3.2354 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right ) x^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{3 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^4}-\frac{b^3 \log (x)}{a^4}-\frac{3 b^2}{a^3 \sqrt [3]{x}}+\frac{3 b}{2 a^2 x^{2/3}}-\frac{1}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0885364, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{3 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^4}-\frac{b^3 \log (x)}{a^4}-\frac{3 b^2}{a^3 \sqrt [3]{x}}+\frac{3 b}{2 a^2 x^{2/3}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^(1/3))*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.9982, size = 65, normalized size = 1.03 \[ - \frac{1}{a x} + \frac{3 b}{2 a^{2} x^{\frac{2}{3}}} - \frac{3 b^{2}}{a^{3} \sqrt [3]{x}} - \frac{3 b^{3} \log{\left (\sqrt [3]{x} \right )}}{a^{4}} + \frac{3 b^{3} \log{\left (a + b \sqrt [3]{x} \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0319644, size = 62, normalized size = 0.98 \[ -\frac{2 a^3-3 a^2 b \sqrt [3]{x}-6 b^3 x \log \left (a+b \sqrt [3]{x}\right )+6 a b^2 x^{2/3}+2 b^3 x \log (x)}{2 a^4 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 56, normalized size = 0.9 \[ -{\frac{1}{ax}}+{\frac{3\,b}{2\,{a}^{2}}{x}^{-{\frac{2}{3}}}}-3\,{\frac{{b}^{2}}{{a}^{3}\sqrt [3]{x}}}+3\,{\frac{{b}^{3}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{4}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))/x^2,x)

[Out]

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

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Maxima [A]  time = 1.4354, size = 76, normalized size = 1.21 \[ \frac{3 \, b^{3} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{4}} - \frac{b^{3} \log \left (x\right )}{a^{4}} - \frac{6 \, b^{2} x^{\frac{2}{3}} - 3 \, a b x^{\frac{1}{3}} + 2 \, a^{2}}{2 \, a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.226166, size = 76, normalized size = 1.21 \[ \frac{6 \, b^{3} x \log \left (b x^{\frac{1}{3}} + a\right ) - 6 \, b^{3} x \log \left (x^{\frac{1}{3}}\right ) - 6 \, a b^{2} x^{\frac{2}{3}} + 3 \, a^{2} b x^{\frac{1}{3}} - 2 \, a^{3}}{2 \, a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.67268, size = 83, normalized size = 1.32 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{4}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{a x} & \text{for}\: b = 0 \\- \frac{3}{4 b x^{\frac{4}{3}}} & \text{for}\: a = 0 \\- \frac{1}{a x} + \frac{3 b}{2 a^{2} x^{\frac{2}{3}}} - \frac{3 b^{2}}{a^{3} \sqrt [3]{x}} - \frac{b^{3} \log{\left (x \right )}}{a^{4}} + \frac{3 b^{3} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{4}} & \text{otherwise} \end{cases} \]

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

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GIAC/XCAS [A]  time = 0.224452, size = 82, normalized size = 1.3 \[ \frac{3 \, b^{3}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{4}} - \frac{b^{3}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x^{\frac{2}{3}} - 3 \, a^{2} b x^{\frac{1}{3}} + 2 \, a^{3}}{2 \, a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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