Optimal. Leaf size=666 \[ \frac {\sqrt {3} \sqrt [3]{b} d p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^2}-\frac {\sqrt {3} b^{2/3} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e}-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}-\frac {\sqrt [3]{b} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^2}-\frac {b^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {3 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}+\frac {\sqrt [3]{b} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^2}+\frac {b^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^3}+\frac {3 d^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^3} \]
[Out]
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Rubi [A]
time = 0.49, antiderivative size = 666, normalized size of antiderivative = 1.00, number of steps
used = 34, number of rules used = 18, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {2516, 2498,
269, 206, 31, 648, 631, 210, 642, 2505, 298, 2512, 266, 2463, 2441, 2352, 2440, 2438}
\begin {gather*} -\frac {d^2 p \text {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \text {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \text {PolyLog}\left (2,\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^3}+\frac {3 d^2 p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^3}-\frac {\sqrt {3} b^{2/3} p \text {ArcTan}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e}+\frac {\sqrt [3]{b} d p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{2 \sqrt [3]{a} e^2}+\frac {b^{2/3} p \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{4 a^{2/3} e}-\frac {b^{2/3} p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{2 a^{2/3} e}+\frac {\sqrt {3} \sqrt [3]{b} d p \text {ArcTan}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^2}+\frac {d^2 \log (d+e x) \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}-\frac {d^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{a} x+(-1)^{2/3} \sqrt [3]{b}\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^3}-\frac {\sqrt [3]{b} d p \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a} e^2}+\frac {3 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 206
Rule 210
Rule 266
Rule 269
Rule 298
Rule 631
Rule 642
Rule 648
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2498
Rule 2505
Rule 2512
Rule 2516
Rubi steps
\begin {align*} \int \frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx &=\int \left (-\frac {d \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {d \int \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \, dx}{e^2}+\frac {d^2 \int \frac {\log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{d+e x} \, dx}{e^2}+\frac {\int x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \, dx}{e}\\ &=-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {\left (3 b d^2 p\right ) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx}{e^3}-\frac {(3 b d p) \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^3} \, dx}{e^2}+\frac {(3 b p) \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^2} \, dx}{2 e}\\ &=-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {\left (3 b d^2 p\right ) \int \left (\frac {\log (d+e x)}{b x}-\frac {a x^2 \log (d+e x)}{b \left (b+a x^3\right )}\right ) \, dx}{e^3}-\frac {(3 b d p) \int \frac {1}{b+a x^3} \, dx}{e^2}+\frac {(3 b p) \int \frac {x}{b+a x^3} \, dx}{2 e}\\ &=-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {\left (3 d^2 p\right ) \int \frac {\log (d+e x)}{x} \, dx}{e^3}-\frac {\left (3 a d^2 p\right ) \int \frac {x^2 \log (d+e x)}{b+a x^3} \, dx}{e^3}-\frac {\left (\sqrt [3]{b} d p\right ) \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^2}-\frac {\left (\sqrt [3]{b} d p\right ) \int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{e^2}-\frac {\left (b^{2/3} p\right ) \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{2 \sqrt [3]{a} e}+\frac {\left (b^{2/3} p\right ) \int \frac {\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 \sqrt [3]{a} e}\\ &=-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}-\frac {\sqrt [3]{b} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^2}-\frac {b^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {3 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {\left (3 a d^2 p\right ) \int \left (\frac {\log (d+e x)}{3 a^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}+\frac {\log (d+e x)}{3 a^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x\right )}+\frac {\log (d+e x)}{3 a^{2/3} \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}\right ) \, dx}{e^3}+\frac {\left (\sqrt [3]{b} d p\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 \sqrt [3]{a} e^2}-\frac {\left (3 b^{2/3} d p\right ) \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 e^2}-\frac {\left (3 d^2 p\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{e^2}+\frac {\left (b^{2/3} p\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{4 a^{2/3} e}+\frac {(3 b p) \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{4 \sqrt [3]{a} e}\\ &=-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}-\frac {\sqrt [3]{b} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^2}-\frac {b^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {3 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}+\frac {\sqrt [3]{b} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^2}+\frac {b^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e}+\frac {3 d^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^3}-\frac {\left (\sqrt [3]{a} d^2 p\right ) \int \frac {\log (d+e x)}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^3}-\frac {\left (\sqrt [3]{a} d^2 p\right ) \int \frac {\log (d+e x)}{-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^3}-\frac {\left (\sqrt [3]{a} d^2 p\right ) \int \frac {\log (d+e x)}{(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{e^3}-\frac {\left (3 \sqrt [3]{b} d p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a} e^2}+\frac {\left (3 b^{2/3} p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{2 a^{2/3} e}\\ &=\frac {\sqrt {3} \sqrt [3]{b} d p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^2}-\frac {\sqrt {3} b^{2/3} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e}-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}-\frac {\sqrt [3]{b} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^2}-\frac {b^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {3 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}+\frac {\sqrt [3]{b} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^2}+\frac {b^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e}+\frac {3 d^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (d^2 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+\sqrt [3]{b} e}\right )}{d+e x} \, dx}{e^2}+\frac {\left (d^2 p\right ) \int \frac {\log \left (\frac {e \left (-\sqrt [3]{-1} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d-\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{d+e x} \, dx}{e^2}+\frac {\left (d^2 p\right ) \int \frac {\log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right )}{d+e x} \, dx}{e^2}\\ &=\frac {\sqrt {3} \sqrt [3]{b} d p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^2}-\frac {\sqrt {3} b^{2/3} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e}-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}-\frac {\sqrt [3]{b} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^2}-\frac {b^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {3 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}+\frac {\sqrt [3]{b} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^2}+\frac {b^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e}+\frac {3 d^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (d^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{a} x}{-\sqrt [3]{a} d+\sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{e^3}+\frac {\left (d^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{a} x}{-\sqrt [3]{a} d-\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{e^3}+\frac {\left (d^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{a} x}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right )}{x} \, dx,x,d+e x\right )}{e^3}\\ &=\frac {\sqrt {3} \sqrt [3]{b} d p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt [3]{a} e^2}-\frac {\sqrt {3} b^{2/3} p \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{2 a^{2/3} e}-\frac {d x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )}{2 e}-\frac {\sqrt [3]{b} d p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} e^2}-\frac {b^{2/3} p \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{2 a^{2/3} e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)}{e^3}+\frac {3 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (\frac {\sqrt [3]{-1} e \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)}{e^3}+\frac {\sqrt [3]{b} d p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{2 \sqrt [3]{a} e^2}+\frac {b^{2/3} p \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{4 a^{2/3} e}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )}{e^3}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )}{e^3}+\frac {3 d^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.17, size = 443, normalized size = 0.67 \begin {gather*} \frac {-3 b e^2 p x \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {b}{a x^3}\right )+3 b d e p \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b}{a x^3}\right )+a x^2 \left (-2 d e x \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )+e^2 x^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right )+2 d^2 \log \left (c \left (a+\frac {b}{x^3}\right )^p\right ) \log (d+e x)+6 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)-2 d^2 p \log \left (\frac {e \left (\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right ) \log (d+e x)-2 d^2 p \log \left (\frac {e \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+\sqrt [3]{b} e}\right ) \log (d+e x)-2 d^2 p \log \left (\frac {e \left ((-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} x\right )}{-\sqrt [3]{a} d+(-1)^{2/3} \sqrt [3]{b} e}\right ) \log (d+e x)-2 d^2 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-\sqrt [3]{b} e}\right )-2 d^2 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} e}\right )-2 d^2 p \text {Li}_2\left (\frac {\sqrt [3]{a} (d+e x)}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} e}\right )+6 d^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )\right )}{2 a e^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \ln \left (c \left (a +\frac {b}{x^{3}}\right )^{p}\right )}{e x +d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,\ln \left (c\,{\left (a+\frac {b}{x^3}\right )}^p\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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