Integrand size = 25, antiderivative size = 667 \[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {2 f p x}{g^2}+\frac {2 d p x}{3 e g}-\frac {2 p x^3}{9 g}-\frac {2 \sqrt {d} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} g^2}-\frac {2 d^{3/2} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2} g}+\frac {2 f^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{g^{5/2}}-\frac {f^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{g^{5/2}}-\frac {f^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{g^{5/2}}-\frac {f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac {f^{3/2} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}-\frac {i f^{3/2} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{g^{5/2}}+\frac {i f^{3/2} p \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 g^{5/2}}+\frac {i f^{3/2} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 g^{5/2}} \] Output:
2*f*p*x/g^2+2/3*d*p*x/e/g-2/9*p*x^3/g-2*d^(1/2)*f*p*arctan(e^(1/2)*x/d^(1/ 2))/e^(1/2)/g^2-2/3*d^(3/2)*p*arctan(e^(1/2)*x/d^(1/2))/e^(3/2)/g+2*f^(3/2 )*p*arctan(g^(1/2)*x/f^(1/2))*ln(2*f^(1/2)/(f^(1/2)-I*g^(1/2)*x))/g^(5/2)- f^(3/2)*p*arctan(g^(1/2)*x/f^(1/2))*ln(-2*f^(1/2)*g^(1/2)*((-d)^(1/2)-e^(1 /2)*x)/(I*e^(1/2)*f^(1/2)-(-d)^(1/2)*g^(1/2))/(f^(1/2)-I*g^(1/2)*x))/g^(5/ 2)-f^(3/2)*p*arctan(g^(1/2)*x/f^(1/2))*ln(2*f^(1/2)*g^(1/2)*((-d)^(1/2)+e^ (1/2)*x)/(I*e^(1/2)*f^(1/2)+(-d)^(1/2)*g^(1/2))/(f^(1/2)-I*g^(1/2)*x))/g^( 5/2)-f*x*ln(c*(e*x^2+d)^p)/g^2+1/3*x^3*ln(c*(e*x^2+d)^p)/g+f^(3/2)*arctan( g^(1/2)*x/f^(1/2))*ln(c*(e*x^2+d)^p)/g^(5/2)-I*f^(3/2)*p*polylog(2,1-2*f^( 1/2)/(f^(1/2)-I*g^(1/2)*x))/g^(5/2)+1/2*I*f^(3/2)*p*polylog(2,1+2*f^(1/2)* g^(1/2)*((-d)^(1/2)-e^(1/2)*x)/(I*e^(1/2)*f^(1/2)-(-d)^(1/2)*g^(1/2))/(f^( 1/2)-I*g^(1/2)*x))/g^(5/2)+1/2*I*f^(3/2)*p*polylog(2,1-2*f^(1/2)*g^(1/2)*( (-d)^(1/2)+e^(1/2)*x)/(I*e^(1/2)*f^(1/2)+(-d)^(1/2)*g^(1/2))/(f^(1/2)-I*g^ (1/2)*x))/g^(5/2)
Time = 0.68 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.04 \[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {2 f p x}{g^2}+\frac {2 d p x}{3 e g}-\frac {2 p x^3}{9 g}-\frac {2 \sqrt {d} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} g^2}-\frac {2 d^{3/2} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2} g}-\frac {f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac {f^{3/2} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}-\frac {i f^{3/2} p \left (\log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )+\log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )-\log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )-\log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )\right )}{2 g^{5/2}} \] Input:
Integrate[(x^4*Log[c*(d + e*x^2)^p])/(f + g*x^2),x]
Output:
(2*f*p*x)/g^2 + (2*d*p*x)/(3*e*g) - (2*p*x^3)/(9*g) - (2*Sqrt[d]*f*p*ArcTa n[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*g^2) - (2*d^(3/2)*p*ArcTan[(Sqrt[e]*x)/Sq rt[d]])/(3*e^(3/2)*g) - (f*x*Log[c*(d + e*x^2)^p])/g^2 + (x^3*Log[c*(d + e *x^2)^p])/(3*g) + (f^(3/2)*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p ])/g^(5/2) - ((I/2)*f^(3/2)*p*(Log[(Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/(I*Sqr t[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] + Log[(Sq rt[g]*(Sqrt[-d] + Sqrt[e]*x))/((-I)*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*L og[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]] + 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, (Sqr t[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])] - Po lyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*S qrt[g])]))/g^(5/2)
Time = 1.63 (sec) , antiderivative size = 667, 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.
| \(\displaystyle \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx\) |
| \(\Big \downarrow \) 2926 |
| \(\displaystyle \int \left (\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{g^2 \left (f+g x^2\right )}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{g}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f^{3/2} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}-\frac {2 d^{3/2} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2} g}-\frac {f^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{g^{5/2}}-\frac {f^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{g^{5/2}}-\frac {2 \sqrt {d} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} g^2}+\frac {2 f^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{g^{5/2}}-\frac {f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac {i f^{3/2} p \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{2 g^{5/2}}+\frac {i f^{3/2} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 g^{5/2}}+\frac {2 d p x}{3 e g}-\frac {i f^{3/2} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{g^{5/2}}+\frac {2 f p x}{g^2}-\frac {2 p x^3}{9 g}\) |
Input:
Int[(x^4*Log[c*(d + e*x^2)^p])/(f + g*x^2),x]
Output:
(2*f*p*x)/g^2 + (2*d*p*x)/(3*e*g) - (2*p*x^3)/(9*g) - (2*Sqrt[d]*f*p*ArcTa n[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*g^2) - (2*d^(3/2)*p*ArcTan[(Sqrt[e]*x)/Sq rt[d]])/(3*e^(3/2)*g) + (2*f^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sq rt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) - (f^(3/2)*p*ArcTan[(Sqrt[g]*x)/S qrt[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))])/g^(5/2) - (f^(3/2)*p*ArcT an[(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))])/g^(5/2) - (f*x*Log[c*(d + e*x^2)^p])/g^2 + (x^3*Log[c*(d + e*x^2)^p])/(3*g) + (f^(3/ 2)*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/g^(5/2) - (I*f^(3/2)* p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) + ((I/2)*f^ (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))])/g^(5/2) + ((I/2 )*f^(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))])/g^(5/2)
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 = 2.84 (sec) , antiderivative size = 616, 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 {\frac {1}{3} g \,x^{3}-f x}{g^{2}}+\frac {f^{2} \arctan \left (\frac {g x}{\sqrt {g f}}\right )}{g^{2} \sqrt {g f}}\right )+\frac {p \,x^{3} \ln \left (e \,x^{2}+d \right )}{3 g}-\frac {2 p \,x^{3}}{9 g}+\frac {2 d p x}{3 e g}-\frac {2 p \,d^{2} \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{3 g e \sqrt {d e}}-\frac {p f x \ln \left (e \,x^{2}+d \right )}{g^{2}}+\frac {2 f p x}{g^{2}}-\frac {2 p f d \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{g^{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 (\textit {\_Z}^{2} e 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 (\textit {\_Z}^{2} e 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 (\textit {\_Z}^{2} e 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 (\textit {\_Z}^{2} e 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 (\textit {\_Z}^{2} e 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 (\textit {\_Z}^{2} e 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 (\textit {\_Z}^{2} e 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 (\textit {\_Z}^{2} e 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 ) f^{2}}{2 g^{3} \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 {\frac {1}{3} g \,x^{3}-f x}{g^{2}}+\frac {f^{2} \arctan \left (\frac {g x}{\sqrt {g f}}\right )}{g^{2} \sqrt {g f}}\right )\) | \(616\) |
Input:
int(x^4*ln(c*(e*x^2+d)^p)/(g*x^2+f),x,method=_RETURNVERBOSE)
Output:
(ln((e*x^2+d)^p)-p*ln(e*x^2+d))*(1/g^2*(1/3*g*x^3-f*x)+f^2/g^2/(g*f)^(1/2) *arctan(g*x/(g*f)^(1/2)))+1/3*p/g*x^3*ln(e*x^2+d)-2/9*p*x^3/g+2/3*d*p*x/e/ g-2/3*p/g*d^2/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-p*f/g^2*x*ln(e*x^2+d)+ 2*f*p*x/g^2-2*p*f/g^2*d/(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*_alp ha*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)/R ootOf(_Z^2*e*g+2*_Z*_alpha*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*_alp ha*e*g+d*g-e*f,index=1))+dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,in dex=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)))/e))*f^ 2/g^3/_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*csg n(I*c)+ln(c))*(1/g^2*(1/3*g*x^3-f*x)+f^2/g^2/(g*f)^(1/2)*arctan(g*x/(g*f)^ (1/2)))
\[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{4} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \] Input:
integrate(x^4*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="fricas")
Output:
integral(x^4*log((e*x^2 + d)^p*c)/(g*x^2 + f), x)
\[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {x^{4} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{f + g x^{2}}\, dx \] Input:
integrate(x**4*ln(c*(e*x**2+d)**p)/(g*x**2+f),x)
Output:
Integral(x**4*log(c*(d + e*x**2)**p)/(f + g*x**2), x)
Exception generated. \[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^4*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="maxima")
Output:
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
\[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{4} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \] Input:
integrate(x^4*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="giac")
Output:
integrate(x^4*log((e*x^2 + d)^p*c)/(g*x^2 + f), x)
Timed out. \[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {x^4\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^2+f} \,d x \] Input:
int((x^4*log(c*(d + e*x^2)^p))/(f + g*x^2),x)
Output:
int((x^4*log(c*(d + e*x^2)^p))/(f + g*x^2), x)
\[ \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {-6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d g p -18 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) e f p +9 \left (\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}{g \,x^{2}+f}d x \right ) e^{2} f^{2}-9 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{2} f x +3 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{2} g \,x^{3}+6 d e g p x +18 e^{2} f p x -2 e^{2} g p \,x^{3}}{9 e^{2} g^{2}} \] Input:
int(x^4*log(c*(e*x^2+d)^p)/(g*x^2+f),x)
Output:
( - 6*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d*g*p - 18*sqrt(e)*sqr t(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*e*f*p + 9*int(log((d + e*x**2)**p*c)/(f + g*x**2),x)*e**2*f**2 - 9*log((d + e*x**2)**p*c)*e**2*f*x + 3*log((d + e *x**2)**p*c)*e**2*g*x**3 + 6*d*e*g*p*x + 18*e**2*f*p*x - 2*e**2*g*p*x**3)/ (9*e**2*g**2)