3.4.56 \(\int \frac {\log (c (d+e x^2)^p)}{x^2 (f+g x^2)^2} \, dx\) [356]

3.4.56.1 Optimal result

Integrand size = 25, antiderivative size = 803 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2 \left (f+g x^2\right )^2} \, dx=\frac {2 \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f^2}-\frac {\sqrt {d} \sqrt {e} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{f^2 (e f-d g)}-\frac {e \sqrt {g} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 (-f)^{3/2} (e f-d g)}-\frac {3 \sqrt {g} 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 {3 \sqrt {g} 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 f^{5/2}}+\frac {3 \sqrt {g} 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 f^{5/2}}+\frac {e \sqrt {g} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{2 (-f)^{3/2} (e f-d g)}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f^2 x}+\frac {\sqrt {g} \log \left (c \left (d+e x^2\right )^p\right )}{4 f^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {g} \log \left (c \left (d+e x^2\right )^p\right )}{4 f^2 \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {3 \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f^{5/2}}+\frac {3 i \sqrt {g} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 f^{5/2}}-\frac {3 i \sqrt {g} 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 f^{5/2}}-\frac {3 i \sqrt {g} 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 f^{5/2}} \]

output

-ln(c*(e*x^2+d)^p)/f^2/x+2*p*arctan(x*e^(1/2)/d^(1/2))*e^(1/2)/f^2/d^(1/2) 
-g*p*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)*e^(1/2)/f^2/(-d*g+e*f)-3/2*arctan(x 
*g^(1/2)/f^(1/2))*ln(c*(e*x^2+d)^p)*g^(1/2)/f^(5/2)-1/2*e*p*ln((-f)^(1/2)- 
x*g^(1/2))*g^(1/2)/(-f)^(3/2)/(-d*g+e*f)-3*p*arctan(x*g^(1/2)/f^(1/2))*ln( 
2*f^(1/2)/(f^(1/2)-I*x*g^(1/2)))*g^(1/2)/f^(5/2)+1/2*e*p*ln((-f)^(1/2)+x*g 
^(1/2))*g^(1/2)/(-f)^(3/2)/(-d*g+e*f)+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)))*g^(1/2)/f^(5/2)+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)))*g^(1/2)/f^(5/2)+3/2*I*p*polylog(2,1-2*f 
^(1/2)/(f^(1/2)-I*x*g^(1/2)))*g^(1/2)/f^(5/2)-3/4*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^(1/2)/f^(5/2)-3/4*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^(1/2)/f^(5/2)+1/4*ln(c*(e*x^2+d)^p)*g^(1/2)/f^2/((-f)^(1/2)- 
x*g^(1/2))-1/4*ln(c*(e*x^2+d)^p)*g^(1/2)/f^2/((-f)^(1/2)+x*g^(1/2))
 

3.4.56.2 Mathematica [A] (verified)

Time = 1.08 (sec) , antiderivative size = 939, normalized size of antiderivative = 1.17 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2 \left (f+g x^2\right )^2} \, dx=\frac {\frac {8 \sqrt {e} \sqrt {f} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {2 \sqrt {-d} \sqrt {e} \sqrt {f} g p \log \left (\sqrt {-d}-\sqrt {e} x\right )}{e f-d g}+\frac {2 \sqrt {-d} \sqrt {e} \sqrt {f} g p \log \left (\sqrt {-d}+\sqrt {e} x\right )}{-e f+d g}-\frac {2 e \sqrt {-f^2} \sqrt {g} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{e f-d g}+\frac {2 e \sqrt {-f^2} \sqrt {g} p \log \left (\sqrt {-f}+\sqrt {g} x\right )}{e f-d g}+3 i \sqrt {g} 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 )+3 i \sqrt {g} 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 )-3 i \sqrt {g} 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 )-3 i \sqrt {g} 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 )-\frac {4 \sqrt {f} \log \left (c \left (d+e x^2\right )^p\right )}{x}+\frac {\sqrt {f} \sqrt {g} \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {-f}-\sqrt {g} x}-\frac {\sqrt {f} \sqrt {g} \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {-f}+\sqrt {g} x}-6 \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )+3 i \sqrt {g} 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 )+3 i \sqrt {g} 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 )-3 i \sqrt {g} 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 )-3 i \sqrt {g} 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 )}{4 f^{5/2}} \]

input

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

output

((8*Sqrt[e]*Sqrt[f]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + (2*Sqrt[-d]*S 
qrt[e]*Sqrt[f]*g*p*Log[Sqrt[-d] - Sqrt[e]*x])/(e*f - d*g) + (2*Sqrt[-d]*Sq 
rt[e]*Sqrt[f]*g*p*Log[Sqrt[-d] + Sqrt[e]*x])/(-(e*f) + d*g) - (2*e*Sqrt[-f 
^2]*Sqrt[g]*p*Log[Sqrt[-f] - Sqrt[g]*x])/(e*f - d*g) + (2*e*Sqrt[-f^2]*Sqr 
t[g]*p*Log[Sqrt[-f] + Sqrt[g]*x])/(e*f - d*g) + (3*I)*Sqrt[g]*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]] + (3*I)*Sqrt[g]*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]] - (3*I)*Sqrt[g]*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]] - (3*I)*Sqrt[g 
]*p*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]] - (4*Sqrt[f]*Log[c*(d + e*x^2)^p])/x 
 + (Sqrt[f]*Sqrt[g]*Log[c*(d + e*x^2)^p])/(Sqrt[-f] - Sqrt[g]*x) - (Sqrt[f 
]*Sqrt[g]*Log[c*(d + e*x^2)^p])/(Sqrt[-f] + Sqrt[g]*x) - 6*Sqrt[g]*ArcTan[ 
(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p] + (3*I)*Sqrt[g]*p*PolyLog[2, (Sq 
rt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])] + ( 
3*I)*Sqrt[g]*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[ 
f] + I*Sqrt[-d]*Sqrt[g])] - (3*I)*Sqrt[g]*p*PolyLog[2, (Sqrt[e]*(Sqrt[f] + 
 I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])] - (3*I)*Sqrt[g]*p*P 
olyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-...
 

3.4.56.3 Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 803, 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.

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

\(\Big \downarrow \) 2926

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {d} \sqrt {e} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{f^2 (e f-d g)}+\frac {2 \sqrt {e} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} f^2}-\frac {e \sqrt {g} p \log \left (\sqrt {-f}-\sqrt {g} x\right )}{2 (-f)^{3/2} (e f-d g)}-\frac {3 \sqrt {g} 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 {3 \sqrt {g} 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 f^{5/2}}+\frac {3 \sqrt {g} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\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 {e \sqrt {g} p \log \left (\sqrt {g} x+\sqrt {-f}\right )}{2 (-f)^{3/2} (e f-d g)}-\frac {3 \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{2 f^{5/2}}-\frac {\log \left (c \left (e x^2+d\right )^p\right )}{f^2 x}+\frac {\sqrt {g} \log \left (c \left (e x^2+d\right )^p\right )}{4 f^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {g} \log \left (c \left (e x^2+d\right )^p\right )}{4 f^2 \left (\sqrt {g} x+\sqrt {-f}\right )}+\frac {3 i \sqrt {g} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{2 f^{5/2}}-\frac {3 i \sqrt {g} 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 f^{5/2}}-\frac {3 i \sqrt {g} 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 f^{5/2}}\)

input

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

output

(2*Sqrt[e]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*f^2) - (Sqrt[d]*Sqrt[e] 
*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(f^2*(e*f - d*g)) - (e*Sqrt[g]*p*Log[Sqr 
t[-f] - Sqrt[g]*x])/(2*(-f)^(3/2)*(e*f - d*g)) - (3*Sqrt[g]*p*ArcTan[(Sqrt 
[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/f^(5/2) + (3*Sqr 
t[g]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(-2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqr 
t[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))] 
)/(2*f^(5/2)) + (3*Sqrt[g]*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sq 
rt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqr 
t[f] - I*Sqrt[g]*x))])/(2*f^(5/2)) + (e*Sqrt[g]*p*Log[Sqrt[-f] + Sqrt[g]*x 
])/(2*(-f)^(3/2)*(e*f - d*g)) - Log[c*(d + e*x^2)^p]/(f^2*x) + (Sqrt[g]*Lo 
g[c*(d + e*x^2)^p])/(4*f^2*(Sqrt[-f] - Sqrt[g]*x)) - (Sqrt[g]*Log[c*(d + e 
*x^2)^p])/(4*f^2*(Sqrt[-f] + Sqrt[g]*x)) - (3*Sqrt[g]*ArcTan[(Sqrt[g]*x)/S 
qrt[f]]*Log[c*(d + e*x^2)^p])/(2*f^(5/2)) + (((3*I)/2)*Sqrt[g]*p*PolyLog[2 
, 1 - (2*Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/f^(5/2) - (((3*I)/4)*Sqrt[g]*p 
*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqr 
t[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/f^(5/2) - (((3*I)/4)*S 
qrt[g]*p*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqr 
t[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/f^(5/2)
 

3.4.56.3.1 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]
 

3.4.56.4 Maple [F]

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

input

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

output

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

3.4.56.5 Fricas [F]

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

input

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

output

integral(log((e*x^2 + d)^p*c)/(g^2*x^6 + 2*f*g*x^4 + f^2*x^2), x)
 

3.4.56.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.4.56.7 Maxima [F(-2)]

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

input

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

3.4.56.8 Giac [F]

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

input

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

output

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

3.4.56.9 Mupad [F(-1)]

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

input

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

output

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