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

Optimal. Leaf size=194 \[ \frac{x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2 n};-p,1;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d (m+1)}-\frac{e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+n+1}{2 n};-p,1;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (m+n+1)} \]

[Out]

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

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Rubi [A]  time = 0.504238, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+1}{2 n};-p,1;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d (m+1)}-\frac{e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{m+n+1}{2 n};-p,1;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (m+n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 88.1715, size = 162, normalized size = 0.84 \[ \frac{x^{- m} x^{m + 1} \left (f x\right )^{m} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m + 1}{2 n},1,- p,1 + \frac{m + 1}{2 n},\frac{e^{2} x^{2 n}}{d^{2}},- \frac{c x^{2 n}}{a} \right )}}{d \left (m + 1\right )} - \frac{e x^{- m} x^{m + n + 1} \left (f x\right )^{m} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m + n + 1}{2 n},1,- p,\frac{m + 3 n + 1}{2 n},\frac{e^{2} x^{2 n}}{d^{2}},- \frac{c x^{2 n}}{a} \right )}}{d^{2} \left (m + n + 1\right )} \]

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

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Mathematica [A]  time = 0.095658, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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

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),x)

[Out]

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

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

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),x, algorithm="maxima")

[Out]

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

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

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),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

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),x, algorithm="giac")

[Out]

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