3.1409 \(\int \frac{x^{17}}{(2+x^6)^{3/2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{1}{9} \left (x^6+2\right )^{3/2}-\frac{4 \sqrt{x^6+2}}{3}-\frac{4}{3 \sqrt{x^6+2}} \]

[Out]

-4/(3*Sqrt[2 + x^6]) - (4*Sqrt[2 + x^6])/3 + (2 + x^6)^(3/2)/9

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Rubi [A]  time = 0.0152648, antiderivative size = 40, 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} \[ \frac{1}{9} \left (x^6+2\right )^{3/2}-\frac{4 \sqrt{x^6+2}}{3}-\frac{4}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Int[x^17/(2 + x^6)^(3/2),x]

[Out]

-4/(3*Sqrt[2 + x^6]) - (4*Sqrt[2 + x^6])/3 + (2 + x^6)^(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^{17}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{x^2}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \left (\frac{4}{(2+x)^{3/2}}-\frac{4}{\sqrt{2+x}}+\sqrt{2+x}\right ) \, dx,x,x^6\right )\\ &=-\frac{4}{3 \sqrt{2+x^6}}-\frac{4 \sqrt{2+x^6}}{3}+\frac{1}{9} \left (2+x^6\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0078036, size = 23, normalized size = 0.57 \[ \frac{x^{12}-8 x^6-32}{9 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^17/(2 + x^6)^(3/2),x]

[Out]

(-32 - 8*x^6 + x^12)/(9*Sqrt[2 + x^6])

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Maple [A]  time = 0.004, size = 20, normalized size = 0.5 \begin{align*}{\frac{{x}^{12}-8\,{x}^{6}-32}{9}{\frac{1}{\sqrt{{x}^{6}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^17/(x^6+2)^(3/2),x)

[Out]

1/9*(x^12-8*x^6-32)/(x^6+2)^(1/2)

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Maxima [A]  time = 0.976307, size = 38, normalized size = 0.95 \begin{align*} \frac{1}{9} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - \frac{4}{3} \, \sqrt{x^{6} + 2} - \frac{4}{3 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17/(x^6+2)^(3/2),x, algorithm="maxima")

[Out]

1/9*(x^6 + 2)^(3/2) - 4/3*sqrt(x^6 + 2) - 4/3/sqrt(x^6 + 2)

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Fricas [A]  time = 1.41152, size = 53, normalized size = 1.32 \begin{align*} \frac{x^{12} - 8 \, x^{6} - 32}{9 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17/(x^6+2)^(3/2),x, algorithm="fricas")

[Out]

1/9*(x^12 - 8*x^6 - 32)/sqrt(x^6 + 2)

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Sympy [A]  time = 5.01411, size = 39, normalized size = 0.98 \begin{align*} \frac{x^{12}}{9 \sqrt{x^{6} + 2}} - \frac{8 x^{6}}{9 \sqrt{x^{6} + 2}} - \frac{32}{9 \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**17/(x**6+2)**(3/2),x)

[Out]

x**12/(9*sqrt(x**6 + 2)) - 8*x**6/(9*sqrt(x**6 + 2)) - 32/(9*sqrt(x**6 + 2))

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Giac [A]  time = 1.14306, size = 38, normalized size = 0.95 \begin{align*} \frac{1}{9} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - \frac{4}{3} \, \sqrt{x^{6} + 2} - \frac{4}{3 \, \sqrt{x^{6} + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17/(x^6+2)^(3/2),x, algorithm="giac")

[Out]

1/9*(x^6 + 2)^(3/2) - 4/3*sqrt(x^6 + 2) - 4/3/sqrt(x^6 + 2)