3.2426 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x^2} \, dx\)

Optimal. Leaf size=52 \[ \frac{6 a \log \left (a \sqrt [3]{x}+b\right )}{b^3}-\frac{2 a \log (x)}{b^3}-\frac{3 a}{b^2 \left (a \sqrt [3]{x}+b\right )}-\frac{3}{b^2 \sqrt [3]{x}} \]

[Out]

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

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Rubi [A]  time = 0.0882315, antiderivative size = 52, 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 \[ \frac{6 a \log \left (a \sqrt [3]{x}+b\right )}{b^3}-\frac{2 a \log (x)}{b^3}-\frac{3 a}{b^2 \left (a \sqrt [3]{x}+b\right )}-\frac{3}{b^2 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.0973, size = 54, normalized size = 1.04 \[ - \frac{3 a}{b^{2} \left (a \sqrt [3]{x} + b\right )} - \frac{6 a \log{\left (\sqrt [3]{x} \right )}}{b^{3}} + \frac{6 a \log{\left (a \sqrt [3]{x} + b \right )}}{b^{3}} - \frac{3}{b^{2} \sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.101208, size = 49, normalized size = 0.94 \[ \frac{3 \left (-\frac{a b}{a \sqrt [3]{x}+b}+2 a \log \left (a \sqrt [3]{x}+b\right )-\frac{2}{3} a \log (x)-\frac{b}{\sqrt [3]{x}}\right )}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 47, normalized size = 0.9 \[ -3\,{\frac{a}{{b}^{2} \left ( b+a\sqrt [3]{x} \right ) }}-3\,{\frac{1}{{b}^{2}\sqrt [3]{x}}}+6\,{\frac{a\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{3}}}-2\,{\frac{a\ln \left ( x \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^2/x^2,x)

[Out]

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

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Maxima [A]  time = 1.43214, size = 59, normalized size = 1.13 \[ \frac{6 \, a \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{3}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}}{b^{3}} + \frac{3 \, a^{2}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237582, size = 101, normalized size = 1.94 \[ -\frac{3 \,{\left (2 \, a b x^{\frac{1}{3}} + b^{2} - 2 \,{\left (a^{2} x^{\frac{2}{3}} + a b x^{\frac{1}{3}}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 2 \,{\left (a^{2} x^{\frac{2}{3}} + a b x^{\frac{1}{3}}\right )} \log \left (x^{\frac{1}{3}}\right )\right )}}{a b^{3} x^{\frac{2}{3}} + b^{4} x^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.258, size = 211, normalized size = 4.06 \[ \begin{cases} \frac{\tilde{\infty }}{\sqrt [3]{x}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{b^{2} \sqrt [3]{x}} & \text{for}\: a = 0 \\- \frac{1}{a^{2} x} & \text{for}\: b = 0 \\- \frac{2 a^{2} x^{2} \log{\left (x \right )}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} + \frac{6 a^{2} x^{2} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} - \frac{2 a b x^{\frac{5}{3}} \log{\left (x \right )}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} + \frac{6 a b x^{\frac{5}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} - \frac{6 a b x^{\frac{5}{3}}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} - \frac{3 b^{2} x^{\frac{4}{3}}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222346, size = 69, normalized size = 1.33 \[ \frac{6 \, a{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{3}} - \frac{2 \, a{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} - \frac{3 \,{\left (2 \, a x^{\frac{1}{3}} + b\right )}}{{\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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