3.14.84 \(\int \frac {x^{17}}{\sqrt {2+x^6}} \, dx\) [1384]

Optimal. Leaf size=40 \[ \frac {4 \sqrt {2+x^6}}{3}-\frac {4}{9} \left (2+x^6\right )^{3/2}+\frac {1}{15} \left (2+x^6\right )^{5/2} \]

[Out]

-4/9*(x^6+2)^(3/2)+1/15*(x^6+2)^(5/2)+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}{15} \left (x^6+2\right )^{5/2}-\frac {4}{9} \left (x^6+2\right )^{3/2}+\frac {4 \sqrt {x^6+2}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^17/Sqrt[2 + x^6],x]

[Out]

(4*Sqrt[2 + x^6])/3 - (4*(2 + x^6)^(3/2))/9 + (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^{17}}{\sqrt {2+x^6}} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+x}} \, dx,x,x^6\right )\\ &=\frac {1}{6} \text {Subst}\left (\int \left (\frac {4}{\sqrt {2+x}}-4 \sqrt {2+x}+(2+x)^{3/2}\right ) \, dx,x,x^6\right )\\ &=\frac {4 \sqrt {2+x^6}}{3}-\frac {4}{9} \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 = 25, normalized size = 0.62 \begin {gather*} \frac {1}{45} \sqrt {2+x^6} \left (32-8 x^6+3 x^{12}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^17/Sqrt[2 + x^6],x]

[Out]

(Sqrt[2 + x^6]*(32 - 8*x^6 + 3*x^12))/45

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Maple [A]
time = 0.17, size = 21, normalized size = 0.52

method	result	size
trager	\(\sqrt {x^{6}+2}\, \left (\frac {1}{15} x^{12}-\frac {8}{45} x^{6}+\frac {32}{45}\right )\)	\(21\)
gosper	\(\frac {\sqrt {x^{6}+2}\, \left (3 x^{12}-8 x^{6}+32\right )}{45}\)	\(22\)
risch	\(\frac {\sqrt {x^{6}+2}\, \left (3 x^{12}-8 x^{6}+32\right )}{45}\)	\(22\)
meijerg	\(\frac {2 \sqrt {2}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {3}{2} x^{12}-4 x^{6}+16\right ) \sqrt {1+\frac {x^{6}}{2}}}{15}\right )}{3 \sqrt {\pi }}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^17/(x^6+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(x^6+2)^(1/2)*(1/15*x^12-8/45*x^6+32/45)

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Maxima [A]
time = 0.29, size = 28, normalized size = 0.70 \begin {gather*} \frac {1}{15} \, {\left (x^{6} + 2\right )}^{\frac {5}{2}} - \frac {4}{9} \, {\left (x^{6} + 2\right )}^{\frac {3}{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)^(1/2),x, algorithm="maxima")

[Out]

1/15*(x^6 + 2)^(5/2) - 4/9*(x^6 + 2)^(3/2) + 4/3*sqrt(x^6 + 2)

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Fricas [A]
time = 0.34, size = 21, normalized size = 0.52 \begin {gather*} \frac {1}{45} \, {\left (3 \, x^{12} - 8 \, x^{6} + 32\right )} \sqrt {x^{6} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17/(x^6+2)^(1/2),x, algorithm="fricas")

[Out]

1/45*(3*x^12 - 8*x^6 + 32)*sqrt(x^6 + 2)

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Sympy [A]
time = 0.53, size = 39, normalized size = 0.98 \begin {gather*} \frac {x^{12} \sqrt {x^{6} + 2}}{15} - \frac {8 x^{6} \sqrt {x^{6} + 2}}{45} + \frac {32 \sqrt {x^{6} + 2}}{45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**17/(x**6+2)**(1/2),x)

[Out]

x**12*sqrt(x**6 + 2)/15 - 8*x**6*sqrt(x**6 + 2)/45 + 32*sqrt(x**6 + 2)/45

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Giac [A]
time = 1.53, size = 28, normalized size = 0.70 \begin {gather*} \frac {1}{15} \, {\left (x^{6} + 2\right )}^{\frac {5}{2}} - \frac {4}{9} \, {\left (x^{6} + 2\right )}^{\frac {3}{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)^(1/2),x, algorithm="giac")

[Out]

1/15*(x^6 + 2)^(5/2) - 4/9*(x^6 + 2)^(3/2) + 4/3*sqrt(x^6 + 2)

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Mupad [B]
time = 1.15, size = 20, normalized size = 0.50 \begin {gather*} \sqrt {x^6+2}\,\left (\frac {x^{12}}{15}-\frac {8\,x^6}{45}+\frac {32}{45}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^17/(x^6 + 2)^(1/2),x)

[Out]

(x^6 + 2)^(1/2)*(x^12/15 - (8*x^6)/45 + 32/45)

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