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

Optimal. Leaf size=302 \[ \frac{x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2 n};-p,2;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (m+1)}+\frac{e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+2 n+1}{2 n};-p,2;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^4 (m+2 n+1)}-\frac{2 e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+n+1}{2 n};-p,2;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^3 (m+n+1)} \]

[Out]

(x*(f*x)^m*(a + c*x^(2*n))^p*AppellF1[(1 + m)/(2*n), -p, 2, 1 + (1 + m)/(2*n), -
((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^2*(1 + m)*(1 + (c*x^(2*n))/a)^p) - (2*e*
x^(1 + n)*(f*x)^m*(a + c*x^(2*n))^p*AppellF1[(1 + m + n)/(2*n), -p, 2, (1 + m +
3*n)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^3*(1 + m + n)*(1 + (c*x^(2*
n))/a)^p) + (e^2*x^(1 + 2*n)*(f*x)^m*(a + c*x^(2*n))^p*AppellF1[(1 + m + 2*n)/(2
*n), -p, 2, (1 + m + 4*n)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^4*(1 +
m + 2*n)*(1 + (c*x^(2*n))/a)^p)

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Rubi [A]  time = 0.763223, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2 n};-p,2;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (m+1)}+\frac{e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+2 n+1}{2 n};-p,2;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^4 (m+2 n+1)}-\frac{2 e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+n+1}{2 n};-p,2;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^3 (m+n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(f*x)^m*(a + c*x^(2*n))^p*AppellF1[(1 + m)/(2*n), -p, 2, 1 + (1 + m)/(2*n), -
((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^2*(1 + m)*(1 + (c*x^(2*n))/a)^p) - (2*e*
x^(1 + n)*(f*x)^m*(a + c*x^(2*n))^p*AppellF1[(1 + m + n)/(2*n), -p, 2, (1 + m +
3*n)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^3*(1 + m + n)*(1 + (c*x^(2*
n))/a)^p) + (e^2*x^(1 + 2*n)*(f*x)^m*(a + c*x^(2*n))^p*AppellF1[(1 + m + 2*n)/(2
*n), -p, 2, (1 + m + 4*n)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^4*(1 +
m + 2*n)*(1 + (c*x^(2*n))/a)^p)

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Rubi in Sympy [A]  time = 155.763, size = 258, normalized size = 0.85 \[ \frac{x^{- m} x^{m + 1} \left (f x\right )^{m} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m + 1}{2 n},2,- p,1 + \frac{m + 1}{2 n},\frac{e^{2} x^{2 n}}{d^{2}},- \frac{c x^{2 n}}{a} \right )}}{d^{2} \left (m + 1\right )} - \frac{2 e x^{- m} x^{m + n + 1} \left (f x\right )^{m} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m + n + 1}{2 n},2,- p,\frac{m + 3 n + 1}{2 n},\frac{e^{2} x^{2 n}}{d^{2}},- \frac{c x^{2 n}}{a} \right )}}{d^{3} \left (m + n + 1\right )} + \frac{e^{2} x^{- m} x^{m + 2 n + 1} \left (f x\right )^{m} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m + 2 n + 1}{2 n},2,- p,\frac{m + 4 n + 1}{2 n},\frac{e^{2} x^{2 n}}{d^{2}},- \frac{c x^{2 n}}{a} \right )}}{d^{4} \left (m + 2 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(-m)*x**(m + 1)*(f*x)**m*(1 + c*x**(2*n)/a)**(-p)*(a + c*x**(2*n))**p*appellf
1((m + 1)/(2*n), 2, -p, 1 + (m + 1)/(2*n), e**2*x**(2*n)/d**2, -c*x**(2*n)/a)/(d
**2*(m + 1)) - 2*e*x**(-m)*x**(m + n + 1)*(f*x)**m*(1 + c*x**(2*n)/a)**(-p)*(a +
c*x**(2*n))**p*appellf1((m + n + 1)/(2*n), 2, -p, (m + 3*n + 1)/(2*n), e**2*x**
(2*n)/d**2, -c*x**(2*n)/a)/(d**3*(m + n + 1)) + e**2*x**(-m)*x**(m + 2*n + 1)*(f
*x)**m*(1 + c*x**(2*n)/a)**(-p)*(a + c*x**(2*n))**p*appellf1((m + 2*n + 1)/(2*n)
, 2, -p, (m + 4*n + 1)/(2*n), e**2*x**(2*n)/d**2, -c*x**(2*n)/a)/(d**4*(m + 2*n
+ 1))

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Mathematica [A]  time = 0.195435, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.131, size = 0, normalized size = 0. \[ \int{\frac{ \left ( fx \right ) ^{m} \left ( a+c{x}^{2\,n} \right ) ^{p}}{ \left ( d+e{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^(2*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+c*x**(2*n))**p/(d+e*x**n)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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