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

Optimal. Leaf size=147 \[ -\frac{3 a \sqrt [3]{x} \left (a+b \sqrt [3]{x}\right )}{b^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 x^{2/3} \left (a+b \sqrt [3]{x}\right )}{2 b \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 a^2 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]

[Out]

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

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

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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]

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

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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]

[Out]

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

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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)

[Out]

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

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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")

[Out]

3*a^2*b^2*log(x^(1/3) + a/b)/(b^2)^(5/2) - 3*a*b*x^(1/3)/(b^2)^(3/2) + 3/2*x^(2/
3)/sqrt(b^2)

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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]

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

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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]

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 = 0.283168, 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]

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