Optimal. Leaf size=1328 \[ -\frac {e^{4/3} \log ^2\left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{4 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) p^2}{4 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} \log ^2\left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{4 d^{4/3}}-\frac {3 \sqrt {3} e^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}-\frac {3 e^{4/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{2 d^{4/3}}-\frac {e^{4/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (-\frac {\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) p^2}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{2 d^{4/3}}-\frac {e^{4/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}+\frac {3 e^{4/3} \log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right ) p^2}{4 d^{4/3}}-\frac {e^{4/3} \text {Li}_2\left (\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} \text {Li}_2\left (-\frac {(-1)^{2/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}-\frac {e^{4/3} \text {Li}_2\left (\frac {2 \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} \text {Li}_2\left (-\frac {\sqrt [3]{-1} \left (\sqrt [3]{e} x+(-1)^{2/3} \sqrt [3]{d}\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} \text {Li}_2\left (\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{2 d^{4/3}}+\frac {e^{4/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} \log \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{2 d^{4/3}}-\frac {3 e \log \left (c \left (e x^3+d\right )^p\right ) p}{2 d x}-\frac {\log ^2\left (c \left (e x^3+d\right )^p\right )}{4 x^4} \]
[Out]
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Rubi [A] time = 1.72, antiderivative size = 1334, normalized size of antiderivative = 1.00, number of steps used = 48, number of rules used = 18, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2457, 2476, 2455, 292, 31, 634, 617, 204, 628, 2462, 260, 2416, 2390, 2301, 2394, 2393, 2391, 12} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 31
Rule 204
Rule 260
Rule 292
Rule 617
Rule 628
Rule 634
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2455
Rule 2457
Rule 2462
Rule 2476
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{x^5} \, dx &=-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {1}{2} (3 e p) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{x^2 \left (d+e x^3\right )} \, dx\\ &=-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {1}{2} (3 e p) \int \left (\frac {\log \left (c \left (d+e x^3\right )^p\right )}{d x^2}-\frac {e x \log \left (c \left (d+e x^3\right )^p\right )}{d \left (d+e x^3\right )}\right ) \, dx\\ &=-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {(3 e p) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{x^2} \, dx}{2 d}-\frac {\left (3 e^2 p\right ) \int \frac {x \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx}{2 d}\\ &=-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}-\frac {\left (3 e^2 p\right ) \int \left (-\frac {\log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}-\frac {(-1)^{2/3} \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (d+e x^3\right )^p\right )}{3 \sqrt [3]{d} \sqrt [3]{e} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}\right ) \, dx}{2 d}+\frac {\left (9 e^2 p^2\right ) \int \frac {x}{d+e x^3} \, dx}{2 d}\\ &=-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {\left (e^{5/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left (\sqrt [3]{-1} e^{5/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{2 d^{4/3}}+\frac {\left ((-1)^{2/3} e^{5/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left (3 e^{5/3} p^2\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}+\frac {\left (3 e^{5/3} p^2\right ) \int \frac {\sqrt [3]{d}+\sqrt [3]{e} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 d^{4/3}}\\ &=-\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {\left (3 e^{4/3} p^2\right ) \int \frac {-\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{4 d^{4/3}}+\frac {\left (9 e^{5/3} p^2\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{4 d}-\frac {\left (3 e^{7/3} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d+e x^3} \, dx}{2 d^{4/3}}+\frac {\left (3 \sqrt [3]{-1} e^{7/3} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{d+e x^3} \, dx}{2 d^{4/3}}-\frac {\left (3 (-1)^{2/3} e^{7/3} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{d+e x^3} \, dx}{2 d^{4/3}}\\ &=-\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {3 e^{4/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {\left (9 e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {\left (3 e^{7/3} p^2\right ) \int \left (\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx}{2 d^{4/3}}+\frac {\left (3 \sqrt [3]{-1} e^{7/3} p^2\right ) \int \left (\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx}{2 d^{4/3}}-\frac {\left (3 (-1)^{2/3} e^{7/3} p^2\right ) \int \left (\frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}+\frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{3 e^{2/3} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}\right ) \, dx}{2 d^{4/3}}\\ &=-\frac {3 \sqrt {3} e^{4/3} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {3 e^{4/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}-\frac {\left (e^{5/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left (e^{5/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left (e^{5/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}+\frac {\left (\sqrt [3]{-1} e^{5/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}+\frac {\left (\sqrt [3]{-1} e^{5/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}+\frac {\left (\sqrt [3]{-1} e^{5/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left ((-1)^{2/3} e^{5/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left ((-1)^{2/3} e^{5/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left ((-1)^{2/3} e^{5/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}\\ &=-\frac {3 \sqrt {3} e^{4/3} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {3 e^{4/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}-\frac {\left (e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {\left (e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1} \log (x)}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {\left (e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {(-1)^{2/3} \log (x)}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {\left (e^{5/3} p^2\right ) \int \frac {\log \left (\frac {\sqrt [3]{e} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}+\frac {\left (e^{5/3} p^2\right ) \int \frac {\log \left (\frac {\sqrt [3]{e} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left (\sqrt [3]{-1} e^{5/3} p^2\right ) \int \frac {\log \left (\frac {(-1)^{2/3} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{2 d^{4/3}}-\frac {\left (\sqrt [3]{-1} e^{5/3} p^2\right ) \int \frac {\log \left (\frac {(-1)^{2/3} \sqrt [3]{e} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x} \, dx}{2 d^{4/3}}+\frac {\left ((-1)^{2/3} e^{5/3} p^2\right ) \int \frac {\log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{e} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{2 d^{4/3}}+\frac {\left ((-1)^{2/3} e^{5/3} p^2\right ) \int \frac {\log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{e} \left (-\sqrt [3]{-1} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x} \, dx}{2 d^{4/3}}\\ &=-\frac {3 \sqrt {3} e^{4/3} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{4 d^{4/3}}-\frac {e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {3 e^{4/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}+\frac {\left (e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {\left (e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {\left (\sqrt [3]{-1} e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {\left (\sqrt [3]{-1} e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {\left (\sqrt [3]{-1} e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {\left ((-1)^{2/3} e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {\left ((-1)^{2/3} e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{e} x}{-\sqrt [3]{d} \sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {\left ((-1)^{2/3} e^{5/3} p^2\right ) \int \frac {\log \left (\frac {\sqrt [3]{e} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\sqrt [3]{d} \sqrt [3]{e}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 d^{4/3}}\\ &=-\frac {3 \sqrt {3} e^{4/3} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{4 d^{4/3}}-\frac {e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{4 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{4 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {3 e^{4/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}-\frac {e^{4/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \text {Li}_2\left (\frac {2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \text {Li}_2\left (-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {\left ((-1)^{2/3} e^{4/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {(-1)^{2/3} \sqrt [3]{e} x}{\sqrt [3]{d} \sqrt [3]{e}+\sqrt [3]{-1} \sqrt [3]{d} \sqrt [3]{e}}\right )}{x} \, dx,x,(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}\\ &=-\frac {3 \sqrt {3} e^{4/3} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {3 e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{4 d^{4/3}}-\frac {e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{4 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{4 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p^2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) \log \left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {3 e^{4/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 d^{4/3}}-\frac {3 e p \log \left (c \left (d+e x^3\right )^p\right )}{2 d x}+\frac {e^{4/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\sqrt [3]{-1} e^{4/3} p \log \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p \log \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{2 d^{4/3}}-\frac {\log ^2\left (c \left (d+e x^3\right )^p\right )}{4 x^4}-\frac {e^{4/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {e^{4/3} p^2 \text {Li}_2\left (\frac {2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (3-i \sqrt {3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \text {Li}_2\left (-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {\sqrt [3]{-1} e^{4/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}+\frac {(-1)^{2/3} e^{4/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}-\frac {(-1)^{2/3} e^{4/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )}{2 d^{4/3}}\\ \end {align*}
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Mathematica [C] time = 1.64, size = 847, normalized size = 0.64 \[ \frac {\frac {e p x^3 \left (9 e p \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {e x^3}{d}\right ) x^3+2 d^{2/3} \sqrt [3]{e} \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) x-2 \sqrt [3]{-1} d^{2/3} \sqrt [3]{e} \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) x+2 (-1)^{2/3} d^{2/3} \sqrt [3]{e} \log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) x+\sqrt [3]{-1} d^{2/3} \sqrt [3]{e} p \left (\log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \left (2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right )+2 \log \left (\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right )+2 \text {Li}_2\left (\frac {\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right ) x-(-1)^{2/3} d^{2/3} \sqrt [3]{e} p \left (\log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \left (2 \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{e} x+\sqrt [3]{d}\right )}{\left (-1+(-1)^{2/3}\right ) \sqrt [3]{d}}\right )+2 \log \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right )\right )+2 \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}}{\left (1-(-1)^{2/3}\right ) \sqrt [3]{d}}\right )\right ) x-d^{2/3} \sqrt [3]{e} p \left (\log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) \left (\log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right )+2 \left (\log \left (\frac {\sqrt [3]{-1} \sqrt [3]{d}-\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+\log \left (\frac {-\frac {2 i \sqrt [3]{e} x}{\sqrt [3]{d}}+\sqrt {3}+i}{3 i+\sqrt {3}}\right )\right )\right )+2 \text {Li}_2\left (\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )+2 \text {Li}_2\left (\frac {2 i \left (\frac {\sqrt [3]{e} x}{\sqrt [3]{d}}+1\right )}{3 i+\sqrt {3}}\right )\right ) x-6 d \log \left (c \left (e x^3+d\right )^p\right )\right )}{d^2}-\log ^2\left (c \left (e x^3+d\right )^p\right )}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{5}}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.05, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (e \,x^{3}+d \right )^{p}\right )^{2}}{x^{5}}\, dx \]
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 \[ -\frac {\log \left ({\left (e x^{3} + d\right )}^{p}\right )^{2}}{4 \, x^{4}} + \int \frac {2 \, e x^{3} \log \relax (c)^{2} + 2 \, d \log \relax (c)^{2} + {\left ({\left (3 \, e p + 4 \, e \log \relax (c)\right )} x^{3} + 4 \, d \log \relax (c)\right )} \log \left ({\left (e x^{3} + d\right )}^{p}\right )}{2 \, {\left (e x^{8} + d x^{5}\right )}}\,{d x} \]
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 \[ \int \frac {{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2}{x^5} \,d x \]
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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