\(\int \frac {x^2 \log (c (d+e x^2)^p)}{f+g x^2} \, dx\) [344]

Optimal result

Integrand size = 25, antiderivative size = 585 \[ \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=-\frac {2 p x}{g}+\frac {2 \sqrt {d} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} g}-\frac {2 \sqrt {f} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{g^{3/2}}+\frac {\sqrt {f} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{g^{3/2}}+\frac {\sqrt {f} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{g^{3/2}}+\frac {x \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {\sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}+\frac {i \sqrt {f} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{g^{3/2}}-\frac {i \sqrt {f} p \operatorname {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 g^{3/2}}-\frac {i \sqrt {f} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 g^{3/2}} \] Output:

-2*p*x/g+2*d^(1/2)*p*arctan(e^(1/2)*x/d^(1/2))/e^(1/2)/g-2*f^(1/2)*p*arcta 
n(g^(1/2)*x/f^(1/2))*ln(2*f^(1/2)/(f^(1/2)-I*g^(1/2)*x))/g^(3/2)+f^(1/2)*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^(3/2)+f^(1/2 
)*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^(3/2)+x*ln 
(c*(e*x^2+d)^p)/g-f^(1/2)*arctan(g^(1/2)*x/f^(1/2))*ln(c*(e*x^2+d)^p)/g^(3 
/2)+I*f^(1/2)*p*polylog(2,1-2*f^(1/2)/(f^(1/2)-I*g^(1/2)*x))/g^(3/2)-1/2*I 
*f^(1/2)*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^(3/2)-1/2*I*f^(1/2)* 
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^(3/2)
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.06 \[ \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {-2 \sqrt {g} p x+\frac {2 \sqrt {d} \sqrt {g} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\sqrt {g} x \log \left (c \left (d+e x^2\right )^p\right )-\sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} i \sqrt {f} p \left (\log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )+\log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )-\log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{-i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )-\log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}}\right ) \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {e} \left (\sqrt {f}+i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )\right )}{g^{3/2}} \] Input:

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

Output:

(-2*Sqrt[g]*p*x + (2*Sqrt[d]*Sqrt[g]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e 
] + Sqrt[g]*x*Log[c*(d + e*x^2)^p] - Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*L 
og[c*(d + e*x^2)^p] + (I/2)*Sqrt[f]*p*(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]*Sqr 
t[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]*Sq 
rt[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, (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*Sq 
rt[-d]*Sqrt[g])]))/g^(3/2)
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 585, 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.

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

\(\Big \downarrow \) 2926

\(\displaystyle \int \left (\frac {\log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{g \left (f+g x^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {f} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{3/2}}+\frac {\sqrt {f} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{g^{3/2}}+\frac {\sqrt {f} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{g^{3/2}}+\frac {2 \sqrt {d} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e} g}-\frac {2 \sqrt {f} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{g^{3/2}}+\frac {x \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {i \sqrt {f} p \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{2 g^{3/2}}-\frac {i \sqrt {f} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 g^{3/2}}+\frac {i \sqrt {f} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{g^{3/2}}-\frac {2 p x}{g}\)

Input:

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

Output:

(-2*p*x)/g + (2*Sqrt[d]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*g) - (2*Sq 
rt[f]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x 
)])/g^(3/2) + (Sqrt[f]*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))])/g^(3/2) + (Sqrt[f]*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]*S 
qrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g^(3/2) + (x*Log[c*(d + e*x^2)^p])/g - 
(Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/g^(3/2) + (I*Sq 
rt[f]*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(3/2) - ((I 
/2)*Sqrt[f]*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^(3/2) - 
 ((I/2)*Sqrt[f]*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^(3/ 
2)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2926

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.19 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.90

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 {x}{g}-\frac {f \arctan \left (\frac {g x}{\sqrt {g f}}\right )}{g \sqrt {g f}}\right )+\frac {p x \ln \left (e \,x^{2}+d \right )}{g}-\frac {2 p x}{g}+\frac {2 p d \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{g \sqrt {d e}}+p \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (g \,\textit {\_Z}^{2}+f \right )}{\sum }\left (-\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 g^{2} \underline {\hspace {1.25 ex}}\alpha }\right )\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 {x}{g}-\frac {f \arctan \left (\frac {g x}{\sqrt {g f}}\right )}{g \sqrt {g f}}\right )\)	\(525\)

Input:

int(x^2*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))*(x/g-f/g/(g*f)^(1/2)*arctan(g*x/(g*f)^(1/2 
)))+p*x/g*ln(e*x^2+d)-2*p*x/g+2*p/g*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*_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))+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*_alpha*e*g+d*g-e*f,index=2)))/e+1/2*(di 
log((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,in 
dex=2)))/e))*f/g^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))*(x/g-f/g/(g*f)^(1/2)*arctan(g*x/(g*f)^(1/2)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(x**2*log(c*(d + e*x**2)**p)/(f + g*x**2), x)
 

Maxima [F(-2)]

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

integrate(x^2*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
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {2 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) p -\left (\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}{g \,x^{2}+f}d x \right ) e f +\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e x -2 e p x}{e g} \] Input:

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

Output:

(2*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*p - int(log((d + e*x**2)* 
*p*c)/(f + g*x**2),x)*e*f + log((d + e*x**2)**p*c)*e*x - 2*e*p*x)/(e*g)