3.309 \(\int x^9 \sqrt{1+x^5+x^{10}} \, dx\)

Optimal. Leaf size=58 \[ \frac{1}{15} \left (x^{10}+x^5+1\right )^{3/2}-\frac{1}{40} \left (2 x^5+1\right ) \sqrt{x^{10}+x^5+1}-\frac{3}{80} \sinh ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right ) \]

[Out]

-((1 + 2*x^5)*Sqrt[1 + x^5 + x^10])/40 + (1 + x^5 + x^10)^(3/2)/15 - (3*ArcSinh[(1 + 2*x^5)/Sqrt[3]])/80

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Rubi [A]  time = 0.0370013, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1357, 640, 612, 619, 215} \[ \frac{1}{15} \left (x^{10}+x^5+1\right )^{3/2}-\frac{1}{40} \left (2 x^5+1\right ) \sqrt{x^{10}+x^5+1}-\frac{3}{80} \sinh ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^9*Sqrt[1 + x^5 + x^10],x]

[Out]

-((1 + 2*x^5)*Sqrt[1 + x^5 + x^10])/40 + (1 + x^5 + x^10)^(3/2)/15 - (3*ArcSinh[(1 + 2*x^5)/Sqrt[3]])/80

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int x^9 \sqrt{1+x^5+x^{10}} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int x \sqrt{1+x+x^2} \, dx,x,x^5\right )\\ &=\frac{1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac{1}{10} \operatorname{Subst}\left (\int \sqrt{1+x+x^2} \, dx,x,x^5\right )\\ &=-\frac{1}{40} \left (1+2 x^5\right ) \sqrt{1+x^5+x^{10}}+\frac{1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac{3}{80} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x+x^2}} \, dx,x,x^5\right )\\ &=-\frac{1}{40} \left (1+2 x^5\right ) \sqrt{1+x^5+x^{10}}+\frac{1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac{1}{80} \sqrt{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{3}}} \, dx,x,1+2 x^5\right )\\ &=-\frac{1}{40} \left (1+2 x^5\right ) \sqrt{1+x^5+x^{10}}+\frac{1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac{3}{80} \sinh ^{-1}\left (\frac{1+2 x^5}{\sqrt{3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0168163, size = 47, normalized size = 0.81 \[ \frac{1}{240} \left (2 \sqrt{x^{10}+x^5+1} \left (8 x^{10}+2 x^5+5\right )-9 \sinh ^{-1}\left (\frac{2 x^5+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^9*Sqrt[1 + x^5 + x^10],x]

[Out]

(2*Sqrt[1 + x^5 + x^10]*(5 + 2*x^5 + 8*x^10) - 9*ArcSinh[(1 + 2*x^5)/Sqrt[3]])/240

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Maple [F]  time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{x}^{9}\sqrt{{x}^{10}+{x}^{5}+1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(x^10+x^5+1)^(1/2),x)

[Out]

int(x^9*(x^10+x^5+1)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{10} + x^{5} + 1} x^{9}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(x^10+x^5+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^10 + x^5 + 1)*x^9, x)

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Fricas [A]  time = 2.37243, size = 131, normalized size = 2.26 \begin{align*} \frac{1}{120} \,{\left (8 \, x^{10} + 2 \, x^{5} + 5\right )} \sqrt{x^{10} + x^{5} + 1} + \frac{3}{80} \, \log \left (-2 \, x^{5} + 2 \, \sqrt{x^{10} + x^{5} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(x^10+x^5+1)^(1/2),x, algorithm="fricas")

[Out]

1/120*(8*x^10 + 2*x^5 + 5)*sqrt(x^10 + x^5 + 1) + 3/80*log(-2*x^5 + 2*sqrt(x^10 + x^5 + 1) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{9} \sqrt{\left (x^{2} + x + 1\right ) \left (x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(x**10+x**5+1)**(1/2),x)

[Out]

Integral(x**9*sqrt((x**2 + x + 1)*(x**8 - x**7 + x**5 - x**4 + x**3 - x + 1)), x)

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Giac [A]  time = 1.07607, size = 66, normalized size = 1.14 \begin{align*} \frac{1}{120} \, \sqrt{x^{10} + x^{5} + 1}{\left (2 \,{\left (4 \, x^{5} + 1\right )} x^{5} + 5\right )} + \frac{3}{80} \, \log \left (-2 \, x^{5} + 2 \, \sqrt{x^{10} + x^{5} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(x^10+x^5+1)^(1/2),x, algorithm="giac")

[Out]

1/120*sqrt(x^10 + x^5 + 1)*(2*(4*x^5 + 1)*x^5 + 5) + 3/80*log(-2*x^5 + 2*sqrt(x^10 + x^5 + 1) - 1)