3.2423 \(\int \frac{1}{a+\frac{b}{\sqrt [3]{x}}} \, dx\)

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

[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

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Rubi [A]  time = 0.0366103, 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, Rules used = {190, 44} \[ \frac{3 b^2 \sqrt [3]{x}}{a^3}-\frac{3 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^4}-\frac{b^3 \log (x)}{a^4}-\frac{3 b x^{2/3}}{2 a^2}+\frac{x}{a} \]

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

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{a+\frac{b}{\sqrt [3]{x}}} \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \left (\frac{1}{a x^4}-\frac{b}{a^2 x^3}+\frac{b^2}{a^3 x^2}-\frac{b^3}{a^4 x}+\frac{b^4}{a^4 (a+b x)}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=\frac{3 b^2 \sqrt [3]{x}}{a^3}-\frac{3 b x^{2/3}}{2 a^2}+\frac{x}{a}-\frac{3 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^4}-\frac{b^3 \log (x)}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.0247504, size = 50, normalized size = 0.83 \[ \frac{3 b^2 \sqrt [3]{x}}{a^3}-\frac{3 b^3 \log \left (a \sqrt [3]{x}+b\right )}{a^4}-\frac{3 b x^{2/3}}{2 a^2}+\frac{x}{a} \]

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

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Maple [A]  time = 0.003, size = 43, normalized size = 0.7 \begin{align*}{\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}}} \end{align*}

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))

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Maxima [A]  time = 1.03379, size = 73, normalized size = 1.22 \begin{align*} -\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}} \end{align*}

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

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Fricas [A]  time = 1.50132, size = 111, normalized size = 1.85 \begin{align*} \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}} \end{align*}

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

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Sympy [A]  time = 0.536891, size = 58, normalized size = 0.97 \begin{align*} \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} \end{align*}

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))

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Giac [A]  time = 1.15367, size = 61, normalized size = 1.02 \begin{align*} -\frac{3 \, b^{3} \log \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}} \end{align*}

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*log(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