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]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {3 \sqrt [3]{a+b x^2}}{2 b} \]

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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fricas [A]  time = 0.76, size = 14, normalized size = 0.78 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{2 \, b} \]

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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giac [A]  time = 0.57, size = 14, normalized size = 0.78 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{2 \, b} \]

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

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \[ \frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{2 b} \]

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 = 1.24, size = 14, normalized size = 0.78 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{2 \, b} \]

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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mupad [B]  time = 4.69, size = 14, normalized size = 0.78 \[ \frac {3\,{\left (b\,x^2+a\right )}^{1/3}}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^2)^(2/3),x)

[Out]

(3*(a + b*x^2)^(1/3))/(2*b)

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sympy [A]  time = 0.41, size = 24, normalized size = 1.33 \[ \begin {cases} \frac {3 \sqrt [3]{a + b x^{2}}}{2 b} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {2}{3}}} & \text {otherwise} \end {cases} \]

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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