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

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

[Out]

(2*Sqrt[a + b*x^3])/(3*b)

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Rubi [A]  time = 0.0045655, 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} \[ \frac{2 \sqrt{a+b x^3}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[a + b*x^3],x]

[Out]

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

Mathematica [A]  time = 0.003546, size = 18, normalized size = 1. \[ \frac{2 \sqrt{a+b x^3}}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[a + b*x^3],x]

[Out]

(2*Sqrt[a + b*x^3])/(3*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.956793, size = 19, normalized size = 1.06 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a}}{3 \, 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]

2/3*sqrt(b*x^3 + a)/b

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Fricas [A]  time = 1.44843, size = 31, normalized size = 1.72 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a}}{3 \, 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]

2/3*sqrt(b*x^3 + a)/b

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Sympy [A]  time = 0.609666, size = 24, normalized size = 1.33 \begin{align*} \begin{cases} \frac{2 \sqrt{a + b x^{3}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 \sqrt{a}} & \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*sqrt(a + b*x**3)/(3*b), Ne(b, 0)), (x**3/(3*sqrt(a)), True))

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Giac [A]  time = 1.09125, size = 19, normalized size = 1.06 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a}}{3 \, 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]

2/3*sqrt(b*x^3 + a)/b