Integrand size = 16, antiderivative size = 58 \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=-\frac {1}{40} \left (1+2 x^5\right ) \sqrt {1+x^5+x^{10}}+\frac {1}{15} \left (1+x^5+x^{10}\right )^{3/2}-\frac {3}{80} \text {arcsinh}\left (\frac {1+2 x^5}{\sqrt {3}}\right ) \]
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)
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\frac {1}{120} \sqrt {1+x^5+x^{10}} \left (5+2 x^5+8 x^{10}\right )+\frac {3}{80} \log \left (-1-2 x^5+2 \sqrt {1+x^5+x^{10}}\right ) \]
(Sqrt[1 + x^5 + x^10]*(5 + 2*x^5 + 8*x^10))/120 + (3*Log[-1 - 2*x^5 + 2*Sq rt[1 + x^5 + x^10]])/80
Time = 0.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1693, 1160, 1087, 1090, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^9 \sqrt {x^{10}+x^5+1} \, dx\) |
\(\Big \downarrow \) 1693 |
\(\displaystyle \frac {1}{5} \int x^5 \sqrt {x^{10}+x^5+1}dx^5\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (x^{10}+x^5+1\right )^{3/2}-\frac {1}{2} \int \sqrt {x^{10}+x^5+1}dx^5\right )\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (-\frac {3}{8} \int \frac {1}{\sqrt {x^{10}+x^5+1}}dx^5-\frac {1}{4} \sqrt {x^{10}+x^5+1} \left (2 x^5+1\right )\right )+\frac {1}{3} \left (x^{10}+x^5+1\right )^{3/2}\right )\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (-\frac {1}{8} \sqrt {3} \int \frac {1}{\sqrt {\frac {x^{10}}{3}+1}}d\left (2 x^5+1\right )-\frac {1}{4} \sqrt {x^{10}+x^5+1} \left (2 x^5+1\right )\right )+\frac {1}{3} \left (x^{10}+x^5+1\right )^{3/2}\right )\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{2} \left (-\frac {3}{8} \text {arcsinh}\left (\frac {2 x^5+1}{\sqrt {3}}\right )-\frac {1}{4} \sqrt {x^{10}+x^5+1} \left (2 x^5+1\right )\right )+\frac {1}{3} \left (x^{10}+x^5+1\right )^{3/2}\right )\) |
((1 + x^5 + x^10)^(3/2)/3 + (-1/4*((1 + 2*x^5)*Sqrt[1 + x^5 + x^10]) - (3* ArcSinh[(1 + 2*x^5)/Sqrt[3]])/8)/2)/5
3.4.9.3.1 Defintions of rubi rules used
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]
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] - Simp[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] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[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]
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] + Simp[(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[p, -1]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(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] && IntegerQ [Simplify[(m + 1)/n]]
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 | \(\frac {\left (8 x^{10}+2 x^{5}+5\right ) \sqrt {x^{10}+x^{5}+1}}{120}-\frac {3 \ln \left (2 x^{5}+2 \sqrt {x^{10}+x^{5}+1}+1\right )}{80}\) | \(48\) |
Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81 \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\frac {1}{120} \, {\left (8 \, x^{10} + 2 \, x^{5} + 5\right )} \sqrt {x^{10} + x^{5} + 1} + \frac {3}{80} \, \log \left (-2 \, x^{5} + 2 \, \sqrt {x^{10} + x^{5} + 1} - 1\right ) \]
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)
\[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\int x^{9} \sqrt {\left (x^{2} + x + 1\right ) \left (x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1\right )}\, dx \]
\[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\int { \sqrt {x^{10} + x^{5} + 1} x^{9} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\frac {1}{120} \, \sqrt {x^{10} + x^{5} + 1} {\left (2 \, {\left (4 \, x^{5} + 1\right )} x^{5} + 5\right )} + \frac {3}{80} \, \log \left (-2 \, x^{5} + 2 \, \sqrt {x^{10} + x^{5} + 1} - 1\right ) \]
1/120*sqrt(x^10 + x^5 + 1)*(2*(4*x^5 + 1)*x^5 + 5) + 3/80*log(-2*x^5 + 2*s qrt(x^10 + x^5 + 1) - 1)
Time = 0.33 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.74 \[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\frac {\sqrt {x^{10}+x^5+1}\,\left (8\,x^{10}+2\,x^5+5\right )}{120}-\frac {3\,\ln \left (\sqrt {x^{10}+x^5+1}+x^5+\frac {1}{2}\right )}{80} \]
((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
\[ \int x^9 \sqrt {1+x^5+x^{10}} \, dx=\frac {\sqrt {x^{10}+x^{5}+1}\, x^{10}}{15}+\frac {\sqrt {x^{10}+x^{5}+1}\, x^{5}}{60}+\frac {7 \sqrt {x^{10}+x^{5}+1}}{60}-\frac {3 \left (\int \frac {\sqrt {x^{10}+x^{5}+1}\, x^{14}}{x^{15}+2 x^{10}+2 x^{5}+1}d x \right )}{8}-\frac {9 \left (\int \frac {\sqrt {x^{10}+x^{5}+1}\, x^{9}}{x^{15}+2 x^{10}+2 x^{5}+1}d x \right )}{16}-\frac {3 \,\mathrm {log}\left (\sqrt {x^{10}+x^{5}+1}+x^{5}\right )}{80}+\frac {3 \,\mathrm {log}\left (\sqrt {x^{10}+x^{5}+1}-x^{5}\right )}{80} \]
(16*sqrt(x**10 + x**5 + 1)*x**10 + 4*sqrt(x**10 + x**5 + 1)*x**5 + 28*sqrt (x**10 + x**5 + 1) - 90*int((sqrt(x**10 + x**5 + 1)*x**14)/(x**15 + 2*x**1 0 + 2*x**5 + 1),x) - 135*int((sqrt(x**10 + x**5 + 1)*x**9)/(x**15 + 2*x**1 0 + 2*x**5 + 1),x) - 9*log(sqrt(x**10 + x**5 + 1) + x**5) + 9*log(sqrt(x** 10 + x**5 + 1) - x**5))/240