3.440 \(\int \frac{x^5}{\sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=27 \[ \frac{2}{9} \left (x^3+1\right )^{3/2}-\frac{2 \sqrt{x^3+1}}{3} \]

[Out]

(-2*Sqrt[1 + x^3])/3 + (2*(1 + x^3)^(3/2))/9

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Rubi [A]  time = 0.0102721, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{2}{9} \left (x^3+1\right )^{3/2}-\frac{2 \sqrt{x^3+1}}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[1 + x^3],x]

[Out]

(-2*Sqrt[1 + x^3])/3 + (2*(1 + x^3)^(3/2))/9

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^5}{\sqrt{1+x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt{1+x}}+\sqrt{1+x}\right ) \, dx,x,x^3\right )\\ &=-\frac{2}{3} \sqrt{1+x^3}+\frac{2}{9} \left (1+x^3\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0049559, size = 18, normalized size = 0.67 \[ \frac{2}{9} \left (x^3-2\right ) \sqrt{x^3+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[1 + x^3],x]

[Out]

(2*(-2 + x^3)*Sqrt[1 + x^3])/9

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Maple [A]  time = 0.004, size = 26, normalized size = 1. \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ({x}^{3}-2 \right ) }{9}{\frac{1}{\sqrt{{x}^{3}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^3+1)^(1/2),x)

[Out]

2/9*(1+x)*(x^2-x+1)*(x^3-2)/(x^3+1)^(1/2)

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Maxima [A]  time = 0.967665, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{9} \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

2/9*(x^3 + 1)^(3/2) - 2/3*sqrt(x^3 + 1)

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Fricas [A]  time = 1.94551, size = 39, normalized size = 1.44 \begin{align*} \frac{2}{9} \, \sqrt{x^{3} + 1}{\left (x^{3} - 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

2/9*sqrt(x^3 + 1)*(x^3 - 2)

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Sympy [A]  time = 0.363514, size = 26, normalized size = 0.96 \begin{align*} \frac{2 x^{3} \sqrt{x^{3} + 1}}{9} - \frac{4 \sqrt{x^{3} + 1}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(x**3+1)**(1/2),x)

[Out]

2*x**3*sqrt(x**3 + 1)/9 - 4*sqrt(x**3 + 1)/9

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Giac [A]  time = 1.11529, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{9} \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

2/9*(x^3 + 1)^(3/2) - 2/3*sqrt(x^3 + 1)