3.234 \(\int \frac{\sqrt{a x^{n/2}}}{\sqrt{1+x^n}} \, dx\)

Optimal. Leaf size=52 \[ \frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left (\frac{1}{2},\frac{1}{4} \left (1+\frac{4}{n}\right );\frac{1}{4} \left (5+\frac{4}{n}\right );-x^n\right )}{n+4} \]

[Out]

(4*x*Sqrt[a*x^(n/2)]*Hypergeometric2F1[1/2, (1 + 4/n)/4, (5 + 4/n)/4, -x^n])/(4
+ n)

_______________________________________________________________________________________

Rubi [A]  time = 0.0361213, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left (\frac{1}{2},\frac{1}{4} \left (1+\frac{4}{n}\right );\frac{1}{4} \left (5+\frac{4}{n}\right );-x^n\right )}{n+4} \]

Antiderivative was successfully verified.

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

[Out]

(4*x*Sqrt[a*x^(n/2)]*Hypergeometric2F1[1/2, (1 + 4/n)/4, (5 + 4/n)/4, -x^n])/(4
+ n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.07168, size = 44, normalized size = 0.85 \[ \frac{4 x^{- \frac{n}{4}} x^{\frac{n}{4} + 1} \sqrt{a x^{\frac{n}{2}}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{n + 4}{4 n} \\ \frac{5}{4} + \frac{1}{n} \end{matrix}\middle |{- x^{n}} \right )}}{n + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*x**(-n/4)*x**(n/4 + 1)*sqrt(a*x**(n/2))*hyper((1/2, (n + 4)/(4*n)), (5/4 + 1/n
,), -x**n)/(n + 4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0293149, size = 44, normalized size = 0.85 \[ \frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left (\frac{1}{2},\frac{1}{4}+\frac{1}{n};\frac{5}{4}+\frac{1}{n};-x^n\right )}{n+4} \]

Antiderivative was successfully verified.

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

[Out]

(4*x*Sqrt[a*x^(n/2)]*Hypergeometric2F1[1/2, 1/4 + n^(-1), 5/4 + n^(-1), -x^n])/(
4 + n)

_______________________________________________________________________________________

Maple [A]  time = 0.089, size = 37, normalized size = 0.7 \[ 4\,{\frac{x{\mbox{$_2$F$_1$}(1/2,1/4+{n}^{-1};\,5/4+{n}^{-1};\,-{x}^{n})}\sqrt{a{x}^{n/2}}}{4+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*x*hypergeom([1/2,1/4+1/n],[5/4+1/n],-x^n)*(a*x^(1/2*n))^(1/2)/(4+n)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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^(1/2*n))/sqrt(x^n + 1),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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