Optimal. Leaf size=1165 \[ -\frac {p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \log \left (-\frac {\sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (-\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {(-1)^{2/3} \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\sqrt [3]{-1} \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.15, antiderivative size = 1165, normalized size of antiderivative = 1.00, number of steps
used = 29, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2521, 2512,
266, 2463, 2441, 2440, 2438} \begin {gather*} -\frac {p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right ) \log \left (-\sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \log \left (-\frac {\sqrt [3]{g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\log \left (c \left (e x^2+d\right )^p\right ) \log \left (-\sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt [3]{-1} \sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{(-1)^{2/3} \sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (-\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-(-1)^{2/3} \sqrt [3]{g} x-\sqrt [3]{f}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{-1} \sqrt [3]{g} x-\sqrt [3]{f}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\sqrt [3]{-1} \log \left (-(-1)^{2/3} \sqrt [3]{g} x-\sqrt [3]{f}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \text {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt [3]{g} x+\sqrt [3]{f}\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \text {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt [3]{g} x+\sqrt [3]{f}\right )}{\sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \text {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \text {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt [3]{-1} \sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \text {PolyLog}\left (2,\frac {\sqrt {e} \left ((-1)^{2/3} \sqrt [3]{g} x+\sqrt [3]{f}\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \text {PolyLog}\left (2,\frac {\sqrt {e} \left ((-1)^{2/3} \sqrt [3]{g} x+\sqrt [3]{f}\right )}{(-1)^{2/3} \sqrt [3]{g} \sqrt {-d}+\sqrt {e} \sqrt [3]{f}}\right )}{3 f^{2/3} \sqrt [3]{g}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2512
Rule 2521
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx &=\int \left (-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{-\sqrt [3]{f}-\sqrt [3]{g} x} \, dx}{3 f^{2/3}}-\frac {\int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x} \, dx}{3 f^{2/3}}-\frac {\int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x} \, dx}{3 f^{2/3}}\\ &=\frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {(-1)^{2/3} \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\sqrt [3]{-1} \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(2 e p) \int \frac {x \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{d+e x^2} \, dx}{3 f^{2/3} \sqrt [3]{g}}+\frac {\left (2 \sqrt [3]{-1} e p\right ) \int \frac {x \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{d+e x^2} \, dx}{3 f^{2/3} \sqrt [3]{g}}-\frac {\left (2 (-1)^{2/3} e p\right ) \int \frac {x \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{d+e x^2} \, dx}{3 f^{2/3} \sqrt [3]{g}}\\ &=\frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {(-1)^{2/3} \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\sqrt [3]{-1} \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(2 e p) \int \left (-\frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{3 f^{2/3} \sqrt [3]{g}}+\frac {\left (2 \sqrt [3]{-1} e p\right ) \int \left (-\frac {\log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{3 f^{2/3} \sqrt [3]{g}}-\frac {\left (2 (-1)^{2/3} e p\right ) \int \left (-\frac {\log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{3 f^{2/3} \sqrt [3]{g}}\\ &=\frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {(-1)^{2/3} \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\sqrt [3]{-1} \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\left (\sqrt {e} p\right ) \int \frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{3 f^{2/3} \sqrt [3]{g}}-\frac {\left (\sqrt {e} p\right ) \int \frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{3 f^{2/3} \sqrt [3]{g}}-\frac {\left (\sqrt [3]{-1} \sqrt {e} p\right ) \int \frac {\log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{3 f^{2/3} \sqrt [3]{g}}+\frac {\left (\sqrt [3]{-1} \sqrt {e} p\right ) \int \frac {\log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{3 f^{2/3} \sqrt [3]{g}}+\frac {\left ((-1)^{2/3} \sqrt {e} p\right ) \int \frac {\log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{3 f^{2/3} \sqrt [3]{g}}-\frac {\left ((-1)^{2/3} \sqrt {e} p\right ) \int \frac {\log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{3 f^{2/3} \sqrt [3]{g}}\\ &=-\frac {p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \log \left (-\frac {\sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (-\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {(-1)^{2/3} \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\sqrt [3]{-1} \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \int \frac {\log \left (-\frac {\sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )}{-\sqrt [3]{f}-\sqrt [3]{g} x} \, dx}{3 f^{2/3}}-\frac {p \int \frac {\log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x} \, dx}{3 f^{2/3}}-\frac {p \int \frac {\log \left (-\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x} \, dx}{3 f^{2/3}}-\frac {p \int \frac {\log \left (-\frac {\sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )}{-\sqrt [3]{f}-\sqrt [3]{g} x} \, dx}{3 f^{2/3}}-\frac {p \int \frac {\log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x} \, dx}{3 f^{2/3}}-\frac {p \int \frac {\log \left (-\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x} \, dx}{3 f^{2/3}}\\ &=-\frac {p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \log \left (-\frac {\sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (-\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {(-1)^{2/3} \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\sqrt [3]{-1} \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {p \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{-\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )}{x} \, dx,x,-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )}{x} \, dx,x,-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\left (\sqrt [3]{-1} p\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{-\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{x} \, dx,x,-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\left (\sqrt [3]{-1} p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{x} \, dx,x,-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\left ((-1)^{2/3} p\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{-\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{x} \, dx,x,-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\left ((-1)^{2/3} p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{x} \, dx,x,-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}\\ &=-\frac {p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \log \left (-\frac {\sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (-\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \log \left (-\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {(-1)^{2/3} \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {\sqrt [3]{-1} \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}-\frac {(-1)^{2/3} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}+\frac {\sqrt [3]{-1} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.59, size = 990, normalized size = 0.85 \begin {gather*} \frac {-p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )-p \log \left (\frac {\sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )-(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )-(-1)^{2/3} p \log \left (\frac {\sqrt [3]{-1} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )+\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )+\sqrt [3]{-1} p \log \left (\frac {(-1)^{2/3} \sqrt [3]{g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right ) \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )+\log \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )+(-1)^{2/3} \log \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )-\sqrt [3]{-1} \log \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )-p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )-p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt {-d} \sqrt [3]{g}}\right )-(-1)^{2/3} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )-(-1)^{2/3} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+\sqrt [3]{-1} \sqrt {-d} \sqrt [3]{g}}\right )+\sqrt [3]{-1} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )+\sqrt [3]{-1} p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}+(-1)^{2/3} \sqrt {-d} \sqrt [3]{g}}\right )}{3 f^{2/3} \sqrt [3]{g}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.07, size = 1180, normalized size = 1.01
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1180\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^3+f} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________