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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{(f x)^m}{\log ^2\left (c \left (d+e x^2\right )^p\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.017193, 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 = {} \[ \int \frac{(f x)^m}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(f*x)^m/Log[c*(d + e*x^2)^p]^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.529331, size = 0, normalized size = 0. \[ \int \frac{(f x)^m}{\log ^2\left (c \left (d+e x^2\right )^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 5.842, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m}}{ \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((f*x)^m/ln(c*(e*x^2+d)^p)^2,x)

[Out]

int((f*x)^m/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{{\left (e f^{m} x^{2} + d f^{m}\right )} x^{m}}{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{{\left (e f^{m}{\left (m + 1\right )} x^{2} + d f^{m}{\left (m - 1\right )}\right )} x^{m}}{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((f*x)^m/log(c*(e*x^2+d)^p)^2,x, algorithm="maxima")

[Out]

-1/2*(e*f^m*x^2 + d*f^m)*x^m/(e*p*x*log((e*x^2 + d)^p) + e*p*x*log(c)) + integrate(1/2*(e*f^m*(m + 1)*x^2 + d*
f^m*(m - 1))*x^m/(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{\left (f x\right )^{m}}{\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((f*x)^m/log(c*(e*x^2+d)^p)^2,x, algorithm="fricas")

[Out]

integral((f*x)^m/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((f*x)**m/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{\left (f x\right )^{m}}{\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((f*x)^m/log(c*(e*x^2+d)^p)^2,x, algorithm="giac")

[Out]

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