\(\int x^9 \sqrt {1+x^5+x^{10}} \, dx\) [309]

Optimal result

Integrand size = 16, antiderivative size = 58 \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=-\frac {1}{40} \left (1+2 x^5\right ) \sqrt {1+x^5+x^{10}}+\frac {1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac {3}{80} \text {arcsinh}\left (\frac {1+2 x^5}{\sqrt {3}}\right ) \]

[Out]

1/15*(x^10+x^5+1)^(3/2)-3/80*arcsinh(1/3*(2*x^5+1)*3^(1/2))-1/40*(2*x^5+1)*(x^10+x^5+1)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1371, 654, 626, 633, 221} \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=-\frac {3}{80} \text {arcsinh}\left (\frac {2 x^5+1}{\sqrt {3}}\right )+\frac {1}{15} \left (x^{10}+x^5+1\right )^{3/2}-\frac {1}{40} \left (2 x^5+1\right ) \sqrt {x^{10}+x^5+1} \]

[In]

Int[x^9*Sqrt[1 + x^5 + x^10],x]

[Out]

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

Rule 221

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]

Rule 626

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 633

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 654

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 1371

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]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {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} \text {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} \text {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} \text {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} \text {arcsinh}\left (\frac {1+2 x^5}{\sqrt {3}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\frac {1}{120} \sqrt {1+x^5+x^{10}} \left (5+2 x^5+8 x^{10}\right )+\frac {3}{80} \log \left (-1-2 x^5+2 \sqrt {1+x^5+x^{10}}\right ) \]

[In]

Integrate[x^9*Sqrt[1 + x^5 + x^10],x]

[Out]

(Sqrt[1 + x^5 + x^10]*(5 + 2*x^5 + 8*x^10))/120 + (3*Log[-1 - 2*x^5 + 2*Sqrt[1 + x^5 + x^10]])/80

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71

method	result	size
pseudoelliptic	\(-\frac {3 \,\operatorname {arcsinh}\left (\frac {\left (2 x^{5}+1\right ) \sqrt {3}}{3}\right )}{80}+\frac {\left (8 x^{10}+2 x^{5}+5\right ) \sqrt {x^{10}+x^{5}+1}}{120}\)	\(41\)
trager	\(\left (\frac {1}{15} x^{10}+\frac {1}{60} x^{5}+\frac {1}{24}\right ) \sqrt {x^{10}+x^{5}+1}+\frac {3 \ln \left (-2 x^{5}+2 \sqrt {x^{10}+x^{5}+1}-1\right )}{80}\)	\(47\)
risch	\(\frac {\left (8 x^{10}+2 x^{5}+5\right ) \sqrt {x^{10}+x^{5}+1}}{120}-\frac {3 \ln \left (2 x^{5}+2 \sqrt {x^{10}+x^{5}+1}+1\right )}{80}\)	\(48\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81 \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\frac {1}{120} \, {\left (8 \, x^{10} + 2 \, x^{5} + 5\right )} \sqrt {x^{10} + x^{5} + 1} + \frac {3}{80} \, \log \left (-2 \, x^{5} + 2 \, \sqrt {x^{10} + x^{5} + 1} - 1\right ) \]

[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)

Sympy [F]

\[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\int x^{9} \sqrt {\left (x^{2} + x + 1\right ) \left (x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1\right )}\, dx \]

[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)

Maxima [F]

\[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\int { \sqrt {x^{10} + x^{5} + 1} x^{9} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\frac {1}{120} \, \sqrt {x^{10} + x^{5} + 1} {\left (2 \, {\left (4 \, x^{5} + 1\right )} x^{5} + 5\right )} + \frac {3}{80} \, \log \left (-2 \, x^{5} + 2 \, \sqrt {x^{10} + x^{5} + 1} - 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.74 \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\frac {\sqrt {x^{10}+x^5+1}\,\left (8\,x^{10}+2\,x^5+5\right )}{120}-\frac {3\,\ln \left (\sqrt {x^{10}+x^5+1}+x^5+\frac {1}{2}\right )}{80} \]

[In]

int(x^9*(x^5 + x^10 + 1)^(1/2),x)

[Out]

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