3.2440 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^3 x} \, dx\)

Optimal. Leaf size=54 \[ -\frac{3 b^2}{2 a^3 \left (a \sqrt [3]{x}+b\right )^2}+\frac{6 b}{a^3 \left (a \sqrt [3]{x}+b\right )}+\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a^3} \]

[Out]

(-3*b^2)/(2*a^3*(b + a*x^(1/3))^2) + (6*b)/(a^3*(b + a*x^(1/3))) + (3*Log[b + a*x^(1/3)])/a^3

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Rubi [A]  time = 0.0321558, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 190, 43} \[ -\frac{3 b^2}{2 a^3 \left (a \sqrt [3]{x}+b\right )^2}+\frac{6 b}{a^3 \left (a \sqrt [3]{x}+b\right )}+\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))^3*x),x]

[Out]

(-3*b^2)/(2*a^3*(b + a*x^(1/3))^2) + (6*b)/(a^3*(b + a*x^(1/3))) + (3*Log[b + a*x^(1/3)])/a^3

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x} \, dx &=\int \frac{1}{\left (b+a \sqrt [3]{x}\right )^3} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x^2}{(b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{b^2}{a^2 (b+a x)^3}-\frac{2 b}{a^2 (b+a x)^2}+\frac{1}{a^2 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{3 b^2}{2 a^3 \left (b+a \sqrt [3]{x}\right )^2}+\frac{6 b}{a^3 \left (b+a \sqrt [3]{x}\right )}+\frac{3 \log \left (b+a \sqrt [3]{x}\right )}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.0384186, size = 45, normalized size = 0.83 \[ \frac{3 \left (\frac{b \left (4 a \sqrt [3]{x}+3 b\right )}{\left (a \sqrt [3]{x}+b\right )^2}+2 \log \left (a \sqrt [3]{x}+b\right )\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^(1/3))^3*x),x]

[Out]

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

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Maple [B]  time = 0.069, size = 237, normalized size = 4.4 \begin{align*} -{\frac{9\,{b}^{6}}{2\, \left ({a}^{3}x+{b}^{3} \right ) ^{2}{a}^{3}}}+{\frac{\ln \left ({a}^{3}x+{b}^{3} \right ) }{{a}^{3}}}+9\,{\frac{{b}^{3}}{{a}^{3} \left ({a}^{3}x+{b}^{3} \right ) }}-{\frac{13\,{b}^{2}}{2\,a}{x}^{{\frac{2}{3}}} \left ({a}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{b}^{2} \right ) ^{-2}}+5\,{\frac{{b}^{3}\sqrt [3]{x}}{{a}^{2} \left ({a}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{b}^{2} \right ) ^{2}}}-3\,{\frac{{b}^{4}}{{a}^{3} \left ({a}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{b}^{2} \right ) ^{2}}}-{\frac{1}{{a}^{3}}\ln \left ({a}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{b}^{2} \right ) }+2\,{\frac{\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{3}}}-{\frac{{b}^{2}}{{a}^{3}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}+2\,{\frac{bx}{ \left ({a}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{b}^{2} \right ) ^{2}}}+4\,{\frac{b}{{a}^{3} \left ( b+a\sqrt [3]{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/2*b^6/(a^3*x+b^3)^2/a^3+1/a^3*ln(a^3*x+b^3)+9/a^3*b^3/(a^3*x+b^3)-13/2/a*b^2/(a^2*x^(2/3)-a*b*x^(1/3)+b^2)^
2*x^(2/3)+5/a^2*b^3/(a^2*x^(2/3)-a*b*x^(1/3)+b^2)^2*x^(1/3)-3/a^3*b^4/(a^2*x^(2/3)-a*b*x^(1/3)+b^2)^2-1/a^3*ln
(a^2*x^(2/3)-a*b*x^(1/3)+b^2)+2*ln(b+a*x^(1/3))/a^3-b^2/a^3/(b+a*x^(1/3))^2+2*b/(a^2*x^(2/3)-a*b*x^(1/3)+b^2)^
2*x+4*b/a^3/(b+a*x^(1/3))

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Maxima [A]  time = 1.00305, size = 77, normalized size = 1.43 \begin{align*} -\frac{3 \,{\left (3 \, a + \frac{2 \, b}{x^{\frac{1}{3}}}\right )}}{2 \,{\left (a^{4} + \frac{2 \, a^{3} b}{x^{\frac{1}{3}}} + \frac{a^{2} b^{2}}{x^{\frac{2}{3}}}\right )}} + \frac{3 \, \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{3}} + \frac{\log \left (x\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.5667, size = 243, normalized size = 4.5 \begin{align*} \frac{3 \,{\left (6 \, a^{3} b^{3} x + 3 \, b^{6} + 2 \,{\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) +{\left (4 \, a^{5} b x + a^{2} b^{4}\right )} x^{\frac{2}{3}} -{\left (5 \, a^{4} b^{2} x + 2 \, a b^{5}\right )} x^{\frac{1}{3}}\right )}}{2 \,{\left (a^{9} x^{2} + 2 \, a^{6} b^{3} x + a^{3} 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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.30311, size = 243, normalized size = 4.5 \begin{align*} \begin{cases} \frac{6 a^{2} x^{\frac{4}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{5} x^{\frac{4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{2}{3}}} - \frac{6 a^{2} x^{\frac{4}{3}}}{2 a^{5} x^{\frac{4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{2}{3}}} + \frac{12 a b x \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{5} x^{\frac{4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{2}{3}}} + \frac{6 b^{2} x^{\frac{2}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{5} x^{\frac{4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{2}{3}}} + \frac{3 b^{2} x^{\frac{2}{3}}}{2 a^{5} x^{\frac{4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{2}{3}}} & \text{for}\: a \neq 0 \\\frac{x}{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,x)

[Out]

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

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Giac [A]  time = 1.16251, size = 59, normalized size = 1.09 \begin{align*} \frac{3 \, \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{3}} + \frac{3 \,{\left (4 \, b x^{\frac{1}{3}} + \frac{3 \, b^{2}}{a}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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