3.2377 \(\int \frac{1}{(a+b \sqrt [3]{x})^3 x} \, dx\)

Optimal. Leaf size=56 \[ \frac{3}{a^2 \left (a+b \sqrt [3]{x}\right )}-\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a^3}+\frac{\log (x)}{a^3}+\frac{3}{2 a \left (a+b \sqrt [3]{x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0320444, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 44} \[ \frac{3}{a^2 \left (a+b \sqrt [3]{x}\right )}-\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a^3}+\frac{\log (x)}{a^3}+\frac{3}{2 a \left (a+b \sqrt [3]{x}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))^3*x),x]

[Out]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 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+b \sqrt [3]{x}\right )^3 x} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x}-\frac{b}{a (a+b x)^3}-\frac{b}{a^2 (a+b x)^2}-\frac{b}{a^3 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3}{2 a \left (a+b \sqrt [3]{x}\right )^2}+\frac{3}{a^2 \left (a+b \sqrt [3]{x}\right )}-\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a^3}+\frac{\log (x)}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.0545641, size = 50, normalized size = 0.89 \[ \frac{\frac{3 a \left (3 a+2 b \sqrt [3]{x}\right )}{\left (a+b \sqrt [3]{x}\right )^2}-6 \log \left (a+b \sqrt [3]{x}\right )+2 \log (x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^(1/3))^3*x),x]

[Out]

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

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Maple [A]  time = 0.008, size = 49, normalized size = 0.9 \begin{align*}{\frac{3}{2\,a} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}+3\,{\frac{1}{{a}^{2} \left ( a+b\sqrt [3]{x} \right ) }}-3\,{\frac{\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{3}}}+{\frac{\ln \left ( x \right ) }{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.959999, size = 77, normalized size = 1.38 \begin{align*} \frac{3 \,{\left (2 \, b x^{\frac{1}{3}} + 3 \, a\right )}}{2 \,{\left (a^{2} b^{2} x^{\frac{2}{3}} + 2 \, a^{3} b x^{\frac{1}{3}} + a^{4}\right )}} - \frac{3 \, \log \left (b x^{\frac{1}{3}} + a\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*(2*b*x^(1/3) + 3*a)/(a^2*b^2*x^(2/3) + 2*a^3*b*x^(1/3) + a^4) - 3*log(b*x^(1/3) + a)/a^3 + log(x)/a^3

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Fricas [B]  time = 1.53715, size = 288, normalized size = 5.14 \begin{align*} \frac{3 \,{\left (3 \, a^{6} - 2 \,{\left (b^{6} x^{2} + 2 \, a^{3} b^{3} x + a^{6}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 2 \,{\left (b^{6} x^{2} + 2 \, a^{3} b^{3} x + a^{6}\right )} \log \left (x^{\frac{1}{3}}\right ) +{\left (2 \, a b^{5} x + 5 \, a^{4} b^{2}\right )} x^{\frac{2}{3}} -{\left (a^{2} b^{4} x + 4 \, a^{5} b\right )} x^{\frac{1}{3}}\right )}}{2 \,{\left (a^{3} b^{6} x^{2} + 2 \, a^{6} b^{3} x + a^{9}\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*(3*a^6 - 2*(b^6*x^2 + 2*a^3*b^3*x + a^6)*log(b*x^(1/3) + a) + 2*(b^6*x^2 + 2*a^3*b^3*x + a^6)*log(x^(1/3))
+ (2*a*b^5*x + 5*a^4*b^2)*x^(2/3) - (a^2*b^4*x + 4*a^5*b)*x^(1/3))/(a^3*b^6*x^2 + 2*a^6*b^3*x + a^9)

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Sympy [A]  time = 1.97934, size = 389, normalized size = 6.95 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{b^{3} x} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a^{3}} & \text{for}\: b = 0 \\\frac{2 a^{2} x^{\frac{2}{3}} \log{\left (x \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} - \frac{6 a^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} + \frac{6 a^{2} x^{\frac{2}{3}}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} + \frac{4 a b x \log{\left (x \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} - \frac{12 a b x \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} + \frac{2 b^{2} x^{\frac{4}{3}} \log{\left (x \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} - \frac{6 b^{2} x^{\frac{4}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} - \frac{3 b^{2} x^{\frac{4}{3}}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{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((zoo/x, Eq(a, 0) & Eq(b, 0)), (-1/(b**3*x), Eq(a, 0)), (log(x)/a**3, Eq(b, 0)), (2*a**2*x**(2/3)*log
(x)/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3)) - 6*a**2*x**(2/3)*log(a/b + x**(1/3))/(2*a**5*x**(2/
3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3)) + 6*a**2*x**(2/3)/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3)
) + 4*a*b*x*log(x)/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3)) - 12*a*b*x*log(a/b + x**(1/3))/(2*a**
5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3)) + 2*b**2*x**(4/3)*log(x)/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**
3*b**2*x**(4/3)) - 6*b**2*x**(4/3)*log(a/b + x**(1/3))/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3)) -
3*b**2*x**(4/3)/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3)), True))

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Giac [A]  time = 1.17679, size = 66, normalized size = 1.18 \begin{align*} -\frac{3 \, \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{3}} + \frac{\log \left ({\left | x \right |}\right )}{a^{3}} + \frac{3 \,{\left (2 \, a b x^{\frac{1}{3}} + 3 \, a^{2}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} 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="giac")

[Out]

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