3.729 \(\int \frac{x}{(a+b x^2)^{4/3}} \, dx\)

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

[Out]

-3/(2*b*(a + b*x^2)^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 0.0035819, 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} \[ -\frac{3}{2 b \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2)^(4/3),x]

[Out]

-3/(2*b*(a + b*x^2)^(1/3))

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 )^{4/3}} \, dx &=-\frac{3}{2 b \sqrt [3]{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0026457, size = 18, normalized size = 1. \[ -\frac{3}{2 b \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2)^(4/3),x]

[Out]

-3/(2*b*(a + b*x^2)^(1/3))

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*} -{\frac{3}{2\,b}{\frac{1}{\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)^(4/3),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.53318, size = 19, normalized size = 1.06 \begin{align*} -\frac{3}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^(4/3),x, algorithm="maxima")

[Out]

-3/2/((b*x^2 + a)^(1/3)*b)

________________________________________________________________________________________

Fricas [A]  time = 1.72524, size = 54, normalized size = 3. \begin{align*} -\frac{3 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{2 \,{\left (b^{2} x^{2} + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^(4/3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.603189, size = 26, normalized size = 1.44 \begin{align*} \begin{cases} - \frac{3}{2 b \sqrt [3]{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{4}{3}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x**2+a)**(4/3),x)

[Out]

Piecewise((-3/(2*b*(a + b*x**2)**(1/3)), Ne(b, 0)), (x**2/(2*a**(4/3)), True))

________________________________________________________________________________________

Giac [A]  time = 2.32453, size = 19, normalized size = 1.06 \begin{align*} -\frac{3}{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^(4/3),x, algorithm="giac")

[Out]

-3/2/((b*x^2 + a)^(1/3)*b)