3.2441 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^3 x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0340394, antiderivative size = 56, 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, 266, 44} \[ \frac{3}{b^2 \left (a \sqrt [3]{x}+b\right )}-\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac{3}{2 b \left (a \sqrt [3]{x}+b\right )^2}+\frac{\log (x)}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))^3*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0550758, size = 50, normalized size = 0.89 \[ \frac{\frac{3 b \left (2 a \sqrt [3]{x}+3 b\right )}{\left (a \sqrt [3]{x}+b\right )^2}-6 \log \left (a \sqrt [3]{x}+b\right )+2 \log (x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^(1/3))^3*x^2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^(1/3))^3/x^2,x)

[Out]

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

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Maxima [A]  time = 0.97323, size = 62, normalized size = 1.11 \begin{align*} -\frac{3 \, \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{3}} - \frac{6 \, a}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{3}} + \frac{3 \, a^{2}}{2 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.54666, size = 288, normalized size = 5.14 \begin{align*} \frac{3 \,{\left (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 ) + 2 \,{\left (a^{6} x^{2} + 2 \, a^{3} b^{3} x + b^{6}\right )} \log \left (x^{\frac{1}{3}}\right ) +{\left (2 \, a^{5} b x + 5 \, a^{2} b^{4}\right )} x^{\frac{2}{3}} -{\left (a^{4} b^{2} x + 4 \, a b^{5}\right )} x^{\frac{1}{3}}\right )}}{2 \,{\left (a^{6} b^{3} x^{2} + 2 \, a^{3} b^{6} x + b^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 5.84419, size = 406, normalized size = 7.25 \begin{align*} \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\log{\left (x \right )}}{b^{3}} & \text{for}\: a = 0 \\- \frac{1}{a^{3} x} & \text{for}\: b = 0 \\\frac{2 a^{2} x^{\frac{7}{3}} \log{\left (x \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} - \frac{6 a^{2} x^{\frac{7}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} - \frac{9 a^{2} x^{\frac{7}{3}}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} + \frac{4 a b x^{2} \log{\left (x \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} - \frac{12 a b x^{2} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} - \frac{12 a b x^{2}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} + \frac{2 b^{2} x^{\frac{5}{3}} \log{\left (x \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{3}}} - \frac{6 b^{2} x^{\frac{5}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{2} b^{3} x^{\frac{7}{3}} + 4 a b^{4} x^{2} + 2 b^{5} x^{\frac{5}{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**2,x)

[Out]

Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0)), (log(x)/b**3, Eq(a, 0)), (-1/(a**3*x), Eq(b, 0)), (2*a**2*x**(7/3
)*log(x)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) - 6*a**2*x**(7/3)*log(x**(1/3) + b/a)/(2*a**
2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) - 9*a**2*x**(7/3)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2
*b**5*x**(5/3)) + 4*a*b*x**2*log(x)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) - 12*a*b*x**2*log
(x**(1/3) + b/a)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)) - 12*a*b*x**2/(2*a**2*b**3*x**(7/3)
+ 4*a*b**4*x**2 + 2*b**5*x**(5/3)) + 2*b**2*x**(5/3)*log(x)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**
(5/3)) - 6*b**2*x**(5/3)*log(x**(1/3) + b/a)/(2*a**2*b**3*x**(7/3) + 4*a*b**4*x**2 + 2*b**5*x**(5/3)), True))

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Giac [A]  time = 1.18034, size = 66, normalized size = 1.18 \begin{align*} -\frac{3 \, \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{3}} + \frac{\log \left ({\left | x \right |}\right )}{b^{3}} + \frac{3 \,{\left (2 \, a b x^{\frac{1}{3}} + 3 \, b^{2}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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