3.2361 \(\int \frac{1}{(a+b \sqrt [3]{x}) x^2} \, dx\)

Optimal. Leaf size=63 \[ -\frac{3 b^2}{a^3 \sqrt [3]{x}}+\frac{3 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^4}-\frac{b^3 \log (x)}{a^4}+\frac{3 b}{2 a^2 x^{2/3}}-\frac{1}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0344333, antiderivative size = 63, 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} \[ -\frac{3 b^2}{a^3 \sqrt [3]{x}}+\frac{3 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^4}-\frac{b^3 \log (x)}{a^4}+\frac{3 b}{2 a^2 x^{2/3}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0303699, size = 63, normalized size = 1. \[ -\frac{3 b^2}{a^3 \sqrt [3]{x}}+\frac{3 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^4}-\frac{b^3 \log (x)}{a^4}+\frac{3 b}{2 a^2 x^{2/3}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^(1/3))*x^2),x]

[Out]

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

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Maple [A]  time = 0.009, size = 56, normalized size = 0.9 \begin{align*} -{\frac{1}{ax}}+{\frac{3\,b}{2\,{a}^{2}}{x}^{-{\frac{2}{3}}}}-3\,{\frac{{b}^{2}}{{a}^{3}\sqrt [3]{x}}}+3\,{\frac{{b}^{3}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{4}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.996957, size = 76, normalized size = 1.21 \begin{align*} \frac{3 \, b^{3} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{4}} - \frac{b^{3} \log \left (x\right )}{a^{4}} - \frac{6 \, b^{2} x^{\frac{2}{3}} - 3 \, a b x^{\frac{1}{3}} + 2 \, a^{2}}{2 \, a^{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="maxima")

[Out]

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

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Fricas [A]  time = 1.52795, size = 147, normalized size = 2.33 \begin{align*} \frac{6 \, b^{3} x \log \left (b x^{\frac{1}{3}} + a\right ) - 6 \, b^{3} x \log \left (x^{\frac{1}{3}}\right ) - 6 \, a b^{2} x^{\frac{2}{3}} + 3 \, a^{2} b x^{\frac{1}{3}} - 2 \, a^{3}}{2 \, a^{4} 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]

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

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Sympy [A]  time = 2.84922, size = 83, normalized size = 1.32 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{4}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{3}{4 b x^{\frac{4}{3}}} & \text{for}\: a = 0 \\- \frac{1}{a x} & \text{for}\: b = 0 \\- \frac{1}{a x} + \frac{3 b}{2 a^{2} x^{\frac{2}{3}}} - \frac{3 b^{2}}{a^{3} \sqrt [3]{x}} - \frac{b^{3} \log{\left (x \right )}}{a^{4}} + \frac{3 b^{3} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{4}} & \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**(4/3), Eq(a, 0) & Eq(b, 0)), (-3/(4*b*x**(4/3)), Eq(a, 0)), (-1/(a*x), Eq(b, 0)), (-1/(a*x)
+ 3*b/(2*a**2*x**(2/3)) - 3*b**2/(a**3*x**(1/3)) - b**3*log(x)/a**4 + 3*b**3*log(a/b + x**(1/3))/a**4, True))

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Giac [A]  time = 1.22319, size = 82, normalized size = 1.3 \begin{align*} \frac{3 \, b^{3} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{4}} - \frac{b^{3} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x^{\frac{2}{3}} - 3 \, a^{2} b x^{\frac{1}{3}} + 2 \, a^{3}}{2 \, a^{4} 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="giac")

[Out]

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