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

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{1}{\left (f+g x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0263288, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (f+g x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.56145, size = 0, normalized size = 0. \[ \int \frac{1}{\left (f+g x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 4.606, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( g{x}^{2}+f \right ) \left ( \ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{e x^{2} + d}{2 \,{\left (e g p x^{3} \log \left (c\right ) + e f p x \log \left (c\right ) +{\left (e g p x^{3} + e f p x\right )} \log \left ({\left (e x^{2} + d\right )}^{p}\right )\right )}} - \int \frac{e g x^{4} -{\left (e f - 3 \, d g\right )} x^{2} + d f}{2 \,{\left (e g^{2} p x^{6} \log \left (c\right ) + 2 \, e f g p x^{4} \log \left (c\right ) + e f^{2} p x^{2} \log \left (c\right ) +{\left (e g^{2} p x^{6} + 2 \, e f g p x^{4} + e f^{2} p x^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p}\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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