Optimal. Leaf size=190 \[ -\frac{3 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^2 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{3 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]
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Rubi [A] time = 0.260605, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{3 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^2 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{3 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)],x]
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Rubi in Sympy [A] time = 43.3507, size = 235, normalized size = 1.24 \[ \frac{x \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{2 a \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} - \frac{3 b x^{\frac{2}{3}} \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{4 a^{2} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} + \frac{3 b^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} \log{\left (\frac{1}{\sqrt [3]{x}} \right )}}{a^{4} \left (a + \frac{b}{\sqrt [3]{x}}\right )} - \frac{3 b^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} \log{\left (a + \frac{b}{\sqrt [3]{x}} \right )}}{a^{4} \left (a + \frac{b}{\sqrt [3]{x}}\right )} + \frac{3 b^{2} \sqrt [3]{x} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a**2+b**2/x**(2/3)+2*a*b/x**(1/3))**(1/2),x)
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Mathematica [A] time = 0.0389314, size = 86, normalized size = 0.45 \[ \frac{\left (a \sqrt [3]{x}+b\right ) \left (2 a^3 x-3 a^2 b x^{2/3}-6 b^3 \log \left (a \sqrt [3]{x}+b\right )+6 a b^2 \sqrt [3]{x}\right )}{2 a^4 \sqrt [3]{x} \sqrt{\frac{\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)],x]
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Maple [A] time = 0.009, size = 78, normalized size = 0.4 \[ -{\frac{1}{2\,{a}^{4}} \left ( b+a\sqrt [3]{x} \right ) \left ( 3\,{x}^{2/3}{a}^{2}b+6\,{b}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) -6\,a{b}^{2}\sqrt [3]{x}-2\,{a}^{3}x \right ){\frac{1}{\sqrt{{1 \left ({a}^{2}{x}^{{\frac{2}{3}}}+2\,ab\sqrt [3]{x}+{b}^{2} \right ){x}^{-{\frac{2}{3}}}}}}}{\frac{1}{\sqrt [3]{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(1/2),x)
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Maxima [A] time = 0.753073, size = 59, normalized size = 0.31 \[ -\frac{3 \, b^{3} \log \left (a x^{\frac{1}{3}} + b\right )}{a^{4}} + \frac{2 \, a^{2} x - 3 \, a b x^{\frac{2}{3}} + 6 \, b^{2} x^{\frac{1}{3}}}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3)),x, algorithm="maxima")
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Fricas [A] time = 0.271788, size = 58, normalized size = 0.31 \[ \frac{2 \, a^{3} x - 6 \, b^{3} \log \left (a x^{\frac{1}{3}} + b\right ) - 3 \, a^{2} b x^{\frac{2}{3}} + 6 \, a b^{2} x^{\frac{1}{3}}}{2 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a**2+b**2/x**(2/3)+2*a*b/x**(1/3))**(1/2),x)
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GIAC/XCAS [A] time = 0.308499, size = 104, normalized size = 0.55 \[ -\frac{3 \, b^{3}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{4}{\rm sign}\left (a x + b x^{\frac{2}{3}}\right ){\rm sign}\left (x\right )} + \frac{2 \, a^{2} x - 3 \, a b x^{\frac{2}{3}} + 6 \, b^{2} x^{\frac{1}{3}}}{2 \, a^{3}{\rm sign}\left (a x + b x^{\frac{2}{3}}\right ){\rm sign}\left (x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3)),x, algorithm="giac")
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