3.2431 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^2} \, dx\)

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

[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 = 0.0515696, 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, Rules used = {190, 44} \[ \frac{3 b^3}{a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{9 b^2 \sqrt [3]{x}}{a^4}-\frac{12 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5}-\frac{4 b^3 \log (x)}{a^5}-\frac{3 b x^{2/3}}{a^3}+\frac{x}{a^2} \]

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

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}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^2} \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^2} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^4}-\frac{2 b}{a^3 x^3}+\frac{3 b^2}{a^4 x^2}-\frac{4 b^3}{a^5 x}+\frac{b^4}{a^4 (a+b x)^2}+\frac{4 b^4}{a^5 (a+b x)}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=\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}-\frac{12 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5}-\frac{4 b^3 \log (x)}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.051744, size = 63, normalized size = 0.82 \[ \frac{-3 a^2 b x^{2/3}+a^3 x-\frac{3 b^4}{a \sqrt [3]{x}+b}+9 a b^2 \sqrt [3]{x}-12 b^3 \log \left (a \sqrt [3]{x}+b\right )}{a^5} \]

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*Log[b + a*x^(1/3)])/a^5

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Maple [A]  time = 0.007, size = 60, normalized size = 0.8 \begin{align*}{\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}}{{a}^{5} \left ( b+a\sqrt [3]{x} \right ) }}-12\,{\frac{{b}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{5}}} \end{align*}

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

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Maxima [A]  time = 0.968717, size = 103, normalized size = 1.34 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.52672, size = 217, normalized size = 2.82 \begin{align*} \frac{a^{6} x^{2} + a^{3} b^{3} x - 3 \, b^{6} - 12 \,{\left (a^{3} b^{3} x + b^{6}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 3 \,{\left (a^{5} b x + 2 \, a^{2} b^{4}\right )} x^{\frac{2}{3}} + 3 \,{\left (3 \, a^{4} b^{2} x + 4 \, a b^{5}\right )} x^{\frac{1}{3}}}{a^{8} x + a^{5} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^6*x^2 + a^3*b^3*x - 3*b^6 - 12*(a^3*b^3*x + b^6)*log(a*x^(1/3) + b) - 3*(a^5*b*x + 2*a^2*b^4)*x^(2/3) + 3*(
3*a^4*b^2*x + 4*a*b^5)*x^(1/3))/(a^8*x + a^5*b^3)

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Sympy [A]  time = 0.805307, size = 165, normalized size = 2.14 \begin{align*} \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} \end{align*}

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 [A]  time = 1.15354, size = 88, normalized size = 1.14 \begin{align*} -\frac{12 \, b^{3} \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12*b^3*log(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