3.15.9 \(\int \frac {x^{17}}{(2+x^6)^{3/2}} \, dx\) [1409]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 40, 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}{9} \left (x^6+2\right )^{3/2}-\frac {4 \sqrt {x^6+2}}{3}-\frac {4}{3 \sqrt {x^6+2}} \end {gather*}

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 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^{17}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {x^2}{(2+x)^{3/2}} \, dx,x,x^6\right )\\ &=\frac {1}{6} \text {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*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.58 \begin {gather*} \frac {-32-8 x^6+x^{12}}{9 \sqrt {2+x^6}} \end {gather*}

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 = 0.17, size = 20, normalized size = 0.50

method	result	size
gosper	\(\frac {x^{12}-8 x^{6}-32}{9 \sqrt {x^{6}+2}}\)	\(20\)
trager	\(\frac {x^{12}-8 x^{6}-32}{9 \sqrt {x^{6}+2}}\)	\(20\)
risch	\(\frac {x^{12}-8 x^{6}-32}{9 \sqrt {x^{6}+2}}\)	\(20\)
meijerg	\(\frac {2 \sqrt {2}\, \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {1}{2} x^{12}+4 x^{6}+16\right )}{6 \sqrt {1+\frac {x^{6}}{2}}}\right )}{3 \sqrt {\pi }}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 28, normalized size = 0.70 \begin {gather*} \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 {gather*}

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 = 0.37, size = 19, normalized size = 0.48 \begin {gather*} \frac {x^{12} - 8 \, x^{6} - 32}{9 \, \sqrt {x^{6} + 2}} \end {gather*}

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 = 0.66, size = 39, normalized size = 0.98 \begin {gather*} \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 {gather*}

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 = 1.38, size = 28, normalized size = 0.70 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 1.18, size = 25, normalized size = 0.62 \begin {gather*} -\frac {12\,x^6-{\left (x^6+2\right )}^2+36}{9\,\sqrt {x^6+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^17/(x^6 + 2)^(3/2),x)

[Out]

-(12*x^6 - (x^6 + 2)^2 + 36)/(9*(x^6 + 2)^(1/2))

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