3.453 \(\int \frac{(f x)^m (a+b \log (c x^n))^p}{(d+e x^r)^2} \, dx\)

Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.101942, 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 = {} \[ \int \frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((f*x)^m*(a + b*Log[c*x^n])^p)/(d + e*x^r)^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.97245, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.881, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p}}{ \left ( d+e{x}^{r} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(a+b*ln(c*x^n))^p/(d+e*x^r)^2,x)

[Out]

int((f*x)^m*(a+b*ln(c*x^n))^p/(d+e*x^r)^2,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*log(c*x^n))^p/(d+e*x^r)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*log(c*x^n))^p/(d+e*x^r)^2,x, algorithm="fricas")

[Out]

integral((f*x)^m*(b*log(c*x^n) + a)^p/(e^2*x^(2*r) + 2*d*e*x^r + d^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*(a+b*ln(c*x**n))**p/(d+e*x**r)**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}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{{\left (e x^{r} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*log(c*x^n))^p/(d+e*x^r)^2,x, algorithm="giac")

[Out]

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