3.207 \(\int \frac{\sqrt{a x^3}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=24 \[ \frac{2 \sqrt{a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}} \]

[Out]

(2*Sqrt[a*x^3]*ArcSinh[x^(5/2)])/(5*x^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0251148, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 \sqrt{a x^3} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^3]/Sqrt[1 + x^5],x]

[Out]

(2*Sqrt[a*x^3]*ArcSinh[x^(5/2)])/(5*x^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.86404, size = 22, normalized size = 0.92 \[ \frac{2 \sqrt{a x^{3}} \operatorname{asinh}{\left (x^{\frac{5}{2}} \right )}}{5 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**3)**(1/2)/(x**5+1)**(1/2),x)

[Out]

2*sqrt(a*x**3)*asinh(x**(5/2))/(5*x**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0338001, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^3}}{\sqrt{1+x^5}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a*x^3]/Sqrt[1 + x^5],x]

[Out]

Integrate[Sqrt[a*x^3]/Sqrt[1 + x^5], x]

_______________________________________________________________________________________

Maple [A]  time = 0.058, size = 17, normalized size = 0.7 \[{\frac{2}{5}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) \sqrt{a{x}^{3}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^3)^(1/2)/(x^5+1)^(1/2),x)

[Out]

2/5*arcsinh(x^(5/2))*(a*x^3)^(1/2)/x^(3/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{3}}}{\sqrt{x^{5} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^3)/sqrt(x^5 + 1),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^3)/sqrt(x^5 + 1), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.358236, size = 1, normalized size = 0.04 \[ \left [\frac{1}{10} \, \sqrt{a} \log \left (-8 \, a x^{10} - 8 \, a x^{5} - 4 \,{\left (2 \, x^{6} + x\right )} \sqrt{x^{5} + 1} \sqrt{a x^{3}} \sqrt{a} - a\right ), -\frac{1}{5} \, \sqrt{-a} \arctan \left (\frac{{\left (2 \, x^{5} + 1\right )} \sqrt{-a}}{2 \, \sqrt{x^{5} + 1} \sqrt{a x^{3}} x}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^3)/sqrt(x^5 + 1),x, algorithm="fricas")

[Out]

[1/10*sqrt(a)*log(-8*a*x^10 - 8*a*x^5 - 4*(2*x^6 + x)*sqrt(x^5 + 1)*sqrt(a*x^3)*
sqrt(a) - a), -1/5*sqrt(-a)*arctan(1/2*(2*x^5 + 1)*sqrt(-a)/(sqrt(x^5 + 1)*sqrt(
a*x^3)*x))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{3}}}{\sqrt{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**3)**(1/2)/(x**5+1)**(1/2),x)

[Out]

Integral(sqrt(a*x**3)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

_______________________________________________________________________________________

GIAC/XCAS [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^3)/sqrt(x^5 + 1),x, algorithm="giac")

[Out]

Exception raised: TypeError