Optimal. Leaf size=88 \[ \frac{3 b x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{2 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{a x \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}} \]
[Out]
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Rubi [A] time = 0.127019, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{3 b x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{2 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{a x \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)],x]
[Out]
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Rubi in Sympy [A] time = 9.80796, size = 70, normalized size = 0.8 \[ - \frac{a x \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}}{2 \left (a + \frac{b}{\sqrt [3]{x}}\right )} + \frac{3 x \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a**2+b**2/x**(2/3)+2*a*b/x**(1/3))**(1/2),x)
[Out]
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Mathematica [A] time = 0.018807, size = 49, normalized size = 0.56 \[ \frac{\sqrt{\frac{\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}} \left (2 a x^{4/3}+3 b x\right )}{2 \left (a \sqrt [3]{x}+b\right )} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)],x]
[Out]
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Maple [A] time = 0.008, size = 50, normalized size = 0.6 \[{\frac{1}{2}\sqrt{{1 \left ({a}^{2}{x}^{{\frac{2}{3}}}+2\,ab\sqrt [3]{x}+{b}^{2} \right ){x}^{-{\frac{2}{3}}}}}\sqrt [3]{x} \left ( 3\,{x}^{2/3}b+2\,ax \right ) \left ( b+a\sqrt [3]{x} \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2),x)
[Out]
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Maxima [A] time = 0.749743, size = 14, normalized size = 0.16 \[ a x + \frac{3}{2} \, b x^{\frac{2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269271, size = 14, normalized size = 0.16 \[ a x + \frac{3}{2} \, b x^{\frac{2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a**2+b**2/x**(2/3)+2*a*b/x**(1/3))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.27529, size = 46, normalized size = 0.52 \[ a x{\rm sign}\left (a x + b x^{\frac{2}{3}}\right ){\rm sign}\left (x\right ) + \frac{3}{2} \, b x^{\frac{2}{3}}{\rm sign}\left (a x + b x^{\frac{2}{3}}\right ){\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3)),x, algorithm="giac")
[Out]