Integrand size = 22, antiderivative size = 1861 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx =\text {Too large to display} \]
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)+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)))-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)+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*l n(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*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))-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...
Time = 6.81 (sec) , antiderivative size = 2168, normalized size of antiderivative = 1.16 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\text {Result too large to show} \]
(x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]))/(3*f*(f + g*x^3)) + (2*Ar cTan[(-f^(1/3) + 2*g^(1/3)*x)/(Sqrt[3]*f^(1/3))]*(-(p*Log[d + e*x^2]) + Lo g[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] - Log[-((-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)) + (Sqrt[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*Sqrt[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)) + ((-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...
Time = 2.73 (sec) , antiderivative size = 1867, 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.
| \(\displaystyle \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2921 |
| \(\displaystyle \int \left (\frac {2 \log \left (c \left (d+e x^2\right )^p\right )}{9 f^{5/3} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )}-\frac {2 (-1)^{5/6} \sqrt {3} \log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^5 f^{5/3} \left (\sqrt [3]{-1} \sqrt [3]{g} x-\sqrt [3]{f}\right )}+\frac {2 (-1)^{2/3} \log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{5/3} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{9 f^{4/3} \left (\sqrt [3]{f}+\sqrt [3]{g} x\right )^2}+\frac {(-1)^{2/3} \log \left (c \left (d+e x^2\right )^p\right )}{\left (1+\sqrt [3]{-1}\right )^4 f^{4/3} \left (\sqrt [3]{-1} \sqrt [3]{g} x-\sqrt [3]{f}\right )^2}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (\sqrt [3]{-1}-1\right )^2 \left (1+\sqrt [3]{-1}\right )^4 f^{4/3} \left (\sqrt [3]{f}+(-1)^{2/3} \sqrt [3]{g} x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \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 (2 e f^{2/3}-\left (1+i \sqrt {3}\right ) d g^{2/3}\right )}+\frac {2 \sqrt [3]{-1} e p \log \left (-\sqrt [3]{g} x-(-1)^{2/3} \sqrt [3]{f}\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 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 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 (i \left (i-\sqrt {3}\right ) g^{2/3} d+2 e f^{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 (i \left (i-\sqrt {3}\right ) g^{2/3} d+2 e f^{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}}\) |
(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*Sqrt [d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*f^(4/3)*(2*e*f^(2/3) - (1 + I*Sqrt[3])*d*g^(2/3))) + (2*(-1)^(1/3)*e*p*Log[-((-1)^(2/3)*f^(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*e*p*Log[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))/(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*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*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*Lo g[f^(1/3) + (-1)^(2/3)*g^(1/3)*x])/(9*f*(2*e*f^(2/3) + I*(I - Sqrt[3])*d*g ^(2/3))*g^(1/3)) - (2*p*Log[((-1)^(2/3)*g^(1/3)*(Sqrt[-d] - Sqrt[e]*x))/(S qrt[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)...
3.3.92.3.1 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]))
\[\int \frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\left (g \,x^{3}+f \right )^{2}}d x\]
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{3} + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{3} + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^3\right )^2} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{{\left (g\,x^3+f\right )}^2} \,d x \]