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

Optimal. Leaf size=23 \[ \frac{2}{3} \sqrt{a} \sinh ^{-1}\left (\frac{(a x)^{3/2}}{a^{3/2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0470897, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{2}{3} \sqrt{a} \sinh ^{-1}\left (\frac{(a x)^{3/2}}{a^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x]/Sqrt[1 + x^3],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.19283, size = 20, normalized size = 0.87 \[ \frac{2 \sqrt{a} \operatorname{asinh}{\left (\frac{\left (a x\right )^{\frac{3}{2}}}{a^{\frac{3}{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0250192, size = 22, normalized size = 0.96 \[ \frac{2 \sqrt{a x} \sinh ^{-1}\left (x^{3/2}\right )}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.086, size = 321, normalized size = 14. \[ -4\,{\frac{\sqrt{ax}\sqrt{{x}^{3}+1}a \left ( i\sqrt{3}+1 \right ) \left ( 1+x \right ) ^{2}}{\sqrt{x \left ({x}^{3}+1 \right ) a} \left ( i\sqrt{3}+3 \right ) \sqrt{-ax \left ( 1+x \right ) \left ( i\sqrt{3}+2\,x-1 \right ) \left ( i\sqrt{3}-2\,x+1 \right ) }}\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) x}{ \left ( i\sqrt{3}+1 \right ) \left ( 1+x \right ) }}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{ \left ( -1+i\sqrt{3} \right ) \left ( 1+x \right ) }}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{ \left ( i\sqrt{3}+1 \right ) \left ( 1+x \right ) }}} \left ({\it EllipticF} \left ( \sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) x}{ \left ( i\sqrt{3}+1 \right ) \left ( 1+x \right ) }}},\sqrt{{\frac{ \left ( -3+i\sqrt{3} \right ) \left ( i\sqrt{3}+1 \right ) }{ \left ( -1+i\sqrt{3} \right ) \left ( i\sqrt{3}+3 \right ) }}} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) x}{ \left ( i\sqrt{3}+1 \right ) \left ( 1+x \right ) }}},{\frac{i\sqrt{3}+1}{i\sqrt{3}+3}},\sqrt{{\frac{ \left ( -3+i\sqrt{3} \right ) \left ( i\sqrt{3}+1 \right ) }{ \left ( -1+i\sqrt{3} \right ) \left ( i\sqrt{3}+3 \right ) }}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*(a*x)^(1/2)*(x^3+1)^(1/2)*a*(I*3^(1/2)+1)*((I*3^(1/2)+3)*x/(I*3^(1/2)+1)/(1+x
))^(1/2)*(1+x)^2*((I*3^(1/2)+2*x-1)/(-1+I*3^(1/2))/(1+x))^(1/2)*((I*3^(1/2)-2*x+
1)/(I*3^(1/2)+1)/(1+x))^(1/2)*(EllipticF(((I*3^(1/2)+3)*x/(I*3^(1/2)+1)/(1+x))^(
1/2),((-3+I*3^(1/2))*(I*3^(1/2)+1)/(-1+I*3^(1/2))/(I*3^(1/2)+3))^(1/2))-Elliptic
Pi(((I*3^(1/2)+3)*x/(I*3^(1/2)+1)/(1+x))^(1/2),(I*3^(1/2)+1)/(I*3^(1/2)+3),((-3+
I*3^(1/2))*(I*3^(1/2)+1)/(-1+I*3^(1/2))/(I*3^(1/2)+3))^(1/2)))/(x*(x^3+1)*a)^(1/
2)/(I*3^(1/2)+3)/(-a*x*(1+x)*(I*3^(1/2)+2*x-1)*(I*3^(1/2)-2*x+1))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a*x)/sqrt(x^3 + 1), x)

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Fricas [A]  time = 0.325601, size = 1, normalized size = 0.04 \[ \left [\frac{1}{6} \, \sqrt{a} \log \left (-8 \, a x^{6} - 8 \, a x^{3} - 4 \,{\left (2 \, x^{4} + x\right )} \sqrt{x^{3} + 1} \sqrt{a x} \sqrt{a} - a\right ), \frac{1}{3} \, \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{x^{3} + 1} \sqrt{a x} x}{{\left (2 \, x^{3} + 1\right )} \sqrt{-a}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.65279, size = 14, normalized size = 0.61 \[ \frac{2 \sqrt{a} \operatorname{asinh}{\left (x^{\frac{3}{2}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.263575, size = 47, normalized size = 2.04 \[ -\frac{2 \, a^{\frac{5}{2}}{\rm ln}\left (-\sqrt{a x} a^{\frac{3}{2}} x + \sqrt{a^{4} x^{3} + a^{4}}\right )}{3 \,{\left | a \right |}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*a^(5/2)*ln(-sqrt(a*x)*a^(3/2)*x + sqrt(a^4*x^3 + a^4))/abs(a)^2