3.2424 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2} \, dx\)

Optimal. Leaf size=77 \[ -\frac{12 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5}-\frac{4 b^3 \log (x)}{a^5}+\frac{3 b^3}{a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{9 b^2 \sqrt [3]{x}}{a^4}-\frac{3 b x^{2/3}}{a^3}+\frac{x}{a^2} \]

[Out]

(3*b^3)/(a^4*(a + b/x^(1/3))) + (9*b^2*x^(1/3))/a^4 - (3*b*x^(2/3))/a^3 + x/a^2
- (12*b^3*Log[a + b/x^(1/3)])/a^5 - (4*b^3*Log[x])/a^5

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Rubi [A]  time = 0.124196, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{12 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5}-\frac{4 b^3 \log (x)}{a^5}+\frac{3 b^3}{a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{9 b^2 \sqrt [3]{x}}{a^4}-\frac{3 b x^{2/3}}{a^3}+\frac{x}{a^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^(-2),x]

[Out]

(3*b^3)/(a^4*(a + b/x^(1/3))) + (9*b^2*x^(1/3))/a^4 - (3*b*x^(2/3))/a^3 + x/a^2
- (12*b^3*Log[a + b/x^(1/3)])/a^5 - (4*b^3*Log[x])/a^5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{12 b^{2} \int ^{\sqrt [3]{x}} \frac{1}{a^{3}}\, dx}{a} - \frac{3 x}{a \left (a + \frac{b}{\sqrt [3]{x}}\right )} + \frac{4 x}{a^{2}} - \frac{12 b \int ^{\sqrt [3]{x}} x\, dx}{a^{3}} - \frac{12 b^{3} \log{\left (a \sqrt [3]{x} + b \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*b**2*Integral(a**(-3), (x, x**(1/3)))/a - 3*x/(a*(a + b/x**(1/3))) + 4*x/a**2
- 12*b*Integral(x, (x, x**(1/3)))/a**3 - 12*b**3*log(a*x**(1/3) + b)/a**5

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Mathematica [A]  time = 0.0431833, size = 63, normalized size = 0.82 \[ \frac{a^3 x-3 a^2 b x^{2/3}-\frac{3 b^4}{a \sqrt [3]{x}+b}-12 b^3 \log \left (a \sqrt [3]{x}+b\right )+9 a b^2 \sqrt [3]{x}}{a^5} \]

Antiderivative was successfully verified.

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

[Out]

((-3*b^4)/(b + a*x^(1/3)) + 9*a*b^2*x^(1/3) - 3*a^2*b*x^(2/3) + a^3*x - 12*b^3*L
og[b + a*x^(1/3)])/a^5

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Maple [A]  time = 0.01, size = 60, normalized size = 0.8 \[{\frac{x}{{a}^{2}}}-3\,{\frac{b{x}^{2/3}}{{a}^{3}}}+9\,{\frac{{b}^{2}\sqrt [3]{x}}{{a}^{4}}}-3\,{\frac{{b}^{4}}{ \left ( b+a\sqrt [3]{x} \right ){a}^{5}}}-12\,{\frac{{b}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^2,x)

[Out]

x/a^2-3*b*x^(2/3)/a^3+9*b^2*x^(1/3)/a^4-3*b^4/(b+a*x^(1/3))/a^5-12/a^5*b^3*ln(b+
a*x^(1/3))

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Maxima [A]  time = 1.44707, size = 103, normalized size = 1.34 \[ \frac{a^{3} - \frac{2 \, a^{2} b}{x^{\frac{1}{3}}} + \frac{6 \, a b^{2}}{x^{\frac{2}{3}}} + \frac{12 \, b^{3}}{x}}{\frac{a^{5}}{x} + \frac{a^{4} b}{x^{\frac{4}{3}}}} - \frac{12 \, b^{3} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{5}} - \frac{4 \, b^{3} \log \left (x\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a^3 - 2*a^2*b/x^(1/3) + 6*a*b^2/x^(2/3) + 12*b^3/x)/(a^5/x + a^4*b/x^(4/3)) - 1
2*b^3*log(a + b/x^(1/3))/a^5 - 4*b^3*log(x)/a^5

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Fricas [A]  time = 0.226975, size = 108, normalized size = 1.4 \[ -\frac{2 \, a^{3} b x - 6 \, a^{2} b^{2} x^{\frac{2}{3}} + 3 \, b^{4} + 12 \,{\left (a b^{3} x^{\frac{1}{3}} + b^{4}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) -{\left (a^{4} x + 9 \, a b^{3}\right )} x^{\frac{1}{3}}}{a^{6} x^{\frac{1}{3}} + a^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a^3*b*x - 6*a^2*b^2*x^(2/3) + 3*b^4 + 12*(a*b^3*x^(1/3) + b^4)*log(a*x^(1/3)
+ b) - (a^4*x + 9*a*b^3)*x^(1/3))/(a^6*x^(1/3) + a^5*b)

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Sympy [A]  time = 2.26063, size = 165, normalized size = 2.14 \[ \begin{cases} \frac{a^{4} x^{\frac{4}{3}}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac{2 a^{3} b x}{a^{6} \sqrt [3]{x} + a^{5} b} + \frac{6 a^{2} b^{2} x^{\frac{2}{3}}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac{12 a b^{3} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac{12 b^{4} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a^{6} \sqrt [3]{x} + a^{5} b} - \frac{12 b^{4}}{a^{6} \sqrt [3]{x} + a^{5} b} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{5}{3}}}{5 b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((a**4*x**(4/3)/(a**6*x**(1/3) + a**5*b) - 2*a**3*b*x/(a**6*x**(1/3) +
a**5*b) + 6*a**2*b**2*x**(2/3)/(a**6*x**(1/3) + a**5*b) - 12*a*b**3*x**(1/3)*log
(x**(1/3) + b/a)/(a**6*x**(1/3) + a**5*b) - 12*b**4*log(x**(1/3) + b/a)/(a**6*x*
*(1/3) + a**5*b) - 12*b**4/(a**6*x**(1/3) + a**5*b), Ne(a, 0)), (3*x**(5/3)/(5*b
**2), True))

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GIAC/XCAS [A]  time = 0.215128, size = 88, normalized size = 1.14 \[ -\frac{12 \, b^{3}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{5}} - \frac{3 \, b^{4}}{{\left (a x^{\frac{1}{3}} + b\right )} a^{5}} + \frac{a^{4} x - 3 \, a^{3} b x^{\frac{2}{3}} + 9 \, a^{2} b^{2} x^{\frac{1}{3}}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12*b^3*ln(abs(a*x^(1/3) + b))/a^5 - 3*b^4/((a*x^(1/3) + b)*a^5) + (a^4*x - 3*a^
3*b*x^(2/3) + 9*a^2*b^2*x^(1/3))/a^6