Integrand size = 25, antiderivative size = 651 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4 \left (f+g x^2\right )} \, dx=-\frac {2 e p}{3 d f x}-\frac {2 e^{3/2} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2} f}-\frac {2 \sqrt {e} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f^2}+\frac {2 g^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{5/2}}-\frac {g^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{f^{5/2}}-\frac {g^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{f^{5/2}}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{3 f x^3}+\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{f^2 x}+\frac {g^{3/2} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{5/2}}-\frac {i g^{3/2} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{5/2}}+\frac {i g^{3/2} p \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{5/2}}+\frac {i g^{3/2} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{5/2}} \]
-2/3*e*p/d/f/x-2/3*e^(3/2)*p*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)/f-1/3*ln(c* (e*x^2+d)^p)/f/x^3+g*ln(c*(e*x^2+d)^p)/f^2/x+g^(3/2)*arctan(x*g^(1/2)/f^(1 /2))*ln(c*(e*x^2+d)^p)/f^(5/2)+2*g^(3/2)*p*arctan(x*g^(1/2)/f^(1/2))*ln(2* f^(1/2)/(f^(1/2)-I*x*g^(1/2)))/f^(5/2)-g^(3/2)*p*arctan(x*g^(1/2)/f^(1/2)) *ln(-2*((-d)^(1/2)-x*e^(1/2))*f^(1/2)*g^(1/2)/(f^(1/2)-I*x*g^(1/2))/(I*e^( 1/2)*f^(1/2)-(-d)^(1/2)*g^(1/2)))/f^(5/2)-g^(3/2)*p*arctan(x*g^(1/2)/f^(1/ 2))*ln(2*((-d)^(1/2)+x*e^(1/2))*f^(1/2)*g^(1/2)/(f^(1/2)-I*x*g^(1/2))/(I*e ^(1/2)*f^(1/2)+(-d)^(1/2)*g^(1/2)))/f^(5/2)-I*g^(3/2)*p*polylog(2,1-2*f^(1 /2)/(f^(1/2)-I*x*g^(1/2)))/f^(5/2)+1/2*I*g^(3/2)*p*polylog(2,1+2*((-d)^(1/ 2)-x*e^(1/2))*f^(1/2)*g^(1/2)/(f^(1/2)-I*x*g^(1/2))/(I*e^(1/2)*f^(1/2)-(-d )^(1/2)*g^(1/2)))/f^(5/2)+1/2*I*g^(3/2)*p*polylog(2,1-2*((-d)^(1/2)+x*e^(1 /2))*f^(1/2)*g^(1/2)/(f^(1/2)-I*x*g^(1/2))/(I*e^(1/2)*f^(1/2)+(-d)^(1/2)*g ^(1/2)))/f^(5/2)-2*g*p*arctan(x*e^(1/2)/d^(1/2))*e^(1/2)/f^2/d^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.32 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.03 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4 \left (f+g x^2\right )} \, dx=\frac {-\frac {12 \sqrt {e} \sqrt {f} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {4 e f^{3/2} p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {e x^2}{d}\right )}{d x}-\frac {2 f^{3/2} \log \left (c \left (d+e x^2\right )^p\right )}{x^3}+\frac {6 \sqrt {f} g \log \left (c \left (d+e x^2\right )^p\right )}{x}+6 g^{3/2} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )-3 i g^{3/2} p \left (\log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )+\log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )-\log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )-\log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )\right )}{6 f^{5/2}} \]
((-12*Sqrt[e]*Sqrt[f]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] - (4*e*f^(3 /2)*p*Hypergeometric2F1[-1/2, 1, 1/2, -((e*x^2)/d)])/(d*x) - (2*f^(3/2)*Lo g[c*(d + e*x^2)^p])/x^3 + (6*Sqrt[f]*g*Log[c*(d + e*x^2)^p])/x + 6*g^(3/2) *ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p] - (3*I)*g^(3/2)*p*(Log[( Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/(I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Lo g[1 - (I*Sqrt[g]*x)/Sqrt[f]] + Log[(Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((-I)* Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] - Log[ (Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((-I)*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g]) ]*Log[1 + (I*Sqrt[g]*x)/Sqrt[f]] - Log[(Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/(I *Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 + (I*Sqrt[g]*x)/Sqrt[f]] + Pol yLog[2, (Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sq rt[g])] + PolyLog[2, (Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])] - PolyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[ e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])] - PolyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[ g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])]))/(6*f^(5/2))
Time = 0.83 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2926, 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.
| \(\displaystyle \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4 \left (f+g x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle \int \left (\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{f^2 \left (f+g x^2\right )}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{f^2 x^2}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {g^{3/2} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f^{5/2}}-\frac {2 e^{3/2} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2} f}-\frac {g^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{f^{5/2}}-\frac {g^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{f^{5/2}}-\frac {2 \sqrt {e} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f^2}+\frac {2 g^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{5/2}}+\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{f^2 x}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{3 f x^3}+\frac {i g^{3/2} p \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{2 f^{5/2}}+\frac {i g^{3/2} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 f^{5/2}}-\frac {2 e p}{3 d f x}-\frac {i g^{3/2} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{f^{5/2}}\) |
(-2*e*p)/(3*d*f*x) - (2*e^(3/2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*d^(3/2)* f) - (2*Sqrt[e]*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*f^2) + (2*g^(3/2 )*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/ f^(5/2) - (g^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(-2*Sqrt[f]*Sqrt[g]*( Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/f^(5/2) - (g^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sq rt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[ g])*(Sqrt[f] - I*Sqrt[g]*x))])/f^(5/2) - Log[c*(d + e*x^2)^p]/(3*f*x^3) + (g*Log[c*(d + e*x^2)^p])/(f^2*x) + (g^(3/2)*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Lo g[c*(d + e*x^2)^p])/f^(5/2) - (I*g^(3/2)*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqr t[f] - I*Sqrt[g]*x)])/f^(5/2) + ((I/2)*g^(3/2)*p*PolyLog[2, 1 + (2*Sqrt[f] *Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*( Sqrt[f] - I*Sqrt[g]*x))])/f^(5/2) + ((I/2)*g^(3/2)*p*PolyLog[2, 1 - (2*Sqr t[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g ])*(Sqrt[f] - I*Sqrt[g]*x))])/f^(5/2)
3.4.47.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b *Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c, d, e , f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] & & IntegerQ[s]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.60 (sec) , antiderivative size = 602, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(\left (\ln \left (\left (e \,x^{2}+d \right )^{p}\right )-p \ln \left (e \,x^{2}+d \right )\right ) \left (-\frac {1}{3 f \,x^{3}}+\frac {g}{f^{2} x}+\frac {g^{2} \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{f^{2} \sqrt {f g}}\right )-\frac {p \ln \left (e \,x^{2}+d \right )}{3 f \,x^{3}}-\frac {2 p \,e^{2} \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{3 f d \sqrt {d e}}-\frac {2 e p}{3 d f x}+\frac {p g \ln \left (e \,x^{2}+d \right )}{f^{2} x}-\frac {2 p g e \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{f^{2} \sqrt {d e}}+p \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (g \,\textit {\_Z}^{2}+f \right )}{\sum }\frac {\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (e \,x^{2}+d \right )-2 e \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e +d g -e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e +d g -e f , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e +d g -e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e +d g -e f , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e +d g -e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e +d g -e f , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e +d g -e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e +d g -e f , \operatorname {index} =2\right )}\right )}{2 e}\right )\right ) g}{2 f^{2} \underline {\hspace {1.25 ex}}\alpha }\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (-\frac {1}{3 f \,x^{3}}+\frac {g}{f^{2} x}+\frac {g^{2} \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{f^{2} \sqrt {f g}}\right )\) | \(602\) |
(ln((e*x^2+d)^p)-p*ln(e*x^2+d))*(-1/3/f/x^3+1/f^2*g/x+g^2/f^2/(f*g)^(1/2)* arctan(g*x/(f*g)^(1/2)))-1/3*p/f/x^3*ln(e*x^2+d)-2/3*p/f*e^2/d/(d*e)^(1/2) *arctan(x*e/(d*e)^(1/2))-2/3*e*p/d/f/x+p*g/f^2/x*ln(e*x^2+d)-2*p*g/f^2*e/( d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+p*Sum(1/2*(ln(x-_alpha)*ln(e*x^2+d)-2*e *(1/2*ln(x-_alpha)*(ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1)-x +_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1))+ln((RootOf(_Z^2 *e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alph a*e*g+d*g-e*f,index=2)))/e+1/2*(dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g -e*f,index=1)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1))+ dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)-x+_alpha)/RootOf(_ Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)))/e))*g/f^2/_alpha,_alpha=RootOf( _Z^2*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/f/x^3+1 /f^2*g/x+g^2/f^2/(f*g)^(1/2)*arctan(g*x/(f*g)^(1/2)))
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4 \left (f+g x^2\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4 \left (f+g x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4 \left (f+g x^2\right )} \, dx=\int { \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4 \left (f+g x^2\right )} \, dx=\int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{x^4\,\left (g\,x^2+f\right )} \,d x \]