3.2.56 \(\int \frac {(f x)^m (a+b x^n+c x^{2 n})^p}{(d+e x^n)^2} \, dx\) [156]

Optimal. Leaf size=34 \[ \text {Int}\left (\frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, 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 = {} \begin {gather*} \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{m} \left (a +b \,x^{n}+c \,x^{2 n}\right )^{p}}{\left (d +e \,x^{n}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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