3.1408 \(\int \frac{x^{23}}{(2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=51 \[ \frac{1}{15} \left (x^6+2\right )^{5/2}-\frac{2}{3} \left (x^6+2\right )^{3/2}+4 \sqrt{x^6+2}+\frac{8}{3 \sqrt{x^6+2}} \]

[Out]

8/(3*Sqrt[2 + x^6]) + 4*Sqrt[2 + x^6] - (2*(2 + x^6)^(3/2))/3 + (2 + x^6)^(5/2)/15

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Rubi [A]  time = 0.0183523, antiderivative size = 51, 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} \[ \frac{1}{15} \left (x^6+2\right )^{5/2}-\frac{2}{3} \left (x^6+2\right )^{3/2}+4 \sqrt{x^6+2}+\frac{8}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Int[x^23/(2 + x^6)^(3/2),x]

[Out]

8/(3*Sqrt[2 + x^6]) + 4*Sqrt[2 + x^6] - (2*(2 + x^6)^(3/2))/3 + (2 + x^6)^(5/2)/15

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^{23}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{x^3}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \left (-\frac{8}{(2+x)^{3/2}}+\frac{12}{\sqrt{2+x}}-6 \sqrt{2+x}+(2+x)^{3/2}\right ) \, dx,x,x^6\right )\\ &=\frac{8}{3 \sqrt{2+x^6}}+4 \sqrt{2+x^6}-\frac{2}{3} \left (2+x^6\right )^{3/2}+\frac{1}{15} \left (2+x^6\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.009383, size = 28, normalized size = 0.55 \[ \frac{x^{18}-4 x^{12}+32 x^6+128}{15 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^23/(2 + x^6)^(3/2),x]

[Out]

(128 + 32*x^6 - 4*x^12 + x^18)/(15*Sqrt[2 + x^6])

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \begin{align*}{\frac{{x}^{18}-4\,{x}^{12}+32\,{x}^{6}+128}{15}{\frac{1}{\sqrt{{x}^{6}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^23/(x^6+2)^(3/2),x)

[Out]

1/15*(x^18-4*x^12+32*x^6+128)/(x^6+2)^(1/2)

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Maxima [A]  time = 0.98113, size = 50, normalized size = 0.98 \begin{align*} \frac{1}{15} \,{\left (x^{6} + 2\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} + 4 \, \sqrt{x^{6} + 2} + \frac{8}{3 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^23/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

1/15*(x^6 + 2)^(5/2) - 2/3*(x^6 + 2)^(3/2) + 4*sqrt(x^6 + 2) + 8/3/sqrt(x^6 + 2)

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Fricas [A]  time = 1.43175, size = 69, normalized size = 1.35 \begin{align*} \frac{x^{18} - 4 \, x^{12} + 32 \, x^{6} + 128}{15 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^23/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

1/15*(x^18 - 4*x^12 + 32*x^6 + 128)/sqrt(x^6 + 2)

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Sympy [A]  time = 13.7386, size = 54, normalized size = 1.06 \begin{align*} \frac{x^{18}}{15 \sqrt{x^{6} + 2}} - \frac{4 x^{12}}{15 \sqrt{x^{6} + 2}} + \frac{32 x^{6}}{15 \sqrt{x^{6} + 2}} + \frac{128}{15 \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**23/(x**6+2)**(3/2),x)

[Out]

x**18/(15*sqrt(x**6 + 2)) - 4*x**12/(15*sqrt(x**6 + 2)) + 32*x**6/(15*sqrt(x**6 + 2)) + 128/(15*sqrt(x**6 + 2)
)

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Giac [A]  time = 1.1653, size = 50, normalized size = 0.98 \begin{align*} \frac{1}{15} \,{\left (x^{6} + 2\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} + 4 \, \sqrt{x^{6} + 2} + \frac{8}{3 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^23/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

1/15*(x^6 + 2)^(5/2) - 2/3*(x^6 + 2)^(3/2) + 4*sqrt(x^6 + 2) + 8/3/sqrt(x^6 + 2)