Integrand size = 22, antiderivative size = 1165 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=-\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 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\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 \operatorname {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 \operatorname {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 {\sqrt [3]{-1} p \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\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]
Time = 1.15 (sec) , antiderivative size = 1165, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2521, 2512, 266, 2463, 2441, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=-\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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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}} \]
[In]
[Out]
Rule 266
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2512
Rule 2521
Rubi steps \begin{align*} \text {integral}& = \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}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 990, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\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 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {e} \sqrt [3]{f}-\sqrt {-d} \sqrt [3]{g}}\right )-p \operatorname {PolyLog}\left (2,\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 \operatorname {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 )-(-1)^{2/3} p \operatorname {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 )+\sqrt [3]{-1} p \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\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}} \]
[In]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.54 (sec) , antiderivative size = 577, normalized size of antiderivative = 0.50
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\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{3} + f} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{3} + f} \,d x } \]
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\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{3} + f} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^3+f} \,d x \]
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