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