3.2425 \(\int \frac{1}{(a+\frac{b}{\sqrt [3]{x}}) x^2} \, dx\)

Optimal. Leaf size=51 \[ -\frac{3 a^2 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac{a^2 \log (x)}{b^3}+\frac{3 a}{b^2 \sqrt [3]{x}}-\frac{3}{2 b x^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0309035, antiderivative size = 51, 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} \[ -\frac{3 a^2 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac{a^2 \log (x)}{b^3}+\frac{3 a}{b^2 \sqrt [3]{x}}-\frac{3}{2 b x^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^(1/3))*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0383975, size = 48, normalized size = 0.94 \[ \frac{-6 a^2 \log \left (a \sqrt [3]{x}+b\right )+2 a^2 \log (x)-\frac{3 b \left (b-2 a \sqrt [3]{x}\right )}{x^{2/3}}}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^(1/3))*x^2),x]

[Out]

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

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Maple [A]  time = 0.008, size = 44, normalized size = 0.9 \begin{align*} -{\frac{3}{2\,b}{x}^{-{\frac{2}{3}}}}+3\,{\frac{a}{{b}^{2}\sqrt [3]{x}}}-3\,{\frac{{a}^{2}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{3}}}+{\frac{{a}^{2}\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))/x^2,x)

[Out]

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

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Maxima [A]  time = 0.954732, size = 59, normalized size = 1.16 \begin{align*} -\frac{3 \, a^{2} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{3}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2}}{2 \, b^{3}} + \frac{6 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.48734, size = 130, normalized size = 2.55 \begin{align*} -\frac{3 \,{\left (2 \, a^{2} x \log \left (a x^{\frac{1}{3}} + b\right ) - 2 \, a^{2} x \log \left (x^{\frac{1}{3}}\right ) - 2 \, a b x^{\frac{2}{3}} + b^{2} x^{\frac{1}{3}}\right )}}{2 \, b^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.1679, size = 73, normalized size = 1.43 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{2}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{2 b x^{\frac{2}{3}}} & \text{for}\: a = 0 \\- \frac{1}{a x} & \text{for}\: b = 0 \\\frac{a^{2} \log{\left (x \right )}}{b^{3}} - \frac{3 a^{2} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{b^{3}} + \frac{3 a}{b^{2} \sqrt [3]{x}} - \frac{3}{2 b x^{\frac{2}{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))/x**2,x)

[Out]

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

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Giac [A]  time = 1.19898, size = 66, normalized size = 1.29 \begin{align*} -\frac{3 \, a^{2} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{3}} + \frac{a^{2} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{3 \,{\left (2 \, a b x^{\frac{1}{3}} - b^{2}\right )}}{2 \, b^{3} x^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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