3.113 \(\int \frac{x^2}{\log ^2(c (a+b x^2)^p)} \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{x^2}{\log ^2\left (c \left (a+b x^2\right )^p\right )},x\right ) \]

[Out]

Unintegrable[x^2/Log[c*(a + b*x^2)^p]^2, x]

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Rubi [A]  time = 0.0168387, 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 = {} \[ \int \frac{x^2}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2/Log[c*(a + b*x^2)^p]^2,x]

[Out]

Defer[Int][x^2/Log[c*(a + b*x^2)^p]^2, x]

Rubi steps

\begin{align*} \int \frac{x^2}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx &=\int \frac{x^2}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.331371, size = 0, normalized size = 0. \[ \int \frac{x^2}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^2/Log[c*(a + b*x^2)^p]^2,x]

[Out]

Integrate[x^2/Log[c*(a + b*x^2)^p]^2, x]

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Maple [A]  time = 3.284, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{ \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/ln(c*(b*x^2+a)^p)^2,x)

[Out]

int(x^2/ln(c*(b*x^2+a)^p)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/log(c*(b*x^2+a)^p)^2,x, algorithm="maxima")

[Out]

-1/2*(b*x^3 + a*x)/(b*p*log((b*x^2 + a)^p) + b*p*log(c)) + integrate(1/2*(3*b*x^2 + a)/(b*p*log((b*x^2 + a)^p)
+ b*p*log(c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/log(c*(b*x^2+a)^p)^2,x, algorithm="fricas")

[Out]

integral(x^2/log((b*x^2 + a)^p*c)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/ln(c*(b*x**2+a)**p)**2,x)

[Out]

Integral(x**2/log(c*(a + b*x**2)**p)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/log(c*(b*x^2+a)^p)^2,x, algorithm="giac")

[Out]

integrate(x^2/log((b*x^2 + a)^p*c)^2, x)