3.235 \(\int \left (\frac{\sqrt{a x^{2 n}}}{\sqrt{1+x^n}}+\frac{2 x^{-n} \sqrt{a x^{2 n}}}{(2+n) \sqrt{1+x^n}}\right ) \, dx\)

Optimal. Leaf size=34 \[ \frac{2 x^{1-n} \sqrt{x^n+1} \sqrt{a x^{2 n}}}{n+2} \]

[Out]

(2*x^(1 - n)*Sqrt[a*x^(2*n)]*Sqrt[1 + x^n])/(2 + n)

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Rubi [C]  time = 0.0645534, antiderivative size = 80, normalized size of antiderivative = 2.35, number of steps used = 5, number of rules used = 3, integrand size = 54, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{2 x^{1-n} \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-x^n\right )}{n+2}+\frac{x \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right )}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a*x^(2*n)]*Hypergeometric2F1[1/2, 1 + n^(-1), 2 + n^(-1), -x^n])/(1 + n)
+ (2*x^(1 - n)*Sqrt[a*x^(2*n)]*Hypergeometric2F1[1/2, n^(-1), 1 + n^(-1), -x^n]
)/(2 + n)

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Rubi in Sympy [A]  time = 21.9847, size = 70, normalized size = 2.06 \[ \frac{2 x x^{- n} \sqrt{a x^{2 n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- x^{n}} \right )}}{n + 2} + \frac{x^{- n} x^{n + 1} \sqrt{a x^{2 n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{n + 1}{n} \\ 2 + \frac{1}{n} \end{matrix}\middle |{- x^{n}} \right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x*x**(-n)*sqrt(a*x**(2*n))*hyper((1/2, 1/n), (1 + 1/n,), -x**n)/(n + 2) + x**(
-n)*x**(n + 1)*sqrt(a*x**(2*n))*hyper((1/2, (n + 1)/n), (2 + 1/n,), -x**n)/(n +
1)

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Mathematica [A]  time = 0.0587505, size = 33, normalized size = 0.97 \[ \frac{2 a x^{n+1} \sqrt{x^n+1}}{(n+2) \sqrt{a x^{2 n}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*x^(1 + n)*Sqrt[1 + x^n])/((2 + n)*Sqrt[a*x^(2*n)])

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Maple [A]  time = 0.037, size = 30, normalized size = 0.9 \[ 2\,{\frac{x\sqrt{1+{x}^{n}}\sqrt{a \left ({x}^{n} \right ) ^{2}}}{ \left ( 2+n \right ){x}^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x*(1+x^n)^(1/2)/(2+n)*(a*(x^n)^2)^(1/2)/(x^n)

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Maxima [A]  time = 0.843897, size = 24, normalized size = 0.71 \[ \frac{2 \, \sqrt{a} \sqrt{x^{n} + 1} x}{n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(a)*sqrt(x^n + 1)*x/(n + 2)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{2 \sqrt{a x^{2 n}}}{\sqrt{x^{n} + 1}}\, dx + \int \frac{n \sqrt{a x^{2 n}}}{\sqrt{x^{n} + 1}}\, dx + \int \frac{2 x^{- n} \sqrt{a x^{2 n}}}{\sqrt{x^{n} + 1}}\, dx}{n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(Integral(2*sqrt(a*x**(2*n))/sqrt(x**n + 1), x) + Integral(n*sqrt(a*x**(2*n))/sq
rt(x**n + 1), x) + Integral(2*x**(-n)*sqrt(a*x**(2*n))/sqrt(x**n + 1), x))/(n +
2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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