3.4.54 \(\int \frac {x^2 \log (c (d+e x^2)^p)}{(f+g x^2)^2} \, dx\) [354]

3.4.54.1 Optimal result

Integrand size = 25, antiderivative size = 746 \[ \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {\sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{g (e f-d g)}-\frac {e \sqrt {-f} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} g^{3/2}}-\frac {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 )}{2 \sqrt {f} g^{3/2}}-\frac {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 )}{2 \sqrt {f} g^{3/2}}+\frac {e \sqrt {-f} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} g^{3/2}}+\frac {i 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 )}{4 \sqrt {f} g^{3/2}}+\frac {i 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 )}{4 \sqrt {f} g^{3/2}} \]

output

-p*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)*e^(1/2)/g/(-d*g+e*f)-1/2*e*p*ln((-f)^ 
(1/2)-x*g^(1/2))*(-f)^(1/2)/g^(3/2)/(-d*g+e*f)+1/2*e*p*ln((-f)^(1/2)+x*g^( 
1/2))*(-f)^(1/2)/g^(3/2)/(-d*g+e*f)+1/2*arctan(x*g^(1/2)/f^(1/2))*ln(c*(e* 
x^2+d)^p)/g^(3/2)/f^(1/2)+p*arctan(x*g^(1/2)/f^(1/2))*ln(2*f^(1/2)/(f^(1/2 
)-I*x*g^(1/2)))/g^(3/2)/f^(1/2)-1/2*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)))/g^(3/2)/f^(1/2)-1/2*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)))/g^(3/2)/f^(1/2)-1/2*I*p*polylog(2,1-2*f^(1/2) 
/(f^(1/2)-I*x*g^(1/2)))/g^(3/2)/f^(1/2)+1/4*I*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)))/g^(3/2)/f^(1/2)+1/4*I*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)))/g^(3/2)/f^(1/2)+1/4*ln(c*(e*x^2+d)^p)/g^(3/2)/((-f)^(1/2)-x*g^(1/2)) 
-1/4*ln(c*(e*x^2+d)^p)/g^(3/2)/((-f)^(1/2)+x*g^(1/2))
 

3.4.54.2 Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 850, normalized size of antiderivative = 1.14 \[ \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {-d} \sqrt {e} \sqrt {g} p \log \left (\sqrt {-d}-\sqrt {e} x\right )}{e f-d g}+\frac {2 \sqrt {-d} \sqrt {e} \sqrt {g} p \log \left (\sqrt {-d}+\sqrt {e} x\right )}{-e f+d g}-\frac {2 e \sqrt {-f} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{e f-d g}+\frac {2 e \sqrt {-f} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{e f-d g}-\frac {i p \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 )}{\sqrt {f}}-\frac {i p \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 )}{\sqrt {f}}+\frac {i p \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 )}{\sqrt {f}}+\frac {i p \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 )}{\sqrt {f}}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {-f}-\sqrt {g} x}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {-f}+\sqrt {g} x}+\frac {2 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f}}-\frac {i p \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 )}{\sqrt {f}}-\frac {i p \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 )}{\sqrt {f}}+\frac {i p \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 )}{\sqrt {f}}+\frac {i p \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 )}{\sqrt {f}}}{4 g^{3/2}} \]

input

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

output

((2*Sqrt[-d]*Sqrt[e]*Sqrt[g]*p*Log[Sqrt[-d] - Sqrt[e]*x])/(e*f - d*g) + (2 
*Sqrt[-d]*Sqrt[e]*Sqrt[g]*p*Log[Sqrt[-d] + Sqrt[e]*x])/(-(e*f) + d*g) - (2 
*e*Sqrt[-f]*p*Log[Sqrt[-f] - Sqrt[g]*x])/(e*f - d*g) + (2*e*Sqrt[-f]*p*Log 
[Sqrt[-f] + Sqrt[g]*x])/(e*f - d*g) - (I*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 
]])/Sqrt[f] - (I*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]])/Sqrt[f] + (I*p*Lo 
g[(Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((-I)*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g 
])]*Log[1 + (I*Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] + (I*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]])/Sqrt[f] + Log[c*(d + e*x^2)^p]/(Sqrt[-f] - Sqrt[g]*x) - Log[c* 
(d + e*x^2)^p]/(Sqrt[-f] + Sqrt[g]*x) + (2*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log 
[c*(d + e*x^2)^p])/Sqrt[f] - (I*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] - I*Sqrt[g] 
*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])])/Sqrt[f] - (I*p*PolyLog[2, (S 
qrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])])/S 
qrt[f] + (I*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f 
] - I*Sqrt[-d]*Sqrt[g])])/Sqrt[f] + (I*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] + I* 
Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])])/Sqrt[f])/(4*g^(3/2))
 

3.4.54.3 Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 746, 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.

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

\(\Big \downarrow \) 2926

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 \sqrt {f} g^{3/2}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 \sqrt {f} g^{3/2}}-\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 \sqrt {f} g^{3/2}}-\frac {\sqrt {d} \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{g (e f-d g)}+\frac {p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} g^{3/2}}+\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {i p \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{4 \sqrt {f} g^{3/2}}+\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{4 \sqrt {f} g^{3/2}}-\frac {e \sqrt {-f} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}+\frac {e \sqrt {-f} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 g^{3/2} (e f-d g)}-\frac {i p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 \sqrt {f} g^{3/2}}\)

input

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

output

-((Sqrt[d]*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(g*(e*f - d*g))) - (e*Sq 
rt[-f]*p*Log[Sqrt[-f] - Sqrt[g]*x])/(2*g^(3/2)*(e*f - d*g)) + (p*ArcTan[(S 
qrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[f]*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 
))])/(2*Sqrt[f]*g^(3/2)) - (p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*S 
qrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sq 
rt[f] - I*Sqrt[g]*x))])/(2*Sqrt[f]*g^(3/2)) + (e*Sqrt[-f]*p*Log[Sqrt[-f] + 
 Sqrt[g]*x])/(2*g^(3/2)*(e*f - d*g)) + Log[c*(d + e*x^2)^p]/(4*g^(3/2)*(Sq 
rt[-f] - Sqrt[g]*x)) - Log[c*(d + e*x^2)^p]/(4*g^(3/2)*(Sqrt[-f] + Sqrt[g] 
*x)) + (ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/(2*Sqrt[f]*g^(3/ 
2)) - ((I/2)*p*PolyLog[2, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/(Sqrt[ 
f]*g^(3/2)) + ((I/4)*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))])/ 
(Sqrt[f]*g^(3/2)) + ((I/4)*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))])/(Sqrt[f]*g^(3/2))
 

3.4.54.3.1 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]
 

3.4.54.4 Maple [F]

\[\int \frac {x^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\left (g \,x^{2}+f \right )^{2}}d x\]

input

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

output

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

3.4.54.5 Fricas [F]

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

input

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

output

integral(x^2*log((e*x^2 + d)^p*c)/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
 

3.4.54.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.4.54.7 Maxima [F(-2)]

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

input

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

3.4.54.8 Giac [F]

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

input

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

output

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

3.4.54.9 Mupad [F(-1)]

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

input

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

output

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