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

Optimal. Leaf size=18 \[ \frac{2 \left (a+b x^3\right )^{3/2}}{9 b} \]

[Out]

(2*(a + b*x^3)^(3/2))/(9*b)

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Rubi [A]  time = 0.0040069, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{2 \left (a+b x^3\right )^{3/2}}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a + b*x^3],x]

[Out]

(2*(a + b*x^3)^(3/2))/(9*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^2 \sqrt{a+b x^3} \, dx &=\frac{2 \left (a+b x^3\right )^{3/2}}{9 b}\\ \end{align*}

Mathematica [A]  time = 0.0043185, size = 18, normalized size = 1. \[ \frac{2 \left (a+b x^3\right )^{3/2}}{9 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[a + b*x^3],x]

[Out]

(2*(a + b*x^3)^(3/2))/(9*b)

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Maple [A]  time = 0.005, size = 15, normalized size = 0.8 \begin{align*}{\frac{2}{9\,b} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(b*x^3+a)^(3/2)/b

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Maxima [A]  time = 1.00323, size = 19, normalized size = 1.06 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(b*x^3 + a)^(3/2)/b

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Fricas [A]  time = 1.42344, size = 34, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(b*x^3 + a)^(3/2)/b

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Sympy [A]  time = 0.25668, size = 42, normalized size = 2.33 \begin{align*} \begin{cases} \frac{2 a \sqrt{a + b x^{3}}}{9 b} + \frac{2 x^{3} \sqrt{a + b x^{3}}}{9} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a*sqrt(a + b*x**3)/(9*b) + 2*x**3*sqrt(a + b*x**3)/9, Ne(b, 0)), (sqrt(a)*x**3/3, True))

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Giac [A]  time = 1.11739, size = 19, normalized size = 1.06 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(b*x^3 + a)^(3/2)/b