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}} \]
1/3*ln(-f^(1/3)-g^(1/3)*x)*ln(c*(e*x^2+d)^p)/f^(2/3)/g^(1/3)+1/3*(-1)^(2/3 )*ln(-f^(1/3)+(-1)^(1/3)*g^(1/3)*x)*ln(c*(e*x^2+d)^p)/f^(2/3)/g^(1/3)-1/3* (-1)^(1/3)*ln(-f^(1/3)-(-1)^(2/3)*g^(1/3)*x)*ln(c*(e*x^2+d)^p)/f^(2/3)/g^( 1/3)-1/3*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^(2/3)/g^(1/3)-1/3*(-1)^(2/3)*p*ln(-f^(1/ 3)+(-1)^(1/3)*g^(1/3)*x)*ln(-(-1)^(1/3)*g^(1/3)*((-d)^(1/2)-x*e^(1/2))/(-( -1)^(1/3)*g^(1/3)*(-d)^(1/2)+f^(1/3)*e^(1/2)))/f^(2/3)/g^(1/3)+1/3*(-1)^(1 /3)*p*ln(-f^(1/3)-(-1)^(2/3)*g^(1/3)*x)*ln((-1)^(2/3)*g^(1/3)*((-d)^(1/2)- x*e^(1/2))/((-1)^(2/3)*g^(1/3)*(-d)^(1/2)+f^(1/3)*e^(1/2)))/f^(2/3)/g^(1/3 )-1/3*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^(2/3)/g^(1/3)-1/3*(-1)^(2/3)*p*ln(-f^(1/3 )+(-1)^(1/3)*g^(1/3)*x)*ln((-1)^(1/3)*g^(1/3)*((-d)^(1/2)+x*e^(1/2))/((-1) ^(1/3)*g^(1/3)*(-d)^(1/2)+f^(1/3)*e^(1/2)))/f^(2/3)/g^(1/3)+1/3*(-1)^(1/3) *p*ln(-f^(1/3)-(-1)^(2/3)*g^(1/3)*x)*ln(-(-1)^(2/3)*g^(1/3)*((-d)^(1/2)+x* e^(1/2))/(-(-1)^(2/3)*g^(1/3)*(-d)^(1/2)+f^(1/3)*e^(1/2)))/f^(2/3)/g^(1/3) -1/3*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^(2/3)/g^(1/3)-1/3*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^(2/3)/g^(1/3)-1/3*(-1)^(2/3)*p*polylog (2,(f^(1/3)-(-1)^(1/3)*g^(1/3)*x)*e^(1/2)/(-(-1)^(1/3)*g^(1/3)*(-d)^(1/2)+ f^(1/3)*e^(1/2)))/f^(2/3)/g^(1/3)-1/3*(-1)^(2/3)*p*polylog(2,(f^(1/3)-(...
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}} \]
(-(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]) - 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] - (-1)^ (2/3)*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)^(2/3)*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)^(1/3)*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)^(1/3)*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] + Log[-f^(1/3) - g^(1/3)*x]*Log[c*(d + e*x^2)^p] + (-1)^(2/3)*Log[-f^(1 /3) + (-1)^(1/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p] - (-1)^(1/3)*Log[-f^(1/3) - (-1)^(2/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p] - p*PolyLog[2, (Sqrt[e]*(f^( 1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) - Sqrt[-d]*g^(1/3))] - p*PolyLog[2, (S qrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))] - (-1) ^(2/3)*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)^(2/3)*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)^(1/3)*p*PolyLog[2, (Sqrt[e]*(f^(1/3) + (-1)^(2/3)*g^(1/3...
Time = 1.72 (sec) , antiderivative size = 1165, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2921, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx\) |
| \(\Big \downarrow \) 2921 |
| \(\displaystyle \int \left (-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \left (-\sqrt [3]{f}-\sqrt [3]{g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{g} x-\sqrt [3]{f}\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{3 f^{2/3} \left (-\sqrt [3]{f}-(-1)^{2/3} \sqrt [3]{g} x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\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}}\) |
-1/3*(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])/(f^(2/3)*g^(1/3)) - (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])/(3*f^(2/3)*g^(1/3)) - ((-1)^(2/3)*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])/(3*f^(2/3)*g^(1/3)) - ((-1)^(2/3 )*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])/(3*f^(2/3 )*g^(1/3)) + ((-1)^(1/3)*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])/(3*f^(2/3)*g^(1/3)) + ((-1)^(1/3)*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])/(3*f^(2/3)*g^(1/3)) + (Log[-f^(1/3 ) - g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(3*f^(2/3)*g^(1/3)) + ((-1)^(2/3)*Log [-f^(1/3) + (-1)^(1/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p])/(3*f^(2/3)*g^(1/3) ) - ((-1)^(1/3)*Log[-f^(1/3) - (-1)^(2/3)*g^(1/3)*x]*Log[c*(d + e*x^2)^p]) /(3*f^(2/3)*g^(1/3)) - (p*PolyLog[2, (Sqrt[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt [e]*f^(1/3) - Sqrt[-d]*g^(1/3))])/(3*f^(2/3)*g^(1/3)) - (p*PolyLog[2, (Sqr t[e]*(f^(1/3) + g^(1/3)*x))/(Sqrt[e]*f^(1/3) + Sqrt[-d]*g^(1/3))])/(3*f^(2 /3)*g^(1/3)) - ((-1)^(2/3)*p*PolyLog[2, (Sqrt[e]*(f^(1/3) - (-1)^(1/3)*...
3.3.91.3.1 Defintions of rubi rules used
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]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && Integ erQ[r] && IntegerQ[s] && (EqQ[q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))
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
(ln((e*x^2+d)^p)-p*ln(e*x^2+d))*(1/3/g/(f/g)^(2/3)*ln(x+(f/g)^(1/3))-1/6/g /(f/g)^(2/3)*ln(x^2-(f/g)^(1/3)*x+(f/g)^(2/3))+1/3/g/(f/g)^(2/3)*3^(1/2)*a rctan(1/3*3^(1/2)*(2/(f/g)^(1/3)*x-1)))+1/3*p/g*sum(1/_alpha^2*(ln(x-_alph a)*ln(e*x^2+d)-ln(x-_alpha)*(ln((RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d, index=1)-x+_alpha)/RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,index=1))+ln(( RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,index=2)-x+_alpha)/RootOf(_Z^2*e+ 2*_Z*_alpha*e+_alpha^2*e+d,index=2)))-dilog((RootOf(_Z^2*e+2*_Z*_alpha*e+_ alpha^2*e+d,index=1)-x+_alpha)/RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,in dex=1))-dilog((RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,index=2)-x+_alpha) /RootOf(_Z^2*e+2*_Z*_alpha*e+_alpha^2*e+d,index=2))),_alpha=RootOf(_Z^3*g+ f))+(1/2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2-1/2*I*Pi*csgn(I* (e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-1/2*I*Pi*csgn(I*c*(e*x^2+d)^p )^3+1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+ln(c))*(1/3/g/(f/g)^(2/3)*l n(x+(f/g)^(1/3))-1/6/g/(f/g)^(2/3)*ln(x^2-(f/g)^(1/3)*x+(f/g)^(2/3))+1/3/g /(f/g)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(f/g)^(1/3)*x-1)))
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{3} + f} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{3} + f} \,d x } \]
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{3} + f} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^3} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^3+f} \,d x \]