3.1384 \(\int \frac{x^{17}}{\sqrt{2+x^6}} \, dx\)

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

[Out]

(4*Sqrt[2 + x^6])/3 - (4*(2 + x^6)^(3/2))/9 + (2 + x^6)^(5/2)/15

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Rubi [A]  time = 0.0147073, 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} \[ \frac{1}{15} \left (x^6+2\right )^{5/2}-\frac{4}{9} \left (x^6+2\right )^{3/2}+\frac{4 \sqrt{x^6+2}}{3} \]

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 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}}{\sqrt{2+x^6}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+x}} \, dx,x,x^6\right )\\ &=\frac{1}{6} \operatorname{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*}

Mathematica [A]  time = 0.0065846, size = 25, normalized size = 0.62 \[ \frac{1}{45} \sqrt{x^6+2} \left (3 x^{12}-8 x^6+32\right ) \]

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 = 0.003, size = 22, normalized size = 0.6 \begin{align*}{\frac{3\,{x}^{12}-8\,{x}^{6}+32}{45}\sqrt{{x}^{6}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.991476, size = 38, normalized size = 0.95 \begin{align*} \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{align*}

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 = 1.47841, size = 57, normalized size = 1.42 \begin{align*} \frac{1}{45} \,{\left (3 \, x^{12} - 8 \, x^{6} + 32\right )} \sqrt{x^{6} + 2} \end{align*}

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 = 4.3443, size = 39, normalized size = 0.98 \begin{align*} \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{align*}

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 = 1.16345, size = 38, normalized size = 0.95 \begin{align*} \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{align*}

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)