3.1383 \(\int \frac{x^{23}}{\sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=53 \[ \frac{1}{21} \left (x^6+2\right )^{7/2}-\frac{2}{5} \left (x^6+2\right )^{5/2}+\frac{4}{3} \left (x^6+2\right )^{3/2}-\frac{8 \sqrt{x^6+2}}{3} \]

[Out]

(-8*Sqrt[2 + x^6])/3 + (4*(2 + x^6)^(3/2))/3 - (2*(2 + x^6)^(5/2))/5 + (2 + x^6)^(7/2)/21

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Rubi [A]  time = 0.0212099, antiderivative size = 53, 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} \[ \frac{1}{21} \left (x^6+2\right )^{7/2}-\frac{2}{5} \left (x^6+2\right )^{5/2}+\frac{4}{3} \left (x^6+2\right )^{3/2}-\frac{8 \sqrt{x^6+2}}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^23/Sqrt[2 + x^6],x]

[Out]

(-8*Sqrt[2 + x^6])/3 + (4*(2 + x^6)^(3/2))/3 - (2*(2 + x^6)^(5/2))/5 + (2 + x^6)^(7/2)/21

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}}{\sqrt{2+x^6}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{2+x}} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \left (-\frac{8}{\sqrt{2+x}}+12 \sqrt{2+x}-6 (2+x)^{3/2}+(2+x)^{5/2}\right ) \, dx,x,x^6\right )\\ &=-\frac{8}{3} \sqrt{2+x^6}+\frac{4}{3} \left (2+x^6\right )^{3/2}-\frac{2}{5} \left (2+x^6\right )^{5/2}+\frac{1}{21} \left (2+x^6\right )^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0087013, size = 30, normalized size = 0.57 \[ \frac{1}{105} \sqrt{x^6+2} \left (5 x^{18}-12 x^{12}+32 x^6-128\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^23/Sqrt[2 + x^6],x]

[Out]

(Sqrt[2 + x^6]*(-128 + 32*x^6 - 12*x^12 + 5*x^18))/105

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Maple [A]  time = 0.005, size = 27, normalized size = 0.5 \begin{align*}{\frac{5\,{x}^{18}-12\,{x}^{12}+32\,{x}^{6}-128}{105}\sqrt{{x}^{6}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^23/(x^6+2)^(1/2),x)

[Out]

1/105*(x^6+2)^(1/2)*(5*x^18-12*x^12+32*x^6-128)

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Maxima [A]  time = 1.02291, size = 50, normalized size = 0.94 \begin{align*} \frac{1}{21} \,{\left (x^{6} + 2\right )}^{\frac{7}{2}} - \frac{2}{5} \,{\left (x^{6} + 2\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (x^{6} + 2\right )}^{\frac{3}{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)^(1/2),x, algorithm="maxima")

[Out]

1/21*(x^6 + 2)^(7/2) - 2/5*(x^6 + 2)^(5/2) + 4/3*(x^6 + 2)^(3/2) - 8/3*sqrt(x^6 + 2)

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Fricas [A]  time = 1.41826, size = 74, normalized size = 1.4 \begin{align*} \frac{1}{105} \,{\left (5 \, x^{18} - 12 \, x^{12} + 32 \, x^{6} - 128\right )} \sqrt{x^{6} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^23/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

1/105*(5*x^18 - 12*x^12 + 32*x^6 - 128)*sqrt(x^6 + 2)

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Sympy [A]  time = 12.0197, size = 54, normalized size = 1.02 \begin{align*} \frac{x^{18} \sqrt{x^{6} + 2}}{21} - \frac{4 x^{12} \sqrt{x^{6} + 2}}{35} + \frac{32 x^{6} \sqrt{x^{6} + 2}}{105} - \frac{128 \sqrt{x^{6} + 2}}{105} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**23/(x**6+2)**(1/2),x)

[Out]

x**18*sqrt(x**6 + 2)/21 - 4*x**12*sqrt(x**6 + 2)/35 + 32*x**6*sqrt(x**6 + 2)/105 - 128*sqrt(x**6 + 2)/105

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Giac [A]  time = 1.18287, size = 50, normalized size = 0.94 \begin{align*} \frac{1}{21} \,{\left (x^{6} + 2\right )}^{\frac{7}{2}} - \frac{2}{5} \,{\left (x^{6} + 2\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (x^{6} + 2\right )}^{\frac{3}{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)^(1/2),x, algorithm="giac")

[Out]

1/21*(x^6 + 2)^(7/2) - 2/5*(x^6 + 2)^(5/2) + 4/3*(x^6 + 2)^(3/2) - 8/3*sqrt(x^6 + 2)