3.2433 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}})^2 x^2} \, dx\)

Optimal. Leaf size=52 \[ -\frac{3 a}{b^2 \left (a \sqrt [3]{x}+b\right )}+\frac{6 a \log \left (a \sqrt [3]{x}+b\right )}{b^3}-\frac{2 a \log (x)}{b^3}-\frac{3}{b^2 \sqrt [3]{x}} \]

[Out]

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

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Rubi [A]  time = 0.0349992, antiderivative size = 52, 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} \[ -\frac{3 a}{b^2 \left (a \sqrt [3]{x}+b\right )}+\frac{6 a \log \left (a \sqrt [3]{x}+b\right )}{b^3}-\frac{2 a \log (x)}{b^3}-\frac{3}{b^2 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))^2*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0477713, size = 46, normalized size = 0.88 \[ \frac{-\frac{3 a b}{a \sqrt [3]{x}+b}+6 a \log \left (a \sqrt [3]{x}+b\right )-2 a \log (x)-\frac{3 b}{\sqrt [3]{x}}}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 47, normalized size = 0.9 \begin{align*} -3\,{\frac{a}{{b}^{2} \left ( b+a\sqrt [3]{x} \right ) }}-3\,{\frac{1}{{b}^{2}\sqrt [3]{x}}}+6\,{\frac{a\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{3}}}-2\,{\frac{a\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))^2/x^2,x)

[Out]

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

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Maxima [A]  time = 0.991677, size = 59, normalized size = 1.13 \begin{align*} \frac{6 \, a \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{3}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}}{b^{3}} + \frac{3 \, a^{2}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.53328, size = 217, normalized size = 4.17 \begin{align*} \frac{3 \,{\left (a^{2} b^{2} x^{\frac{4}{3}} - a b^{3} x + 2 \,{\left (a^{4} x^{2} + a b^{3} x\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 2 \,{\left (a^{4} x^{2} + a b^{3} x\right )} \log \left (x^{\frac{1}{3}}\right ) -{\left (2 \, a^{3} b x + b^{4}\right )} x^{\frac{2}{3}}\right )}}{a^{3} b^{3} x^{2} + b^{6} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.34589, size = 209, normalized size = 4.02 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt [3]{x}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{b^{2} \sqrt [3]{x}} & \text{for}\: a = 0 \\- \frac{1}{a^{2} x} & \text{for}\: b = 0 \\- \frac{2 a^{2} x^{2} \log{\left (x \right )}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} + \frac{6 a^{2} x^{2} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} + \frac{6 a^{2} x^{2}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} - \frac{2 a b x^{\frac{5}{3}} \log{\left (x \right )}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} + \frac{6 a b x^{\frac{5}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{a b^{3} x^{2} + b^{4} x^{\frac{5}{3}}} - \frac{3 b^{2} x^{\frac{4}{3}}}{a b^{3} x^{2} + b^{4} 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))**2/x**2,x)

[Out]

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

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Giac [A]  time = 1.22513, size = 69, normalized size = 1.33 \begin{align*} \frac{6 \, a \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{3}} - \frac{2 \, a \log \left ({\left | x \right |}\right )}{b^{3}} - \frac{3 \,{\left (2 \, a x^{\frac{1}{3}} + b\right )}}{{\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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