3.485 \(\int \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}} \, dx\)

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]

(3*b*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x^(2/3))/(2*(a + b/x^(1/3))) + (a
*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x)/(a + b/x^(1/3))

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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]

(3*b*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x^(2/3))/(2*(a + b/x^(1/3))) + (a
*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x)/(a + b/x^(1/3))

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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]

-a*x*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))/(2*(a + b/x**(1/3))) + 3*x*sqrt
(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))/2

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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]

(Sqrt[(b + a*x^(1/3))^2/x^(2/3)]*(3*b*x + 2*a*x^(4/3)))/(2*(b + a*x^(1/3)))

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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]

1/2*((a^2*x^(2/3)+2*a*b*x^(1/3)+b^2)/x^(2/3))^(1/2)*x^(1/3)*(3*x^(2/3)*b+2*a*x)/
(b+a*x^(1/3))

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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]

a*x + 3/2*b*x^(2/3)

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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]

a*x + 3/2*b*x^(2/3)

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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]

Integral(sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3)), x)

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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]

a*x*sign(a*x + b*x^(2/3))*sign(x) + 3/2*b*x^(2/3)*sign(a*x + b*x^(2/3))*sign(x)