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

Optimal. Leaf size=18 \[ \frac{3 \left (a+b x^2\right )^{4/3}}{8 b} \]

[Out]

(3*(a + b*x^2)^(4/3))/(8*b)

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Rubi [A]  time = 0.003245, 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} \[ \frac{3 \left (a+b x^2\right )^{4/3}}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^(1/3),x]

[Out]

(3*(a + b*x^2)^(4/3))/(8*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 x \sqrt [3]{a+b x^2} \, dx &=\frac{3 \left (a+b x^2\right )^{4/3}}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.0031798, size = 18, normalized size = 1. \[ \frac{3 \left (a+b x^2\right )^{4/3}}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^(1/3),x]

[Out]

(3*(a + b*x^2)^(4/3))/(8*b)

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Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \begin{align*}{\frac{3}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(b*x^2+a)^(4/3)/b

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Maxima [A]  time = 1.14705, size = 19, normalized size = 1.06 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(b*x^2 + a)^(4/3)/b

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Fricas [A]  time = 1.41901, size = 34, normalized size = 1.89 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(b*x^2 + a)^(4/3)/b

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Sympy [A]  time = 0.183809, size = 42, normalized size = 2.33 \begin{align*} \begin{cases} \frac{3 a \sqrt [3]{a + b x^{2}}}{8 b} + \frac{3 x^{2} \sqrt [3]{a + b x^{2}}}{8} & \text{for}\: b \neq 0 \\\frac{\sqrt [3]{a} x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*a*(a + b*x**2)**(1/3)/(8*b) + 3*x**2*(a + b*x**2)**(1/3)/8, Ne(b, 0)), (a**(1/3)*x**2/2, True))

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Giac [A]  time = 2.35255, size = 19, normalized size = 1.06 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(b*x^2 + a)^(4/3)/b