3.2416 \(\int \frac{1}{a+\frac{b}{\sqrt [3]{x}}} \, dx\)

Optimal. Leaf size=60 \[ -\frac{3 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^4}-\frac{b^3 \log (x)}{a^4}+\frac{3 b^2 \sqrt [3]{x}}{a^3}-\frac{3 b x^{2/3}}{2 a^2}+\frac{x}{a} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0881969, antiderivative size = 60, 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 \[ -\frac{3 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^4}-\frac{b^3 \log (x)}{a^4}+\frac{3 b^2 \sqrt [3]{x}}{a^3}-\frac{3 b x^{2/3}}{2 a^2}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^(-1),x]

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0132086, size = 50, normalized size = 0.83 \[ -\frac{3 b^3 \log \left (a \sqrt [3]{x}+b\right )}{a^4}+\frac{3 b^2 \sqrt [3]{x}}{a^3}-\frac{3 b x^{2/3}}{2 a^2}+\frac{x}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 43, normalized size = 0.7 \[{\frac{x}{a}}-{\frac{3\,b}{2\,{a}^{2}}{x}^{{\frac{2}{3}}}}+3\,{\frac{{b}^{2}\sqrt [3]{x}}{{a}^{3}}}-3\,{\frac{{b}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3)),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.45889, size = 73, normalized size = 1.22 \[ -\frac{3 \, b^{3} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{4}} - \frac{b^{3} \log \left (x\right )}{a^{4}} + \frac{{\left (2 \, a^{2} - \frac{3 \, a b}{x^{\frac{1}{3}}} + \frac{6 \, b^{2}}{x^{\frac{2}{3}}}\right )} x}{2 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.224799, size = 58, normalized size = 0.97 \[ \frac{2 \, a^{3} x - 6 \, b^{3} \log \left (a x^{\frac{1}{3}} + b\right ) - 3 \, a^{2} b x^{\frac{2}{3}} + 6 \, a b^{2} x^{\frac{1}{3}}}{2 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.57456, size = 58, normalized size = 0.97 \[ \begin{cases} \frac{x}{a} - \frac{3 b x^{\frac{2}{3}}}{2 a^{2}} + \frac{3 b^{2} \sqrt [3]{x}}{a^{3}} - \frac{3 b^{3} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a^{4}} & \text{for}\: a \neq 0 \\\frac{3 x^{\frac{4}{3}}}{4 b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**(1/3)),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213628, size = 61, normalized size = 1.02 \[ -\frac{3 \, b^{3}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{4}} + \frac{2 \, a^{2} x - 3 \, a b x^{\frac{2}{3}} + 6 \, b^{2} x^{\frac{1}{3}}}{2 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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