Optimal. Leaf size=147 \[ -\frac{3 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{b^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{2 b \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rubi [A] time = 0.15305, antiderivative size = 147, 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 \[ -\frac{3 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{b^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{2 b \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)],x]
[Out]
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Rubi in Sympy [A] time = 16.1625, size = 131, normalized size = 0.89 \[ \frac{3 a^{2} \left (a + b \sqrt [3]{x}\right ) \log{\left (a + b \sqrt [3]{x} \right )}}{b^{3} \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}} - \frac{3 a \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}}{b^{3}} + \frac{3 x^{\frac{2}{3}} \left (2 a + 2 b \sqrt [3]{x}\right )}{4 b \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0406836, size = 65, normalized size = 0.44 \[ \frac{3 \left (a+b \sqrt [3]{x}\right ) \left (2 a^2 \log \left (a+b \sqrt [3]{x}\right )+b \sqrt [3]{x} \left (b \sqrt [3]{x}-2 a\right )\right )}{2 b^3 \sqrt{\left (a+b \sqrt [3]{x}\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)],x]
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Maple [A] time = 0.044, size = 103, normalized size = 0.7 \[{\frac{1}{2\,{b}^{3}}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 3\,{b}^{2}{x}^{2/3}+2\,{a}^{2}\ln \left ({b}^{3}x+{a}^{3} \right ) -6\,ab\sqrt [3]{x}+4\,{a}^{2}\ln \left ( a+b\sqrt [3]{x} \right ) -2\,{a}^{2}\ln \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2),x)
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Maxima [A] time = 0.743531, size = 62, normalized size = 0.42 \[ \frac{3 \, a^{2} b^{2} \log \left (x^{\frac{1}{3}} + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{3 \, a b x^{\frac{1}{3}}}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \, x^{\frac{2}{3}}}{2 \, \sqrt{b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2),x, algorithm="maxima")
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Fricas [A] time = 0.276755, size = 45, normalized size = 0.31 \[ \frac{3 \,{\left (2 \, a^{2} \log \left (b x^{\frac{1}{3}} + a\right ) + b^{2} x^{\frac{2}{3}} - 2 \, a b x^{\frac{1}{3}}\right )}}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.283168, size = 82, normalized size = 0.56 \[ \frac{3 \,{\left (b x^{\frac{2}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) - 2 \, a x^{\frac{1}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right )\right )}}{2 \, b^{2}} + \frac{3 \, a^{2}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{3}{\rm sign}\left (b x^{\frac{1}{3}} + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2),x, algorithm="giac")
[Out]