3.2439 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^3} \, dx\)

Optimal. Leaf size=100 \[ \frac{12 b^3}{a^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{3 b^3}{2 a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )^2}+\frac{18 b^2 \sqrt [3]{x}}{a^5}-\frac{30 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}-\frac{9 b x^{2/3}}{2 a^4}+\frac{x}{a^3} \]

[Out]

(3*b^3)/(2*a^4*(a + b/x^(1/3))^2) + (12*b^3)/(a^5*(a + b/x^(1/3))) + (18*b^2*x^(1/3))/a^5 - (9*b*x^(2/3))/(2*a
^4) + x/a^3 - (30*b^3*Log[a + b/x^(1/3)])/a^6 - (10*b^3*Log[x])/a^6

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Rubi [A]  time = 0.0683635, antiderivative size = 100, 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} \[ \frac{12 b^3}{a^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{3 b^3}{2 a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )^2}+\frac{18 b^2 \sqrt [3]{x}}{a^5}-\frac{30 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}-\frac{9 b x^{2/3}}{2 a^4}+\frac{x}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^(1/3))^(-3),x]

[Out]

(3*b^3)/(2*a^4*(a + b/x^(1/3))^2) + (12*b^3)/(a^5*(a + b/x^(1/3))) + (18*b^2*x^(1/3))/a^5 - (9*b*x^(2/3))/(2*a
^4) + x/a^3 - (30*b^3*Log[a + b/x^(1/3)])/a^6 - (10*b^3*Log[x])/a^6

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 )^3} \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^3} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^4}-\frac{3 b}{a^4 x^3}+\frac{6 b^2}{a^5 x^2}-\frac{10 b^3}{a^6 x}+\frac{b^4}{a^4 (a+b x)^3}+\frac{4 b^4}{a^5 (a+b x)^2}+\frac{10 b^4}{a^6 (a+b x)}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=\frac{3 b^3}{2 a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )^2}+\frac{12 b^3}{a^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{18 b^2 \sqrt [3]{x}}{a^5}-\frac{9 b x^{2/3}}{2 a^4}+\frac{x}{a^3}-\frac{30 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}\\ \end{align*}

Mathematica [A]  time = 0.0740524, size = 83, normalized size = 0.83 \[ \frac{-9 a^2 b x^{2/3}+2 a^3 x+\frac{3 b^5}{\left (a \sqrt [3]{x}+b\right )^2}-\frac{30 b^4}{a \sqrt [3]{x}+b}+36 a b^2 \sqrt [3]{x}-60 b^3 \log \left (a \sqrt [3]{x}+b\right )}{2 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^(1/3))^(-3),x]

[Out]

((3*b^5)/(b + a*x^(1/3))^2 - (30*b^4)/(b + a*x^(1/3)) + 36*a*b^2*x^(1/3) - 9*a^2*b*x^(2/3) + 2*a^3*x - 60*b^3*
Log[b + a*x^(1/3)])/(2*a^6)

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Maple [A]  time = 0.007, size = 77, normalized size = 0.8 \begin{align*}{\frac{x}{{a}^{3}}}-{\frac{9\,b}{2\,{a}^{4}}{x}^{{\frac{2}{3}}}}+18\,{\frac{{b}^{2}\sqrt [3]{x}}{{a}^{5}}}-15\,{\frac{{b}^{4}}{{a}^{6} \left ( b+a\sqrt [3]{x} \right ) }}+{\frac{3\,{b}^{5}}{2\,{a}^{6}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}-30\,{\frac{{b}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^(1/3))^3,x)

[Out]

x/a^3-9/2*b*x^(2/3)/a^4+18*b^2*x^(1/3)/a^5-15/a^6*b^4/(b+a*x^(1/3))+3/2*b^5/a^6/(b+a*x^(1/3))^2-30/a^6*b^3*ln(
b+a*x^(1/3))

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Maxima [A]  time = 1.04606, size = 136, normalized size = 1.36 \begin{align*} \frac{2 \, a^{4} - \frac{5 \, a^{3} b}{x^{\frac{1}{3}}} + \frac{20 \, a^{2} b^{2}}{x^{\frac{2}{3}}} + \frac{90 \, a b^{3}}{x} + \frac{60 \, b^{4}}{x^{\frac{4}{3}}}}{2 \,{\left (\frac{a^{7}}{x} + \frac{2 \, a^{6} b}{x^{\frac{4}{3}}} + \frac{a^{5} b^{2}}{x^{\frac{5}{3}}}\right )}} - \frac{30 \, b^{3} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{6}} - \frac{10 \, b^{3} \log \left (x\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a^4 - 5*a^3*b/x^(1/3) + 20*a^2*b^2/x^(2/3) + 90*a*b^3/x + 60*b^4/x^(4/3))/(a^7/x + 2*a^6*b/x^(4/3) + a^
5*b^2/x^(5/3)) - 30*b^3*log(a + b/x^(1/3))/a^6 - 10*b^3*log(x)/a^6

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Fricas [A]  time = 1.53351, size = 350, normalized size = 3.5 \begin{align*} \frac{2 \, a^{9} x^{3} + 4 \, a^{6} b^{3} x^{2} - 34 \, a^{3} b^{6} x - 27 \, b^{9} - 60 \,{\left (a^{6} b^{3} x^{2} + 2 \, a^{3} b^{6} x + b^{9}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 3 \,{\left (3 \, a^{8} b x^{2} + 16 \, a^{5} b^{4} x + 10 \, a^{2} b^{7}\right )} x^{\frac{2}{3}} + 3 \,{\left (12 \, a^{7} b^{2} x^{2} + 35 \, a^{4} b^{5} x + 20 \, a b^{8}\right )} x^{\frac{1}{3}}}{2 \,{\left (a^{12} x^{2} + 2 \, a^{9} b^{3} x + a^{6} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a^9*x^3 + 4*a^6*b^3*x^2 - 34*a^3*b^6*x - 27*b^9 - 60*(a^6*b^3*x^2 + 2*a^3*b^6*x + b^9)*log(a*x^(1/3) +
b) - 3*(3*a^8*b*x^2 + 16*a^5*b^4*x + 10*a^2*b^7)*x^(2/3) + 3*(12*a^7*b^2*x^2 + 35*a^4*b^5*x + 20*a*b^8)*x^(1/3
))/(a^12*x^2 + 2*a^9*b^3*x + a^6*b^6)

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Sympy [A]  time = 1.09564, size = 362, normalized size = 3.62 \begin{align*} \begin{cases} \frac{2 a^{5} x^{\frac{5}{3}}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{5 a^{4} b x^{\frac{4}{3}}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} + \frac{20 a^{3} b^{2} x}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{60 a^{2} b^{3} x^{\frac{2}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{120 a b^{4} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{120 a b^{4} \sqrt [3]{x}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{60 b^{5} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{90 b^{5}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} & \text{for}\: a \neq 0 \\\frac{x^{2}}{2 b^{3}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**(1/3))**3,x)

[Out]

Piecewise((2*a**5*x**(5/3)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 5*a**4*b*x**(4/3)/(2*a**8*x**
(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) + 20*a**3*b**2*x/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2)
- 60*a**2*b**3*x**(2/3)*log(x**(1/3) + b/a)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 120*a*b**4*x
**(1/3)*log(x**(1/3) + b/a)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 120*a*b**4*x**(1/3)/(2*a**8*
x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 60*b**5*log(x**(1/3) + b/a)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3)
+ 2*a**6*b**2) - 90*b**5/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2), Ne(a, 0)), (x**2/(2*b**3), True
))

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Giac [A]  time = 1.17975, size = 107, normalized size = 1.07 \begin{align*} -\frac{30 \, b^{3} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{6}} - \frac{3 \,{\left (10 \, a b^{4} x^{\frac{1}{3}} + 9 \, b^{5}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} a^{6}} + \frac{2 \, a^{6} x - 9 \, a^{5} b x^{\frac{2}{3}} + 36 \, a^{4} b^{2} x^{\frac{1}{3}}}{2 \, a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-30*b^3*log(abs(a*x^(1/3) + b))/a^6 - 3/2*(10*a*b^4*x^(1/3) + 9*b^5)/((a*x^(1/3) + b)^2*a^6) + 1/2*(2*a^6*x -
9*a^5*b*x^(2/3) + 36*a^4*b^2*x^(1/3))/a^9