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