3.58 \(\int \frac{1}{\left (d+e x^n\right ) \sqrt{a+c x^{2 n}}} \, dx\)

Optimal. Leaf size=171 \[ \frac{x \sqrt{\frac{c x^{2 n}}{a}+1} F_1\left (\frac{1}{2 n};\frac{1}{2},1;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d \sqrt{a+c x^{2 n}}}-\frac{e x^{n+1} \sqrt{\frac{c x^{2 n}}{a}+1} F_1\left (\frac{n+1}{2 n};\frac{1}{2},1;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1) \sqrt{a+c x^{2 n}}} \]

[Out]

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

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Rubi [A]  time = 0.425131, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{x \sqrt{\frac{c x^{2 n}}{a}+1} F_1\left (\frac{1}{2 n};\frac{1}{2},1;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d \sqrt{a+c x^{2 n}}}-\frac{e x^{n+1} \sqrt{\frac{c x^{2 n}}{a}+1} F_1\left (\frac{n+1}{2 n};\frac{1}{2},1;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1) \sqrt{a+c x^{2 n}}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x^n)*Sqrt[a + c*x^(2*n)]),x]

[Out]

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

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Rubi in Sympy [A]  time = 73.6189, size = 138, normalized size = 0.81 \[ \frac{x \sqrt{a + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{1}{2 n},\frac{1}{2},1,\frac{n + \frac{1}{2}}{n},- \frac{c x^{2 n}}{a},\frac{e^{2} x^{2 n}}{d^{2}} \right )}}{a d \sqrt{1 + \frac{c x^{2 n}}{a}}} - \frac{e x^{n + 1} \sqrt{a + c x^{2 n}} \operatorname{appellf_{1}}{\left (\frac{n + 1}{2 n},\frac{1}{2},1,\frac{3 n + 1}{2 n},- \frac{c x^{2 n}}{a},\frac{e^{2} x^{2 n}}{d^{2}} \right )}}{a d^{2} \sqrt{1 + \frac{c x^{2 n}}{a}} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d+e*x**n)/(a+c*x**(2*n))**(1/2),x)

[Out]

x*sqrt(a + c*x**(2*n))*appellf1(1/(2*n), 1/2, 1, (n + 1/2)/n, -c*x**(2*n)/a, e**
2*x**(2*n)/d**2)/(a*d*sqrt(1 + c*x**(2*n)/a)) - e*x**(n + 1)*sqrt(a + c*x**(2*n)
)*appellf1((n + 1)/(2*n), 1/2, 1, (3*n + 1)/(2*n), -c*x**(2*n)/a, e**2*x**(2*n)/
d**2)/(a*d**2*sqrt(1 + c*x**(2*n)/a)*(n + 1))

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Mathematica [A]  time = 0.120825, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x^n\right ) \sqrt{a+c x^{2 n}}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/((d + e*x^n)*Sqrt[a + c*x^(2*n)]),x]

[Out]

Integrate[1/((d + e*x^n)*Sqrt[a + c*x^(2*n)]), x]

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{1}{d+e{x}^{n}}{\frac{1}{\sqrt{a+c{x}^{2\,n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d+e*x^n)/(a+c*x^(2*n))^(1/2),x)

[Out]

int(1/(d+e*x^n)/(a+c*x^(2*n))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2 \, n} + a}{\left (e x^{n} + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^(2*n) + a)*(e*x^n + d)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^(2*n) + a)*(e*x^n + d)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{c x^{2 \, n} + a}{\left (e x^{n} + d\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^(2*n) + a)*(e*x^n + d)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(c*x^(2*n) + a)*(e*x^n + d)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + c x^{2 n}} \left (d + e x^{n}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d+e*x**n)/(a+c*x**(2*n))**(1/2),x)

[Out]

Integral(1/(sqrt(a + c*x**(2*n))*(d + e*x**n)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2 \, n} + a}{\left (e x^{n} + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^(2*n) + a)*(e*x^n + d)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^(2*n) + a)*(e*x^n + d)), x)