3.717 \(\int \frac{x}{(a+b x^2)^{2/3}} \, dx\)

Optimal. Leaf size=18 \[ \frac{3 \sqrt [3]{a+b x^2}}{2 b} \]

[Out]

(3*(a + b*x^2)^(1/3))/(2*b)

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Rubi [A]  time = 0.0035487, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ \frac{3 \sqrt [3]{a+b x^2}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2)^(2/3),x]

[Out]

(3*(a + b*x^2)^(1/3))/(2*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{x}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac{3 \sqrt [3]{a+b x^2}}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0023881, size = 18, normalized size = 1. \[ \frac{3 \sqrt [3]{a+b x^2}}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2)^(2/3),x]

[Out]

(3*(a + b*x^2)^(1/3))/(2*b)

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Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \begin{align*}{\frac{3}{2\,b}\sqrt [3]{b{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.15942, size = 19, normalized size = 1.06 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(b*x^2 + a)^(1/3)/b

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Fricas [A]  time = 1.86528, size = 34, normalized size = 1.89 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(b*x^2 + a)^(1/3)/b

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Sympy [A]  time = 0.412376, size = 24, normalized size = 1.33 \begin{align*} \begin{cases} \frac{3 \sqrt [3]{a + b x^{2}}}{2 b} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{2}{3}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x**2+a)**(2/3),x)

[Out]

Piecewise((3*(a + b*x**2)**(1/3)/(2*b), Ne(b, 0)), (x**2/(2*a**(2/3)), True))

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Giac [A]  time = 1.5118, size = 19, normalized size = 1.06 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(b*x^2 + a)^(1/3)/b