3.40 \(\int \frac{x^5}{\sqrt{5+x^2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{1}{5} \left (x^2+5\right )^{5/2}-\frac{10}{3} \left (x^2+5\right )^{3/2}+25 \sqrt{x^2+5} \]

[Out]

25*Sqrt[5 + x^2] - (10*(5 + x^2)^(3/2))/3 + (5 + x^2)^(5/2)/5

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Rubi [A]  time = 0.0129593, antiderivative size = 38, 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} \[ \frac{1}{5} \left (x^2+5\right )^{5/2}-\frac{10}{3} \left (x^2+5\right )^{3/2}+25 \sqrt{x^2+5} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[5 + x^2],x]

[Out]

25*Sqrt[5 + x^2] - (10*(5 + x^2)^(3/2))/3 + (5 + x^2)^(5/2)/5

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{5+x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{5+x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{25}{\sqrt{5+x}}-10 \sqrt{5+x}+(5+x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=25 \sqrt{5+x^2}-\frac{10}{3} \left (5+x^2\right )^{3/2}+\frac{1}{5} \left (5+x^2\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0063945, size = 25, normalized size = 0.66 \[ \frac{1}{15} \sqrt{x^2+5} \left (3 x^4-20 x^2+200\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[5 + x^2],x]

[Out]

(Sqrt[5 + x^2]*(200 - 20*x^2 + 3*x^4))/15

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Maple [A]  time = 0.003, size = 22, normalized size = 0.6 \begin{align*}{\frac{3\,{x}^{4}-20\,{x}^{2}+200}{15}\sqrt{{x}^{2}+5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(x^2+5)^(1/2)*(3*x^4-20*x^2+200)

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Maxima [A]  time = 1.43064, size = 46, normalized size = 1.21 \begin{align*} \frac{1}{5} \, \sqrt{x^{2} + 5} x^{4} - \frac{4}{3} \, \sqrt{x^{2} + 5} x^{2} + \frac{40}{3} \, \sqrt{x^{2} + 5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(x^2 + 5)*x^4 - 4/3*sqrt(x^2 + 5)*x^2 + 40/3*sqrt(x^2 + 5)

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Fricas [A]  time = 0.425337, size = 58, normalized size = 1.53 \begin{align*} \frac{1}{15} \,{\left (3 \, x^{4} - 20 \, x^{2} + 200\right )} \sqrt{x^{2} + 5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*x^4 - 20*x^2 + 200)*sqrt(x^2 + 5)

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Sympy [A]  time = 1.24128, size = 39, normalized size = 1.03 \begin{align*} \frac{x^{4} \sqrt{x^{2} + 5}}{5} - \frac{4 x^{2} \sqrt{x^{2} + 5}}{3} + \frac{40 \sqrt{x^{2} + 5}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4*sqrt(x**2 + 5)/5 - 4*x**2*sqrt(x**2 + 5)/3 + 40*sqrt(x**2 + 5)/3

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Giac [A]  time = 1.08353, size = 38, normalized size = 1. \begin{align*} \frac{1}{5} \,{\left (x^{2} + 5\right )}^{\frac{5}{2}} - \frac{10}{3} \,{\left (x^{2} + 5\right )}^{\frac{3}{2}} + 25 \, \sqrt{x^{2} + 5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(x^2 + 5)^(5/2) - 10/3*(x^2 + 5)^(3/2) + 25*sqrt(x^2 + 5)