3.156 \(\int \frac{(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx\)

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 = 0.0704311, 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. \[ \text{Int}\left (\frac{(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2},x\right ) \]

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]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.26374, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+b x^n+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]

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 = 0.122, size = 0, normalized size = 0. \[ \int{\frac{ \left ( fx \right ) ^{m} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{p}}{ \left ( d+e{x}^{n} \right ) ^{2}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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