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

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 3.25, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )^p}{\left (d+e x^r\right )^2} \, dx \]

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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fricas [A]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (f x\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{e^{2} x^{2 \, r} + 2 \, d e x^{r} + d^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{{\left (e x^{r} + d\right )}^{2}}\,{d x} \]

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)

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maple [A]  time = 0.88, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x \right )^{m} \left (b \ln \left (c \,x^{n}\right )+a \right )^{p}}{\left (e \,x^{r}+d \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (f\,x\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p}{{\left (d+e\,x^r\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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