3.2434 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x^2} \, dx\)

Optimal. Leaf size=56 \[ -\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac{3}{b^2 \left (a \sqrt [3]{x}+b\right )}+\frac{3}{2 b \left (a \sqrt [3]{x}+b\right )^2}+\frac{\log (x)}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.084505, antiderivative size = 56, 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 \[ -\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac{3}{b^2 \left (a \sqrt [3]{x}+b\right )}+\frac{3}{2 b \left (a \sqrt [3]{x}+b\right )^2}+\frac{\log (x)}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.0205, size = 54, normalized size = 0.96 \[ \frac{3}{2 b \left (a \sqrt [3]{x} + b\right )^{2}} + \frac{3}{b^{2} \left (a \sqrt [3]{x} + b\right )} + \frac{3 \log{\left (\sqrt [3]{x} \right )}}{b^{3}} - \frac{3 \log{\left (a \sqrt [3]{x} + b \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0683004, size = 51, normalized size = 0.91 \[ \frac{3 \left (\frac{b \left (2 a \sqrt [3]{x}+3 b\right )}{\left (a \sqrt [3]{x}+b\right )^2}-2 \log \left (a \sqrt [3]{x}+b\right )+\frac{2 \log (x)}{3}\right )}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 49, normalized size = 0.9 \[{\frac{3}{2\,b} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}+3\,{\frac{1}{{b}^{2} \left ( b+a\sqrt [3]{x} \right ) }}-3\,{\frac{\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{3}}}+{\frac{\ln \left ( x \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^3/x^2,x)

[Out]

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

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Maxima [A]  time = 1.42439, size = 62, normalized size = 1.11 \[ -\frac{3 \, \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{3}} - \frac{6 \, a}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{3}} + \frac{3 \, a^{2}}{2 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.241925, size = 124, normalized size = 2.21 \[ \frac{3 \,{\left (2 \, a b x^{\frac{1}{3}} + 3 \, b^{2} - 2 \,{\left (a^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + b^{2}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 2 \,{\left (a^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + b^{2}\right )} \log \left (x^{\frac{1}{3}}\right )\right )}}{2 \,{\left (a^{2} b^{3} x^{\frac{2}{3}} + 2 \, a b^{4} x^{\frac{1}{3}} + b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*(2*a*b*x^(1/3) + 3*b^2 - 2*(a^2*x^(2/3) + 2*a*b*x^(1/3) + b^2)*log(a*x^(1/3)
+ b) + 2*(a^2*x^(2/3) + 2*a*b*x^(1/3) + b^2)*log(x^(1/3)))/(a^2*b^3*x^(2/3) + 2
*a*b^4*x^(1/3) + b^5)

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Sympy [A]  time = 27.4345, size = 406, normalized size = 7.25 \[ \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\log{\left (x \right )}}{b^{3}} & \text{for}\: a = 0 \\- \frac{1}{a^{3} x} & \text{for}\: b = 0 \\\frac{2 a^{2} x^{\frac{7}{3}} \log{\left (x \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} - \frac{6 a^{2} x^{\frac{7}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} + \frac{4 a b x^{2} \log{\left (x \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} - \frac{12 a b x^{2} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} + \frac{6 a b x^{2}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} + \frac{2 b^{2} x^{\frac{5}{3}} \log{\left (x \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} - \frac{6 b^{2} x^{\frac{5}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} + \frac{9 b^{2} x^{\frac{5}{3}}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0)), (log(x)/b**3, Eq(a, 0)), (-1/(a**3*
x), Eq(b, 0)), (2*a**2*x**(7/3)*log(x)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2
*b**5*x**(5/3)) - 6*a**2*x**(7/3)*log(x**(1/3) + b/a)/(2*a**2*b**3*x**(7/3) + 4*
a*b**4*x**2 + 2*b**5*x**(5/3)) + 4*a*b*x**2*log(x)/(2*a**2*b**3*x**(7/3) + 4*a*b
**4*x**2 + 2*b**5*x**(5/3)) - 12*a*b*x**2*log(x**(1/3) + b/a)/(2*a**2*b**3*x**(7
/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) + 6*a*b*x**2/(2*a**2*b**3*x**(7/3) + 4*a*
b**4*x**2 + 2*b**5*x**(5/3)) + 2*b**2*x**(5/3)*log(x)/(2*a**2*b**3*x**(7/3) + 4*
a*b**4*x**2 + 2*b**5*x**(5/3)) - 6*b**2*x**(5/3)*log(x**(1/3) + b/a)/(2*a**2*b**
3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) + 9*b**2*x**(5/3)/(2*a**2*b**3*x**
(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)), True))

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GIAC/XCAS [A]  time = 0.217232, size = 66, normalized size = 1.18 \[ -\frac{3 \,{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{3}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} + \frac{3 \,{\left (2 \, a b x^{\frac{1}{3}} + 3 \, b^{2}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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