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

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

[Out]

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

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Rubi [A]  time = 0.0240136, 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 = {} \[ \int \frac{\left (f+g x^2\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\left (f+g x^2\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx &=\int \frac{\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 = 0.864631, size = 0, normalized size = 0. \[ \int \frac{\left (f+g x^2\right )^2}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 3.822, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( g{x}^{2}+f \right ) ^{2}}{ \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((g*x^2+f)^2/ln(c*(e*x^2+d)^p)^2,x)

[Out]

int((g*x^2+f)^2/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 g^{2} x^{6} +{\left (2 \, e f g + d g^{2}\right )} x^{4} + d f^{2} +{\left (e f^{2} + 2 \, d f g\right )} x^{2}}{2 \,{\left (e p x \log \left ({\left (e x^{2} + d\right )}^{p}\right ) + e p x \log \left (c\right )\right )}} + \int \frac{5 \, e g^{2} x^{6} + 3 \,{\left (2 \, e f g + d g^{2}\right )} x^{4} - d f^{2} +{\left (e f^{2} + 2 \, d f g\right )} x^{2}}{2 \,{\left (e p x^{2} \log \left ({\left (e x^{2} + d\right )}^{p}\right ) + e p x^{2} \log \left (c\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}{\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((g*x^2+f)^2/log(c*(e*x^2+d)^p)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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