\(\int \frac {\log (c (d+e x^2)^p)}{(f+g x^3)^2} \, dx\) [292]

Optimal result

Integrand size = 22, antiderivative size = 1861 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\frac {2 \sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 f^{4/3} \left (e f^{2/3}+d g^{2/3}\right )}+\frac {2 (-1)^{2/3} \sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{4/3} \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right )}+\frac {4 \sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 f^{4/3} \left (2 e f^{2/3}-\left (1+i \sqrt {3}\right ) d g^{2/3}\right )}-\frac {2 e p \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac {2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac {2 \sqrt [3]{-1} e p \log \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\left (1+\sqrt [3]{-1}\right )^4 f \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}+\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {4 \sqrt [3]{-1} e p \log \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{9 f \left (2 e f^{2/3}-\left (1+i \sqrt {3}\right ) d g^{2/3}\right ) \sqrt [3]{g}}-\frac {2 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 )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \sqrt [3]{g}}+\frac {e p \log \left (d+e x^2\right )}{9 f \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac {\sqrt [3]{-1} e p \log \left (d+e x^2\right )}{\left (1+\sqrt [3]{-1}\right )^4 f \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}-\frac {2 \sqrt [3]{-1} e p \log \left (d+e x^2\right )}{9 f \left (2 e f^{2/3}-\left (1+i \sqrt {3}\right ) d g^{2/3}\right ) \sqrt [3]{g}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac {2 \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {2 \log \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \sqrt [3]{g}} \]

[Out]

2*(-1)^(2/3)*p*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)*e^(1/2)/(1+(-1)^(1/3))^4/f^(4/3)/(e*f^(2/3)+(-1)^(2/3)*d*g^(2
/3))+2*(-1)^(1/3)*e*p*ln(f^(1/3)-(-1)^(1/3)*g^(1/3)*x)/(1+(-1)^(1/3))^4/f/(e*f^(2/3)+(-1)^(2/3)*d*g^(2/3))/g^(
1/3)-ln(c*(e*x^2+d)^p)/(1+(-1)^(1/3))^4/f^(4/3)/g^(1/3)/((-1)^(2/3)*f^(1/3)+g^(1/3)*x)+1/9*(-1)^(1/3)*ln(c*(e*
x^2+d)^p)/f^(4/3)/g^(1/3)/(f^(1/3)+(-1)^(2/3)*g^(1/3)*x)-2/9*p*ln(f^(1/3)+g^(1/3)*x)*ln(g^(1/3)*((-d)^(1/2)-x*
e^(1/2))/(g^(1/3)*(-d)^(1/2)+f^(1/3)*e^(1/2)))/f^(5/3)/g^(1/3)-2/9*p*ln(f^(1/3)+g^(1/3)*x)*ln(-g^(1/3)*((-d)^(
1/2)+x*e^(1/2))/(-g^(1/3)*(-d)^(1/2)+f^(1/3)*e^(1/2)))/f^(5/3)/g^(1/3)-4/9*ln(c*(e*x^2+d)^p)*ln(2*f^(1/3)-g^(1
/3)*x*(1-I*3^(1/2)))/f^(5/3)/g^(1/3)/(1-I*3^(1/2))+4/9*p*polylog(2,(2*f^(1/3)-g^(1/3)*x*(1-I*3^(1/2)))*e^(1/2)
/(g^(1/3)*(1-I*3^(1/2))*(-d)^(1/2)+2*f^(1/3)*e^(1/2)))/f^(5/3)/g^(1/3)/(1-I*3^(1/2))+4/9*p*polylog(2,(2*f^(1/3
)-g^(1/3)*x*(1-I*3^(1/2)))*e^(1/2)/(I*g^(1/3)*(3^(1/2)+I)*(-d)^(1/2)+2*f^(1/3)*e^(1/2)))/f^(5/3)/g^(1/3)/(1-I*
3^(1/2))-4/9*ln(c*(e*x^2+d)^p)*ln(2*f^(1/3)-g^(1/3)*x*(1+I*3^(1/2)))/f^(5/3)/g^(1/3)/(1+I*3^(1/2))+4/9*p*polyl
og(2,(2*f^(1/3)-g^(1/3)*x*(1+I*3^(1/2)))*e^(1/2)/(-g^(1/3)*(1+I*3^(1/2))*(-d)^(1/2)+2*f^(1/3)*e^(1/2)))/f^(5/3
)/g^(1/3)/(1+I*3^(1/2))+4/9*p*polylog(2,(2*f^(1/3)-g^(1/3)*x*(1+I*3^(1/2)))*e^(1/2)/(g^(1/3)*(1+I*3^(1/2))*(-d
)^(1/2)+2*f^(1/3)*e^(1/2)))/f^(5/3)/g^(1/3)/(1+I*3^(1/2))-(-1)^(1/3)*e*p*ln(e*x^2+d)/(1+(-1)^(1/3))^4/f/(e*f^(
2/3)+(-1)^(2/3)*d*g^(2/3))/g^(1/3)-2/9*e*p*ln(f^(1/3)+g^(1/3)*x)/f/(e*f^(2/3)+d*g^(2/3))/g^(1/3)+1/9*e*p*ln(e*
x^2+d)/f/(e*f^(2/3)+d*g^(2/3))/g^(1/3)+4/9*p*ln(2*f^(1/3)-g^(1/3)*x*(1-I*3^(1/2)))*ln(-g^(1/3)*(3^(1/2)+I)*((-
d)^(1/2)-x*e^(1/2))/(-g^(1/3)*(3^(1/2)+I)*(-d)^(1/2)+2*I*f^(1/3)*e^(1/2)))/f^(5/3)/g^(1/3)/(1-I*3^(1/2))+4/9*p
*ln(2*f^(1/3)-g^(1/3)*x*(1-I*3^(1/2)))*ln(g^(1/3)*(3^(1/2)+I)*((-d)^(1/2)+x*e^(1/2))/(g^(1/3)*(3^(1/2)+I)*(-d)
^(1/2)+2*I*f^(1/3)*e^(1/2)))/f^(5/3)/g^(1/3)/(1-I*3^(1/2))+4/9*p*ln(2*f^(1/3)-g^(1/3)*x*(1+I*3^(1/2)))*ln(-g^(
1/3)*(1+I*3^(1/2))*((-d)^(1/2)-x*e^(1/2))/(-g^(1/3)*(1+I*3^(1/2))*(-d)^(1/2)+2*f^(1/3)*e^(1/2)))/f^(5/3)/g^(1/
3)/(1+I*3^(1/2))+4/9*p*ln(2*f^(1/3)-g^(1/3)*x*(1+I*3^(1/2)))*ln(g^(1/3)*(1+I*3^(1/2))*((-d)^(1/2)+x*e^(1/2))/(
g^(1/3)*(1+I*3^(1/2))*(-d)^(1/2)+2*f^(1/3)*e^(1/2)))/f^(5/3)/g^(1/3)/(1+I*3^(1/2))+2/9*p*arctan(x*e^(1/2)/d^(1
/2))*d^(1/2)*e^(1/2)/f^(4/3)/(e*f^(2/3)+d*g^(2/3))+4/9*p*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)*e^(1/2)/f^(4/3)/(2*
e*f^(2/3)-d*g^(2/3)*(1+I*3^(1/2)))-1/9*ln(c*(e*x^2+d)^p)/f^(4/3)/g^(1/3)/(f^(1/3)+g^(1/3)*x)+2/9*ln(f^(1/3)+g^
(1/3)*x)*ln(c*(e*x^2+d)^p)/f^(5/3)/g^(1/3)-2/9*p*polylog(2,(f^(1/3)+g^(1/3)*x)*e^(1/2)/(-g^(1/3)*(-d)^(1/2)+f^
(1/3)*e^(1/2)))/f^(5/3)/g^(1/3)-2/9*p*polylog(2,(f^(1/3)+g^(1/3)*x)*e^(1/2)/(g^(1/3)*(-d)^(1/2)+f^(1/3)*e^(1/2
)))/f^(5/3)/g^(1/3)+4/9*(-1)^(1/3)*e*p*ln(f^(1/3)+(-1)^(2/3)*g^(1/3)*x)/f/g^(1/3)/(2*e*f^(2/3)-d*g^(2/3)*(1+I*
3^(1/2)))-2/9*(-1)^(1/3)*e*p*ln(e*x^2+d)/f/g^(1/3)/(2*e*f^(2/3)-d*g^(2/3)*(1+I*3^(1/2)))

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 1863, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2521, 2513, 815, 649, 211, 266, 2512, 2463, 2441, 2440, 2438} \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\frac {2 \sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 f^{4/3} \left (g^{2/3} d+e f^{2/3}\right )}+\frac {2 (-1)^{2/3} \sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{4/3} \left ((-1)^{2/3} g^{2/3} d+e f^{2/3}\right )}+\frac {4 \sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 f^{4/3} \left (i \left (i-\sqrt {3}\right ) g^{2/3} d+2 e f^{2/3}\right )}-\frac {2 e p \log \left (\sqrt [3]{g} x+\sqrt [3]{f}\right )}{9 f \left (g^{2/3} d+e f^{2/3}\right ) \sqrt [3]{g}}-\frac {2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac {2 \sqrt [3]{-1} e p \log \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{\left (1+\sqrt [3]{-1}\right )^4 f \left ((-1)^{2/3} g^{2/3} d+e f^{2/3}\right ) \sqrt [3]{g}}+\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {4 \sqrt [3]{-1} e p \log \left ((-1)^{2/3} \sqrt [3]{g} x+\sqrt [3]{f}\right )}{9 f \left (2 e f^{2/3}-\left (1+i \sqrt {3}\right ) d g^{2/3}\right ) \sqrt [3]{g}}-\frac {2 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 )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \sqrt [3]{g}}+\frac {e p \log \left (e x^2+d\right )}{9 f \left (g^{2/3} d+e f^{2/3}\right ) \sqrt [3]{g}}-\frac {\sqrt [3]{-1} e p \log \left (e x^2+d\right )}{\left (1+\sqrt [3]{-1}\right )^4 f \left ((-1)^{2/3} g^{2/3} d+e f^{2/3}\right ) \sqrt [3]{g}}-\frac {2 \sqrt [3]{-1} e p \log \left (e x^2+d\right )}{9 f \left (2 e f^{2/3}-\left (1+i \sqrt {3}\right ) d g^{2/3}\right ) \sqrt [3]{g}}+\frac {2 \log \left (\sqrt [3]{g} x+\sqrt [3]{f}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {2 \log \left ((-1)^{2/3} \sqrt [3]{g} x+\sqrt [3]{f}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \sqrt [3]{g}}-\frac {\log \left (c \left (e x^2+d\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{g} x+\sqrt [3]{f}\right )}-\frac {\log \left (c \left (e x^2+d\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{g} x+(-1)^{2/3} \sqrt [3]{f}\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (e x^2+d\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{g} x+\sqrt [3]{f}\right )}-\frac {2 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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \sqrt [3]{g}}-\frac {2 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 )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \sqrt [3]{g}} \]

[In]

Int[Log[c*(d + e*x^2)^p]/(f + g*x^3)^2,x]

[Out]

(2*Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(e*f^(2/3) + d*g^(2/3))) + (2*(-1)^(2/3)*Sqrt[d]*
Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/((1 + (-1)^(1/3))^4*f^(4/3)*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))) + (4*Sq
rt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(2*e*f^(2/3) + I*(I - Sqrt[3])*d*g^(2/3))) - (2*e*p*Lo
g[f^(1/3) + g^(1/3)*x])/(9*f*(e*f^(2/3) + d*g^(2/3))*g^(1/3)) - (2*p*Log[(g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqr
t[e]*f^(1/3) + Sqrt[-d]*g^(1/3))]*Log[f^(1/3) + g^(1/3)*x])/(9*f^(5/3)*g^(1/3)) - (2*p*Log[-((g^(1/3)*(Sqrt[-d
] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3)))]*Log[f^(1/3) + g^(1/3)*x])/(9*f^(5/3)*g^(1/3)) + (2*(-1)
^(1/3)*e*p*Log[f^(1/3) - (-1)^(1/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^4*f*(e*f^(2/3) + (-1)^(2/3)*d*g^(2/3))*g^(1/
3)) + ((2*I)*Sqrt[3]*p*Log[-(((-1)^(1/3)*g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - (-1)^(1/3)*Sqrt[-d
]*g^(1/3)))]*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(1/3)) + ((2*I)*Sqrt[3]*p*Log
[((-1)^(1/3)*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) + (-1)^(1/3)*Sqrt[-d]*g^(1/3))]*Log[-f^(1/3) + (
-1)^(1/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(1/3)) + (4*(-1)^(1/3)*e*p*Log[f^(1/3) + (-1)^(2/3)*g^(1/3
)*x])/(9*f*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))*g^(1/3)) - (2*p*Log[((-1)^(2/3)*g^(1/3)*(Sqrt[-d] - Sqrt[
e]*x))/(Sqrt[e]*f^(1/3) + (-1)^(2/3)*Sqrt[-d]*g^(1/3))]*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x])/((1 + (-1)^(1/3))
^4*f^(5/3)*g^(1/3)) - (2*p*Log[-(((-1)^(2/3)*g^(1/3)*(Sqrt[-d] + Sqrt[e]*x))/(Sqrt[e]*f^(1/3) - (-1)^(2/3)*Sqr
t[-d]*g^(1/3)))]*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)) + (e*p*Log[d + e*x^
2])/(9*f*(e*f^(2/3) + d*g^(2/3))*g^(1/3)) - ((-1)^(1/3)*e*p*Log[d + e*x^2])/((1 + (-1)^(1/3))^4*f*(e*f^(2/3) +
(-1)^(2/3)*d*g^(2/3))*g^(1/3)) - (2*(-1)^(1/3)*e*p*Log[d + e*x^2])/(9*f*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2
/3))*g^(1/3)) - Log[c*(d + e*x^2)^p]/(9*f^(4/3)*g^(1/3)*(f^(1/3) + g^(1/3)*x)) - Log[c*(d + e*x^2)^p]/((1 + (-
1)^(1/3))^4*f^(4/3)*g^(1/3)*((-1)^(2/3)*f^(1/3) + g^(1/3)*x)) + ((-1)^(1/3)*Log[c*(d + e*x^2)^p])/(9*f^(4/3)*g
^(1/3)*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x)) + (2*Log[f^(1/3) + g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(9*f^(5/3)*g^(1/3
)) - ((2*I)*Sqrt[3]*Log[-f^(1/3) + (-1)^(1/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(
1/3)) + (2*Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3)) - (2
*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3))])/(9*f^(5/3)*g^(1/3)) - (2*
p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))])/(9*f^(5/3)*g^(1/3)) + ((2*
I)*Sqrt[3]*p*PolyLog[2, (Sqrt[e]*(f^(1/3) - (-1)^(1/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) - (-1)^(1/3)*Sqrt[-d]*g^(1
/3))])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(1/3)) + ((2*I)*Sqrt[3]*p*PolyLog[2, (Sqrt[e]*(f^(1/3) - (-1)^(1/3)*g^(1/
3)*x))/(Sqrt[e]*f^(1/3) + (-1)^(1/3)*Sqrt[-d]*g^(1/3))])/((1 + (-1)^(1/3))^5*f^(5/3)*g^(1/3)) - (2*p*PolyLog[2
, (Sqrt[e]*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) - (-1)^(2/3)*Sqrt[-d]*g^(1/3))])/((1 + (-1)^(1/3
))^4*f^(5/3)*g^(1/3)) - (2*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + (-1)^(2/3)*g^(1/3)*x))/(Sqrt[e]*f^(1/3) + (-1)^(2/
3)*Sqrt[-d]*g^(1/3))])/((1 + (-1)^(1/3))^4*f^(5/3)*g^(1/3))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2512

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[f +
g*x]*((a + b*Log[c*(d + e*x^n)^p])/g), x] - Dist[b*e*n*(p/g), Int[x^(n - 1)*(Log[f + g*x]/(d + e*x^n)), x], x]
/; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]

Rule 2513

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Simp[(f
+ g*x)^(r + 1)*((a + b*Log[c*(d + e*x^n)^p])/(g*(r + 1))), x] - Dist[b*e*n*(p/(g*(r + 1))), Int[x^(n - 1)*((f
+ g*x)^(r + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n
]) && NeQ[r, -1]

Rule 2521

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )^2}+\frac {2 \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac {(-1)^{2/3} \log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{4/3} \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )^2}-\frac {2 (-1)^{5/6} \sqrt {3} \log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 f^{4/3} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )^2}+\frac {2 (-1)^{2/3} \log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}\right ) \, dx \\ & = \frac {2 \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\sqrt [3]{f}+\sqrt [3]{g} x} \, dx}{9 f^{5/3}}+\frac {2 \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x} \, dx}{9 f^{5/3}}-\frac {\left (2 (-1)^{5/6} \sqrt {3}\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3}}+\frac {\int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt [3]{f}+\sqrt [3]{g} x\right )^2} \, dx}{9 f^{4/3}}+\frac {\int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right )^2} \, dx}{9 f^{4/3}}+\frac {\int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )^2} \, dx}{9 f^{4/3}} \\ & = -\frac {\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac {2 \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac {2 \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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac {(4 e p) \int \frac {x \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{d+e x^2} \, dx}{9 f^{5/3} \sqrt [3]{g}}+\frac {\left (4 \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}{9 f^{5/3} \sqrt [3]{g}}+\frac {\left (4 i \sqrt {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}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {(2 e p) \int \frac {x}{\left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \left (d+e x^2\right )} \, dx}{9 f^{4/3} \sqrt [3]{g}}-\frac {\left (2 \sqrt [3]{-1} e p\right ) \int \frac {x}{\left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right ) \left (d+e x^2\right )} \, dx}{9 f^{4/3} \sqrt [3]{g}}-\frac {\left (2 (-1)^{2/3} e p\right ) \int \frac {x}{\left (-\sqrt [3]{f}+\sqrt [3]{-1} \sqrt [3]{g} x\right ) \left (d+e x^2\right )} \, dx}{9 f^{4/3} \sqrt [3]{g}} \\ & = -\frac {\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac {2 \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac {2 \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 )}{9 f^{5/3} \sqrt [3]{g}}-\frac {(4 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}{9 f^{5/3} \sqrt [3]{g}}+\frac {\left (4 \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}{9 f^{5/3} \sqrt [3]{g}}+\frac {\left (4 i \sqrt {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}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {(2 e p) \int \left (-\frac {\sqrt [3]{f} \sqrt [3]{g}}{\left (e f^{2/3}+d g^{2/3}\right ) \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac {d \sqrt [3]{g}+e \sqrt [3]{f} x}{\left (e f^{2/3}+d g^{2/3}\right ) \left (d+e x^2\right )}\right ) \, dx}{9 f^{4/3} \sqrt [3]{g}}-\frac {\left (2 \sqrt [3]{-1} e p\right ) \int \left (-\frac {(-1)^{2/3} \sqrt [3]{f} \sqrt [3]{g}}{\left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac {(-1)^{2/3} d \sqrt [3]{g}+e \sqrt [3]{f} x}{\left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \left (d+e x^2\right )}\right ) \, dx}{9 f^{4/3} \sqrt [3]{g}}-\frac {\left (2 (-1)^{2/3} e p\right ) \int \left (-\frac {\sqrt [3]{-1} \sqrt [3]{f} \sqrt [3]{g}}{\left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}+\frac {\sqrt [3]{-1} d \sqrt [3]{g}-e \sqrt [3]{f} x}{\left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \left (d+e x^2\right )}\right ) \, dx}{9 f^{4/3} \sqrt [3]{g}} \\ & = -\frac {2 e p \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac {2 (-1)^{2/3} e p \log \left (\sqrt [3]{f}-\sqrt [3]{-1} \sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}}+\frac {2 \sqrt [3]{-1} e p \log \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}{9 f \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \sqrt [3]{g}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left ((-1)^{2/3} \sqrt [3]{f}+\sqrt [3]{g} x\right )}+\frac {\sqrt [3]{-1} \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \sqrt [3]{g} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac {2 \log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right ) \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \sqrt [3]{g}}-\frac {2 i \sqrt {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 )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}-\frac {2 \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 )}{9 f^{5/3} \sqrt [3]{g}}+\frac {\left (2 \sqrt {e} p\right ) \int \frac {\log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{9 f^{5/3} \sqrt [3]{g}}-\frac {\left (2 \sqrt {e} p\right ) \int \frac {\log \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{9 f^{5/3} \sqrt [3]{g}}-\frac {\left (2 \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}{9 f^{5/3} \sqrt [3]{g}}+\frac {\left (2 \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}{9 f^{5/3} \sqrt [3]{g}}-\frac {\left (2 i \sqrt {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}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {\left (2 i \sqrt {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}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \sqrt [3]{g}}+\frac {(2 e p) \int \frac {d \sqrt [3]{g}+e \sqrt [3]{f} x}{d+e x^2} \, dx}{9 f^{4/3} \left (e f^{2/3}+d g^{2/3}\right ) \sqrt [3]{g}}-\frac {\left (2 \sqrt [3]{-1} e p\right ) \int \frac {(-1)^{2/3} d \sqrt [3]{g}+e \sqrt [3]{f} x}{d+e x^2} \, dx}{9 f^{4/3} \left (e f^{2/3}-\sqrt [3]{-1} d g^{2/3}\right ) \sqrt [3]{g}}-\frac {\left (2 (-1)^{2/3} e p\right ) \int \frac {\sqrt [3]{-1} d \sqrt [3]{g}-e \sqrt [3]{f} x}{d+e x^2} \, dx}{9 f^{4/3} \left (e f^{2/3}+(-1)^{2/3} d g^{2/3}\right ) \sqrt [3]{g}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 6.81 (sec) , antiderivative size = 2168, normalized size of antiderivative = 1.16 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\text {Result too large to show} \]

[In]

Integrate[Log[c*(d + e*x^2)^p]/(f + g*x^3)^2,x]

[Out]

(x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/(3*f*(f + g*x^3)) + (2*ArcTan[(-f^(1/3) + 2*g^(1/3)*x)/(Sqrt[
3]*f^(1/3))]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/(3*Sqrt[3]*f^(5/3)*g^(1/3)) + (2*Log[f^(1/3) + g^(1
/3)*x]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/(9*f^(5/3)*g^(1/3)) - ((-(p*Log[d + e*x^2]) + Log[c*(d +
e*x^2)^p])*Log[f^(2/3) - f^(1/3)*g^(1/3)*x + g^(2/3)*x^2])/(9*f^(5/3)*g^(1/3)) + p*(-1/3*((-1 + (-1)^(1/3))*(-
(Log[((-I)*Sqrt[d])/Sqrt[e] + x]/((-1)^(2/3)*f^(1/3) + g^(1/3)*x)) + (Sqrt[e]*(Log[I*Sqrt[d] - Sqrt[e]*x] - Lo
g[-((-1)^(2/3)*f^(1/3)) - g^(1/3)*x]))/((-1)^(2/3)*Sqrt[e]*f^(1/3) + I*Sqrt[d]*g^(1/3))))/((1 + (-1)^(1/3))^2*
f^(4/3)*g^(1/3)) - ((-1 + (-1)^(1/3))*(-(Log[(I*Sqrt[d])/Sqrt[e] + x]/((-1)^(2/3)*f^(1/3) + g^(1/3)*x)) + (Sqr
t[e]*(Log[I*Sqrt[d] + Sqrt[e]*x] - Log[-((-1)^(2/3)*f^(1/3)) - g^(1/3)*x]))/((-1)^(2/3)*Sqrt[e]*f^(1/3) - I*Sq
rt[d]*g^(1/3))))/(3*(1 + (-1)^(1/3))^2*f^(4/3)*g^(1/3)) + ((-1)^(1/3)*(-(Log[((-I)*Sqrt[d])/Sqrt[e] + x]/(f^(1
/3) + g^(1/3)*x)) + (Sqrt[e]*(Log[I*Sqrt[d] - Sqrt[e]*x] - Log[f^(1/3) + g^(1/3)*x]))/(Sqrt[e]*f^(1/3) + I*Sqr
t[d]*g^(1/3))))/(3*(1 + (-1)^(1/3))^2*f^(4/3)*g^(1/3)) + ((-1)^(1/3)*(-(Log[(I*Sqrt[d])/Sqrt[e] + x]/(f^(1/3)
+ g^(1/3)*x)) + (Sqrt[e]*(Log[I*Sqrt[d] + Sqrt[e]*x] - Log[f^(1/3) + g^(1/3)*x]))/(Sqrt[e]*f^(1/3) - I*Sqrt[d]
*g^(1/3))))/(3*(1 + (-1)^(1/3))^2*f^(4/3)*g^(1/3)) - (Log[((-I)*Sqrt[d])/Sqrt[e] + x]/((-1)^(1/3)*f^(1/3) - g^
(1/3)*x) + (Sqrt[e]*(-Log[I*Sqrt[d] - Sqrt[e]*x] + Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x]))/((-1)^(1/3)*Sqrt[e]*f
^(1/3) - I*Sqrt[d]*g^(1/3)))/(3*(1 + (-1)^(1/3))^2*f^(4/3)*g^(1/3)) - (Log[(I*Sqrt[d])/Sqrt[e] + x]/((-1)^(1/3
)*f^(1/3) - g^(1/3)*x) + (Sqrt[e]*(-Log[I*Sqrt[d] + Sqrt[e]*x] + Log[f^(1/3) + (-1)^(2/3)*g^(1/3)*x]))/((-1)^(
1/3)*Sqrt[e]*f^(1/3) + I*Sqrt[d]*g^(1/3)))/(3*(1 + (-1)^(1/3))^2*f^(4/3)*g^(1/3)) + ((-Log[((-I)*Sqrt[d])/Sqrt
[e] + x] - Log[(I*Sqrt[d])/Sqrt[e] + x] + Log[d + e*x^2])*((3*f^(2/3)*x)/(f + g*x^3) - (2*Sqrt[3]*ArcTan[(1 -
(2*g^(1/3)*x)/f^(1/3))/Sqrt[3]])/g^(1/3) + (2*Log[f^(1/3) + g^(1/3)*x])/g^(1/3) - Log[f^(2/3) - f^(1/3)*g^(1/3
)*x + g^(2/3)*x^2]/g^(1/3)))/(9*f^(5/3)) - (2*(Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[((-1)^(2/3)*f^(1/3) + g^(1/3)*
x)/((-1)^(2/3)*f^(1/3) - (I*Sqrt[d]*g^(1/3))/Sqrt[e])] + PolyLog[2, -((g^(1/3)*(Sqrt[d] - I*Sqrt[e]*x))/((-1)^
(1/6)*Sqrt[e]*f^(1/3) - Sqrt[d]*g^(1/3)))]))/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)) - (2*(-1 + (-1)^(1/3))*(Lo
g[(I*Sqrt[d])/Sqrt[e] + x]*Log[-((-((-1)^(1/3)*f^(1/3)) + g^(1/3)*x)/((-1)^(1/3)*f^(1/3) + (I*Sqrt[d]*g^(1/3))
/Sqrt[e]))] + PolyLog[2, -((g^(1/3)*(Sqrt[d] - I*Sqrt[e]*x))/((-1)^(5/6)*Sqrt[e]*f^(1/3) - Sqrt[d]*g^(1/3)))])
)/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)) + (2*(-1)^(1/3)*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(f^(1/3) + g^(1/
3)*x)/(f^(1/3) + (I*Sqrt[d]*g^(1/3))/Sqrt[e])] + PolyLog[2, (I*g^(1/3)*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[e]*f^(1/
3) + I*Sqrt[d]*g^(1/3))]))/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)) - (2*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[((
-1)^(2/3)*f^(1/3) + g^(1/3)*x)/((-1)^(2/3)*f^(1/3) + (I*Sqrt[d]*g^(1/3))/Sqrt[e])] + PolyLog[2, (g^(1/3)*(Sqrt
[d] + I*Sqrt[e]*x))/((-1)^(1/6)*Sqrt[e]*f^(1/3) + Sqrt[d]*g^(1/3))]))/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)) -
(2*(-1 + (-1)^(1/3))*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(-((-1)^(1/3)*f^(1/3)) + g^(1/3)*x)/(-((-1)^(1/3)*f
^(1/3)) + (I*Sqrt[d]*g^(1/3))/Sqrt[e])] + PolyLog[2, (g^(1/3)*(Sqrt[d] + I*Sqrt[e]*x))/((-1)^(5/6)*Sqrt[e]*f^(
1/3) + Sqrt[d]*g^(1/3))]))/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)) + (2*(-1)^(1/3)*(Log[(I*Sqrt[d])/Sqrt[e] + x
]*Log[(f^(1/3) + g^(1/3)*x)/(f^(1/3) - (I*Sqrt[d]*g^(1/3))/Sqrt[e])] + PolyLog[2, -((g^(1/3)*(I*Sqrt[d] + Sqrt
[e]*x))/(Sqrt[e]*f^(1/3) - I*Sqrt[d]*g^(1/3)))]))/(3*(1 + (-1)^(1/3))^2*f^(5/3)*g^(1/3)))

Maple [F]

\[\int \frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\left (g \,x^{3}+f \right )^{2}}d x\]

[In]

int(ln(c*(e*x^2+d)^p)/(g*x^3+f)^2,x)

[Out]

int(ln(c*(e*x^2+d)^p)/(g*x^3+f)^2,x)

Fricas [F]

\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{3} + f\right )}^{2}} \,d x } \]

[In]

integrate(log(c*(e*x^2+d)^p)/(g*x^3+f)^2,x, algorithm="fricas")

[Out]

integral(log((e*x^2 + d)^p*c)/(g^2*x^6 + 2*f*g*x^3 + f^2), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(ln(c*(e*x**2+d)**p)/(g*x**3+f)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(log(c*(e*x^2+d)^p)/(g*x^3+f)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{3} + f\right )}^{2}} \,d x } \]

[In]

integrate(log(c*(e*x^2+d)^p)/(g*x^3+f)^2,x, algorithm="giac")

[Out]

integrate(log((e*x^2 + d)^p*c)/(g*x^3 + f)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{{\left (g\,x^3+f\right )}^2} \,d x \]

[In]

int(log(c*(d + e*x^2)^p)/(f + g*x^3)^2,x)

[Out]

int(log(c*(d + e*x^2)^p)/(f + g*x^3)^2, x)