3.441 \(\int \frac{x^2}{\sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=13 \[ \frac{2 \sqrt{x^3+1}}{3} \]

[Out]

(2*Sqrt[1 + x^3])/3

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Rubi [A]  time = 0.0025572, antiderivative size = 13, 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} \[ \frac{2 \sqrt{x^3+1}}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[1 + x^3])/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^2}{\sqrt{1+x^3}} \, dx &=\frac{2 \sqrt{1+x^3}}{3}\\ \end{align*}

Mathematica [A]  time = 0.0020025, size = 13, normalized size = 1. \[ \frac{2 \sqrt{x^3+1}}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[1 + x^3])/3

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Maple [B]  time = 0.004, size = 21, normalized size = 1.6 \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) }{3}{\frac{1}{\sqrt{{x}^{3}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^3+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.960236, size = 12, normalized size = 0.92 \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(x^3 + 1)

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Fricas [A]  time = 1.49133, size = 26, normalized size = 2. \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(x^3 + 1)

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Sympy [A]  time = 0.170545, size = 10, normalized size = 0.77 \begin{align*} \frac{2 \sqrt{x^{3} + 1}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x**3 + 1)/3

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Giac [A]  time = 1.11704, size = 12, normalized size = 0.92 \begin{align*} \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(x^3 + 1)