3.287 \(\int \frac {1}{(f+g x^2)^2 \log ^2(c (d+e x^2)^p)} \, dx\)

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{\left (f+g x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )},x\right ) \]

[Out]

Unintegrable(1/(g*x^2+f)^2/ln(c*(e*x^2+d)^p)^2,x)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (f+g x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((f + g*x^2)^2*Log[c*(d + e*x^2)^p]^2),x]

[Out]

Defer[Int][1/((f + g*x^2)^2*Log[c*(d + e*x^2)^p]^2), x]

Rubi steps

\begin {align*} \int \frac {1}{\left (f+g x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx &=\int \frac {1}{\left (f+g x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 7.73, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (f+g x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((f + g*x^2)^2*Log[c*(d + e*x^2)^p]^2),x]

[Out]

Integrate[1/((f + g*x^2)^2*Log[c*(d + e*x^2)^p]^2), x]

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (g^{2} x^{4} + 2 \, f g x^{2} + f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x^2+f)^2/log(c*(e*x^2+d)^p)^2,x, algorithm="fricas")

[Out]

integral(1/((g^2*x^4 + 2*f*g*x^2 + f^2)*log((e*x^2 + d)^p*c)^2), x)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (g x^{2} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x^2+f)^2/log(c*(e*x^2+d)^p)^2,x, algorithm="giac")

[Out]

integrate(1/((g*x^2 + f)^2*log((e*x^2 + d)^p*c)^2), x)

________________________________________________________________________________________

maple [A]  time = 4.73, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (g \,x^{2}+f \right )^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(g*x^2+f)^2/ln(c*(e*x^2+d)^p)^2,x)

[Out]

int(1/(g*x^2+f)^2/ln(c*(e*x^2+d)^p)^2,x)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {e x^{2} + d}{2 \, {\left (e g^{2} p x^{5} \log \relax (c) + 2 \, e f g p x^{3} \log \relax (c) + e f^{2} p x \log \relax (c) + {\left (e g^{2} p^{2} x^{5} + 2 \, e f g p^{2} x^{3} + e f^{2} p^{2} x\right )} \log \left (e x^{2} + d\right )\right )}} - \int \frac {3 \, e g x^{4} - {\left (e f - 5 \, d g\right )} x^{2} + d f}{2 \, {\left (e g^{3} p x^{8} \log \relax (c) + 3 \, e f g^{2} p x^{6} \log \relax (c) + 3 \, e f^{2} g p x^{4} \log \relax (c) + e f^{3} p x^{2} \log \relax (c) + {\left (e g^{3} p^{2} x^{8} + 3 \, e f g^{2} p^{2} x^{6} + 3 \, e f^{2} g p^{2} x^{4} + e f^{3} p^{2} x^{2}\right )} \log \left (e x^{2} + d\right )\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x^2+f)^2/log(c*(e*x^2+d)^p)^2,x, algorithm="maxima")

[Out]

-1/2*(e*x^2 + d)/(e*g^2*p*x^5*log(c) + 2*e*f*g*p*x^3*log(c) + e*f^2*p*x*log(c) + (e*g^2*p^2*x^5 + 2*e*f*g*p^2*
x^3 + e*f^2*p^2*x)*log(e*x^2 + d)) - integrate(1/2*(3*e*g*x^4 - (e*f - 5*d*g)*x^2 + d*f)/(e*g^3*p*x^8*log(c) +
3*e*f*g^2*p*x^6*log(c) + 3*e*f^2*g*p*x^4*log(c) + e*f^3*p*x^2*log(c) + (e*g^3*p^2*x^8 + 3*e*f*g^2*p^2*x^6 + 3
*e*f^2*g*p^2*x^4 + e*f^3*p^2*x^2)*log(e*x^2 + d)), x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (g\,x^2+f\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(log(c*(d + e*x^2)^p)^2*(f + g*x^2)^2),x)

[Out]

int(1/(log(c*(d + e*x^2)^p)^2*(f + g*x^2)^2), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x**2+f)**2/ln(c*(e*x**2+d)**p)**2,x)

[Out]

Timed out

________________________________________________________________________________________