3.2418 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right ) x^2} \, dx\)

Optimal. Leaf size=51 \[ -\frac{3 a^2 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac{a^2 \log (x)}{b^3}+\frac{3 a}{b^2 \sqrt [3]{x}}-\frac{3}{2 b x^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0822004, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 a^2 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac{a^2 \log (x)}{b^3}+\frac{3 a}{b^2 \sqrt [3]{x}}-\frac{3}{2 b x^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^(1/3))*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.2909, size = 54, normalized size = 1.06 \[ \frac{3 a^{2} \log{\left (\sqrt [3]{x} \right )}}{b^{3}} - \frac{3 a^{2} \log{\left (a \sqrt [3]{x} + b \right )}}{b^{3}} + \frac{3 a}{b^{2} \sqrt [3]{x}} - \frac{3}{2 b x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**(1/3))/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.0545129, size = 48, normalized size = 0.94 \[ \frac{-6 a^2 \log \left (a \sqrt [3]{x}+b\right )+2 a^2 \log (x)-\frac{3 b \left (b-2 a \sqrt [3]{x}\right )}{x^{2/3}}}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^(1/3))*x^2),x]

[Out]

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

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Maple [A]  time = 0.013, size = 44, normalized size = 0.9 \[ -{\frac{3}{2\,b}{x}^{-{\frac{2}{3}}}}+3\,{\frac{a}{{b}^{2}\sqrt [3]{x}}}-3\,{\frac{{a}^{2}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{3}}}+{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))/x^2,x)

[Out]

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

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Maxima [A]  time = 1.44236, size = 59, normalized size = 1.16 \[ -\frac{3 \, a^{2} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{3}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2}}{2 \, b^{3}} + \frac{6 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.235754, size = 63, normalized size = 1.24 \[ -\frac{3 \,{\left (2 \, a^{2} x^{\frac{2}{3}} \log \left (a x^{\frac{1}{3}} + b\right ) - 2 \, a^{2} x^{\frac{2}{3}} \log \left (x^{\frac{1}{3}}\right ) - 2 \, a b x^{\frac{1}{3}} + b^{2}\right )}}{2 \, b^{3} x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.86471, size = 73, normalized size = 1.43 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{2}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{a x} & \text{for}\: b = 0 \\- \frac{3}{2 b x^{\frac{2}{3}}} & \text{for}\: a = 0 \\\frac{a^{2} \log{\left (x \right )}}{b^{3}} - \frac{3 a^{2} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{b^{3}} + \frac{3 a}{b^{2} \sqrt [3]{x}} - \frac{3}{2 b x^{\frac{2}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**(1/3))/x**2,x)

[Out]

Piecewise((zoo/x**(2/3), Eq(a, 0) & Eq(b, 0)), (-1/(a*x), Eq(b, 0)), (-3/(2*b*x*
*(2/3)), Eq(a, 0)), (a**2*log(x)/b**3 - 3*a**2*log(x**(1/3) + b/a)/b**3 + 3*a/(b
**2*x**(1/3)) - 3/(2*b*x**(2/3)), True))

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GIAC/XCAS [A]  time = 0.215069, size = 66, normalized size = 1.29 \[ -\frac{3 \, a^{2}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{3}} + \frac{a^{2}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} + \frac{3 \,{\left (2 \, a b x^{\frac{1}{3}} - b^{2}\right )}}{2 \, b^{3} x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^(1/3))*x^2),x, algorithm="giac")

[Out]

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