3.15.8 \(\int \frac {x^{23}}{(2+x^6)^{3/2}} \, dx\) [1408]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} \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}} \end {gather*}

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 45

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])

Rule 272

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]]

Rubi steps

\begin {align*} \int \frac {x^{23}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {x^3}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6} \text {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*}

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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.55 \begin {gather*} \frac {128+32 x^6-4 x^{12}+x^{18}}{15 \sqrt {2+x^6}} \end {gather*}

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 = 0.18, size = 25, normalized size = 0.49

method	result	size
gosper	\(\frac {x^{18}-4 x^{12}+32 x^{6}+128}{15 \sqrt {x^{6}+2}}\)	\(25\)
trager	\(\frac {x^{18}-4 x^{12}+32 x^{6}+128}{15 \sqrt {x^{6}+2}}\)	\(25\)
risch	\(\frac {x^{18}-4 x^{12}+32 x^{6}+128}{15 \sqrt {x^{6}+2}}\)	\(25\)
meijerg	\(\frac {4 \sqrt {2}\, \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (x^{18}-4 x^{12}+32 x^{6}+128\right )}{40 \sqrt {1+\frac {x^{6}}{2}}}\right )}{3 \sqrt {\pi }}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 37, normalized size = 0.73 \begin {gather*} \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 {gather*}

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 = 0.35, size = 24, normalized size = 0.47 \begin {gather*} \frac {x^{18} - 4 \, x^{12} + 32 \, x^{6} + 128}{15 \, \sqrt {x^{6} + 2}} \end {gather*}

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 = 1.27, size = 54, normalized size = 1.06 \begin {gather*} \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 {gather*}

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 = 1.38, size = 37, normalized size = 0.73 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 1.22, size = 32, normalized size = 0.63 \begin {gather*} \frac {{\left (x^6+2\right )}^3-10\,{\left (x^6+2\right )}^2+60\,x^6+160}{15\,\sqrt {x^6+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^23/(x^6 + 2)^(3/2),x)

[Out]

((x^6 + 2)^3 - 10*(x^6 + 2)^2 + 60*x^6 + 160)/(15*(x^6 + 2)^(1/2))

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