{\left (a^{2/3} d p\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{2 \sqrt [3]{b} e^2}-\frac{\left (a^{2/3} d p\right ) \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{b} e^2}\\ &=-\frac{3 d^2 p x}{e^3}+\frac{3 d p x^2}{4 e^2}-\frac{p x^3}{3 e}+\frac{\sqrt [3]{a} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^3}+\frac{a^{2/3} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e^2}+\frac{d^3 p \log \left (-\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}+\frac{d^3 p \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}+\frac{d^3 p \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}+\frac{d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac{\left (3 a^{2/3} d^2 p\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 e^3}-\frac{\left (\sqrt [3]{a} d^2 p\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{b} e^3}-\frac{\left (d^3 p\right ) \int \frac{\log \left (\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right )}{d+e x} \, dx}{e^3}-\frac{\left (d^3 p\right ) \int \frac{\log \left (\frac{e \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d-\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{d+e x} \, dx}{e^3}-\frac{\left (d^3 p\right ) \int \frac{\log \left (\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right )}{d+e x} \, dx}{e^3}-\frac{\left (a^{2/3} d p\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{4 b^{2/3} e^2}-\frac{(3 a d p) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{4 \sqrt [3]{b} e^2}\\ &=-\frac{3 d^2 p x}{e^3}+\frac{3 d p x^2}{4 e^2}-\frac{p x^3}{3 e}+\frac{\sqrt [3]{a} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^3}+\frac{a^{2/3} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e^2}+\frac{d^3 p \log \left (-\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}+\frac{d^3 p \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}+\frac{d^3 p \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}-\frac{\sqrt [3]{a} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^3}-\frac{a^{2/3} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e^2}+\frac{d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}-\frac{\left (d^3 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{b} x}{-\sqrt [3]{b} d+\sqrt [3]{a} e}\right )}{x} \, dx,x,d+e x\right )}{e^4}-\frac{\left (d^3 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{b} x}{-\sqrt [3]{b} d-\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{x} \, dx,x,d+e x\right )}{e^4}-\frac{\left (d^3 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{b} x}{-\sqrt [3]{b} d+(-1)^{2/3} \sqrt [3]{a} e}\right )}{x} \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 \sqrt [3]{a} d^2 p\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b} e^3}-\frac{\left (3 a^{2/3} d p\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{2 b^{2/3} e^2}\\ &=-\frac{3 d^2 p x}{e^3}+\frac{3 d p x^2}{4 e^2}-\frac{p x^3}{3 e}-\frac{\sqrt{3} \sqrt [3]{a} d^2 p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{b} e^3}+\frac{\sqrt{3} a^{2/3} d p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 b^{2/3} e^2}+\frac{\sqrt [3]{a} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^3}+\frac{a^{2/3} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e^2}+\frac{d^3 p \log \left (-\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}+\frac{d^3 p \log \left (-\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}+\frac{d^3 p \log \left (\frac{\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right ) \log (d+e x)}{e^4}-\frac{\sqrt [3]{a} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^3}-\frac{a^{2/3} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e^2}+\frac{d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac{\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac{d^3 p \text{Li}_2\left (\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \text{Li}_2\left (\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e^4}+\frac{d^3 p \text{Li}_2\left (\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}\\ \end{align*}
Mathematica [C] time = 0.601297, size = 497, normalized size = 0.72 \[ -\frac{-12 d^3 p \left (\text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )+\text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )+\text{PolyLog}\left (2,\frac{\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )+\log (d+e x) \log \left (\frac{e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )+\log (d+e x) \log \left (\frac{e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} e-\sqrt [3]{b} d}\right )+\log (d+e x) \log \left (\frac{e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{a} e-\sqrt [3]{b} d}\right )\right )+\frac{6 d^2 e p \left (\sqrt [3]{a} \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+6 \sqrt [3]{b} x\right )}{\sqrt [3]{b}}+12 d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )-12 d^2 e x \log \left (c \left (a+b x^3\right )^p\right )+6 d e^2 x^2 \log \left (c \left (a+b x^3\right )^p\right )+\frac{4 e^3 \left (b p x^3-\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )\right )}{b}+9 d e^2 p x^2 \left (\, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a}\right )-1\right )}{12 e^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.665, size = 912, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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{x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d}, 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{x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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