3.486 \(\int \frac{1}{\sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}} \, dx\)

Optimal. Leaf size=190 \[ -\frac{3 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^2 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{3 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]

[Out]

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

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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]

[Out]

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

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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)

[Out]

x*(2*a + 2*b/x**(1/3))/(2*a*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))) - 3*b*x
**(2/3)*(2*a + 2*b/x**(1/3))/(4*a**2*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))
) + 3*b**3*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))*log(x**(-1/3))/(a**4*(a +
b/x**(1/3))) - 3*b**3*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))*log(a + b/x**
(1/3))/(a**4*(a + b/x**(1/3))) + 3*b**2*x**(1/3)*sqrt(a**2 + 2*a*b/x**(1/3) + b*
*2/x**(2/3))/a**4

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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]

[Out]

((b + a*x^(1/3))*(6*a*b^2*x^(1/3) - 3*a^2*b*x^(2/3) + 2*a^3*x - 6*b^3*Log[b + a*
x^(1/3)]))/(2*a^4*Sqrt[(b + a*x^(1/3))^2/x^(2/3)]*x^(1/3))

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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)

[Out]

-1/2/((a^2*x^(2/3)+2*a*b*x^(1/3)+b^2)/x^(2/3))^(1/2)/x^(1/3)*(b+a*x^(1/3))*(3*x^
(2/3)*a^2*b+6*b^3*ln(b+a*x^(1/3))-6*a*b^2*x^(1/3)-2*a^3*x)/a^4

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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")

[Out]

-3*b^3*log(a*x^(1/3) + b)/a^4 + 1/2*(2*a^2*x - 3*a*b*x^(2/3) + 6*b^2*x^(1/3))/a^
3

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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")

[Out]

1/2*(2*a^3*x - 6*b^3*log(a*x^(1/3) + b) - 3*a^2*b*x^(2/3) + 6*a*b^2*x^(1/3))/a^4

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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)

[Out]

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

[Out]

-3*b^3*ln(abs(a*x^(1/3) + b))/(a^4*sign(a*x + b*x^(2/3))*sign(x)) + 1/2*(2*a^2*x
- 3*a*b*x^(2/3) + 6*b^2*x^(1/3))/(a^3*sign(a*x + b*x^(2/3))*sign(x))