Optimal. Leaf size=1294 \[ \frac {9 x^2 p^2}{4}+\frac {d^{2/3} \log ^2\left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{2 e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right ) p^2}{2 e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} \log ^2\left ((-1)^{2/3} \sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{2 e^{2/3}}+\frac {3 \sqrt {3} d^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) p^2}{2 e^{2/3}}+\frac {3 d^{2/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) p^2}{2 e^{2/3}}+\frac {d^{2/3} \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}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} \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}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} \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}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} \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}{e^{2/3}}+\frac {d^{2/3} \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}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} \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}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} \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}{e^{2/3}}-\frac {3 d^{2/3} \log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right ) p^2}{4 e^{2/3}}+\frac {d^{2/3} \text {Li}_2\left (\frac {\sqrt [3]{e} x+\sqrt [3]{d}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right ) p^2}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} \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}{e^{2/3}}+\frac {d^{2/3} \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}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} \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}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} \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}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} \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}{e^{2/3}}-\frac {3}{2} x^2 \log \left (c \left (e x^3+d\right )^p\right ) p-\frac {d^{2/3} \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right ) p}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} \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}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} \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}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (e x^3+d\right )^p\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.92, antiderivative size = 1300, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 19, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.187, Rules used = {2457, 2476, 2455, 321, 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 321
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 x \log ^2\left (c \left (d+e x^3\right )^p\right ) \, dx &=\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-(3 e p) \int \frac {x^4 \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx\\ &=\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-(3 e p) \int \left (\frac {x \log \left (c \left (d+e x^3\right )^p\right )}{e}-\frac {d x \log \left (c \left (d+e x^3\right )^p\right )}{e \left (d+e x^3\right )}\right ) \, dx\\ &=\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-(3 p) \int x \log \left (c \left (d+e x^3\right )^p\right ) \, dx+(3 d p) \int \frac {x \log \left (c \left (d+e x^3\right )^p\right )}{d+e x^3} \, dx\\ &=-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+(3 d p) \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+\frac {1}{2} \left (9 e p^2\right ) \int \frac {x^4}{d+e x^3} \, dx\\ &=\frac {9 p^2 x^2}{4}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {\left (d^{2/3} p\right ) \int \frac {\log \left (c \left (d+e x^3\right )^p\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac {\left (\sqrt [3]{-1} d^{2/3} 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}{\sqrt [3]{e}}-\frac {\left ((-1)^{2/3} d^{2/3} 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}{\sqrt [3]{e}}-\frac {1}{2} \left (9 d p^2\right ) \int \frac {x}{d+e x^3} \, dx\\ &=\frac {9 p^2 x^2}{4}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac {\left (3 d^{2/3} p^2\right ) \int \frac {1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{2 \sqrt [3]{e}}-\frac {\left (3 d^{2/3} 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 \sqrt [3]{e}}+\left (3 d^{2/3} \sqrt [3]{e} p^2\right ) \int \frac {x^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d+e x^3} \, dx-\left (3 \sqrt [3]{-1} d^{2/3} \sqrt [3]{e} 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+\left (3 (-1)^{2/3} d^{2/3} \sqrt [3]{e} 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\\ &=\frac {9 p^2 x^2}{4}+\frac {3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {\left (3 d^{2/3} 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 e^{2/3}}-\frac {\left (9 d p^2\right ) \int \frac {1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{4 \sqrt [3]{e}}+\left (3 d^{2/3} \sqrt [3]{e} 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-\left (3 \sqrt [3]{-1} d^{2/3} \sqrt [3]{e} 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+\left (3 (-1)^{2/3} d^{2/3} \sqrt [3]{e} 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\\ &=\frac {9 p^2 x^2}{4}+\frac {3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}-\frac {3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {\left (9 d^{2/3} 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 e^{2/3}}+\frac {\left (d^{2/3} p^2\right ) \int \frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{\sqrt [3]{e}}+\frac {\left (d^{2/3} 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}{\sqrt [3]{e}}+\frac {\left (d^{2/3} 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}{\sqrt [3]{e}}-\frac {\left (\sqrt [3]{-1} d^{2/3} 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}{\sqrt [3]{e}}-\frac {\left (\sqrt [3]{-1} d^{2/3} 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}{\sqrt [3]{e}}-\frac {\left (\sqrt [3]{-1} d^{2/3} 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}{\sqrt [3]{e}}+\frac {\left ((-1)^{2/3} d^{2/3} 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}{\sqrt [3]{e}}+\frac {\left ((-1)^{2/3} d^{2/3} 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}{\sqrt [3]{e}}+\frac {\left ((-1)^{2/3} d^{2/3} 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}{\sqrt [3]{e}}\\ &=\frac {9 p^2 x^2}{4}+\frac {3 \sqrt {3} d^{2/3} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac {3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac {\left (d^{2/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt [3]{d}+\sqrt [3]{e} x\right )}{e^{2/3}}-\frac {\left (d^{2/3} 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 )}{e^{2/3}}+\frac {\left (d^{2/3} 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 )}{e^{2/3}}-\frac {\left (d^{2/3} 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}{\sqrt [3]{e}}-\frac {\left (d^{2/3} 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}{\sqrt [3]{e}}+\frac {\left (\sqrt [3]{-1} d^{2/3} 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}{\sqrt [3]{e}}+\frac {\left (\sqrt [3]{-1} d^{2/3} 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}{\sqrt [3]{e}}-\frac {\left ((-1)^{2/3} d^{2/3} 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}{\sqrt [3]{e}}-\frac {\left ((-1)^{2/3} d^{2/3} 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}{\sqrt [3]{e}}\\ &=\frac {9 p^2 x^2}{4}+\frac {3 \sqrt {3} d^{2/3} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac {3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )-\frac {\left (d^{2/3} 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 )}{e^{2/3}}-\frac {\left (d^{2/3} 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 )}{e^{2/3}}-\frac {\left (\sqrt [3]{-1} d^{2/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{e^{2/3}}+\frac {\left (\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}+\frac {\left (\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}+\frac {\left ((-1)^{2/3} d^{2/3} p^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{e^{2/3}}-\frac {\left ((-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {\left ((-1)^{2/3} d^{2/3} 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}{\sqrt [3]{e}}\\ &=\frac {9 p^2 x^2}{4}+\frac {3 \sqrt {3} d^{2/3} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac {3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac {d^{2/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac {d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {\left ((-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}\\ &=\frac {9 p^2 x^2}{4}+\frac {3 \sqrt {3} d^{2/3} p^2 \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{2 e^{2/3}}+\frac {3 d^{2/3} p^2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{e} x\right )}{2 e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} p^2 \log ^2\left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{e} x\right )}{2 e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {3 d^{2/3} p^2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{4 e^{2/3}}-\frac {3}{2} p x^2 \log \left (c \left (d+e x^3\right )^p\right )-\frac {d^{2/3} p \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \log \left (c \left (d+e x^3\right )^p\right )}{e^{2/3}}+\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {1}{2} x^2 \log ^2\left (c \left (d+e x^3\right )^p\right )+\frac {d^{2/3} p^2 \text {Li}_2\left (\frac {\sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{d}}\right )}{e^{2/3}}+\frac {d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {\sqrt [3]{-1} d^{2/3} 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 )}{e^{2/3}}-\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}+\frac {(-1)^{2/3} d^{2/3} 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 )}{e^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.23, size = 823, normalized size = 0.64 \[ \frac {1}{4} \left (2 x^2 \log ^2\left (c \left (e x^3+d\right )^p\right )+\frac {p \left (-9 e^{2/3} p \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {e x^3}{d}\right )-1\right ) x^2-6 e^{2/3} \log \left (c \left (e x^3+d\right )^p\right ) x^2-4 d^{2/3} \log \left (-\sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right )+4 \sqrt [3]{-1} d^{2/3} \log \left (\sqrt [3]{-1} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right )-4 (-1)^{2/3} d^{2/3} \log \left (-(-1)^{2/3} \sqrt [3]{e} x-\sqrt [3]{d}\right ) \log \left (c \left (e x^3+d\right )^p\right )-2 \sqrt [3]{-1} d^{2/3} 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 )+2 (-1)^{2/3} d^{2/3} 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 )+2 d^{2/3} 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 )\right )}{e^{2/3}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \log \left ({\left (e x^{3} + d\right )}^{p} c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int x \ln \left (c \left (e \,x^{3}+d \right )^{p}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \log \left ({\left (e x^{3} + d\right )}^{p}\right )^{2} + \int \frac {e x^{4} \log \relax (c)^{2} + d x \log \relax (c)^{2} - {\left ({\left (3 \, e p - 2 \, e \log \relax (c)\right )} x^{4} - 2 \, d x \log \relax (c)\right )} \log \left ({\left (e x^{3} + d\right )}^{p}\right )}{e x^{3} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\ln \left (c\,{\left (e\,x^3+d\right )}^p\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \log {\left (c \left (d + e x^{3}\right )^{p} \right )}^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________