Optimal. Leaf size=692 \[ -\frac {\sqrt [3]{a} d^2 p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b} e^3}-\frac {a^{2/3} d p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 b^{2/3} e^2}+\frac {a^{2/3} d p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 b^{2/3} e^2}+\frac {\sqrt {3} a^{2/3} d p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{2 b^{2/3} e^2}-\frac {d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )}{e^4}+\frac {d^2 x \log \left (c \left (a+b x^3\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+b x^3\right )^p\right )}{2 e^2}+\frac {\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )}{3 b e}+\frac {d^3 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )}{e^4}+\frac {\sqrt [3]{a} d^2 p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} e^3}-\frac {\sqrt {3} \sqrt [3]{a} d^2 p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b} e^3}-\frac {3 d^2 p x}{e^3}+\frac {3 d p x^2}{4 e^2}-\frac {p x^3}{3 e} \]
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Rubi [A] time = 0.89, antiderivative size = 692, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 20, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, 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 31
Rule 200
Rule 204
Rule 260
Rule 292
Rule 321
Rule 617
Rule 628
Rule 634
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2448
Rule 2454
Rule 2455
Rule 2462
Rule 2466
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*}
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Mathematica [C] time = 0.65, size = 497, normalized size = 0.72 \[ -\frac {\frac {6 d^2 e p \left (\sqrt [3]{a} \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+6 \sqrt [3]{b} x\right )}{\sqrt [3]{b}}+12 d^3 \log (d+e x) \log \left (c \left (a+b x^3\right )^p\right )-12 d^2 e x \log \left (c \left (a+b x^3\right )^p\right )+6 d e^2 x^2 \log \left (c \left (a+b x^3\right )^p\right )+\frac {4 e^3 \left (b p x^3-\left (a+b x^3\right ) \log \left (c \left (a+b x^3\right )^p\right )\right )}{b}-12 d^3 p \left (\text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )+\text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )+\text {Li}_2\left (\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )+\log (d+e x) \log \left (\frac {e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d}\right )+\log (d+e x) \log \left (\frac {e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} e-\sqrt [3]{b} d}\right )+\log (d+e x) \log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{a} e-\sqrt [3]{b} d}\right )\right )+9 d e^2 p x^2 \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b x^3}{a}\right )-1\right )}{12 e^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d}, x\right ) \]
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 \[ \int \frac {x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.56, size = 912, normalized size = 1.32 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{3} \log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{e x + d}\,{d x} \]
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 \[ \int \frac {x^3\,\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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