Integrand size = 38, antiderivative size = 143 \[ \int (5+2 x) \sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=-\frac {51435 (1-4 x) \sqrt {3-x+2 x^2}}{32768}+\frac {11433 (5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}}{4480}-\frac {823 (5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}}{1344}+\frac {5}{112} (5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}-\frac {(1005757+295276 x) \left (3-x+2 x^2\right )^{3/2}}{71680}-\frac {1183005 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}} \]
11433/4480*(5+2*x)^2*(2*x^2-x+3)^(3/2)-823/1344*(5+2*x)^3*(2*x^2-x+3)^(3/2 )+5/112*(5+2*x)^4*(2*x^2-x+3)^(3/2)-1/71680*(1005757+295276*x)*(2*x^2-x+3) ^(3/2)-1183005/131072*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-51435/32768*( 1-4*x)*(2*x^2-x+3)^(1/2)
Time = 0.89 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.56 \[ \int (5+2 x) \sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (6231117+14742332 x+11357024 x^2+20304768 x^3+1390592 x^4+12984320 x^5+4915200 x^6\right )-124215525 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{13762560} \]
(4*Sqrt[3 - x + 2*x^2]*(6231117 + 14742332*x + 11357024*x^2 + 20304768*x^3 + 1390592*x^4 + 12984320*x^5 + 4915200*x^6) - 124215525*Sqrt[2]*Log[1 - 4 *x + 2*Sqrt[6 - 2*x + 4*x^2]])/13762560
Time = 0.46 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2184, 25, 2184, 27, 2184, 27, 1225, 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.
| \(\displaystyle \int (2 x+5) \sqrt {2 x^2-x+3} \left (5 x^4-x^3+3 x^2+x+2\right ) \, dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{224} \int -\left ((2 x+5) \sqrt {2 x^2-x+3} \left (6584 x^3+10788 x^2+7826 x+3677\right )\right )dx+\frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{112} (2 x+5)^4 \left (2 x^2-x+3\right )^{3/2}-\frac {1}{224} \int (2 x+5) \sqrt {2 x^2-x+3} \left (6584 x^3+10788 x^2+7826 x+3677\right )dx\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{224} \left (-\frac {1}{96} \int -24 (2 x+5) \sqrt {2 x^2-x+3} \left (45732 x^2+37828 x+14097\right )dx-\frac {823}{6} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^3\right )+\frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{224} \left (\frac {1}{4} \int (2 x+5) \sqrt {2 x^2-x+3} \left (45732 x^2+37828 x+14097\right )dx-\frac {823}{6} (2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {1}{224} \left (\frac {1}{4} \left (\frac {1}{40} \int 4 (38073-147638 x) (2 x+5) \sqrt {2 x^2-x+3}dx+\frac {11433}{5} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^2\right )-\frac {823}{6} (2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{224} \left (\frac {1}{4} \left (\frac {1}{10} \int (38073-147638 x) (2 x+5) \sqrt {2 x^2-x+3}dx+\frac {11433}{5} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^2\right )-\frac {823}{6} (2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{224} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {1800225}{16} \int \sqrt {2 x^2-x+3}dx-\frac {1}{8} (295276 x+1005757) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {11433}{5} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^2\right )-\frac {823}{6} (2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{224} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {1800225}{16} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{8} (295276 x+1005757) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {11433}{5} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^2\right )-\frac {823}{6} (2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{224} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {1800225}{16} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{8} (295276 x+1005757) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {11433}{5} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^2\right )-\frac {823}{6} (2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{224} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {1800225}{16} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{8} (295276 x+1005757) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {11433}{5} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^2\right )-\frac {823}{6} (2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {5}{112} \left (2 x^2-x+3\right )^{3/2} (2 x+5)^4\) |
(5*(5 + 2*x)^4*(3 - x + 2*x^2)^(3/2))/112 + ((-823*(5 + 2*x)^3*(3 - x + 2* x^2)^(3/2))/6 + ((11433*(5 + 2*x)^2*(3 - x + 2*x^2)^(3/2))/5 + (-1/8*((100 5757 + 295276*x)*(3 - x + 2*x^2)^(3/2)) + (1800225*(-1/8*((1 - 4*x)*Sqrt[3 - x + 2*x^2]) + (23*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[2])))/16)/10)/ 4)/224
3.4.24.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.80 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.42
| method | result | size |
| risch | \(\frac {\left (4915200 x^{6}+12984320 x^{5}+1390592 x^{4}+20304768 x^{3}+11357024 x^{2}+14742332 x +6231117\right ) \sqrt {2 x^{2}-x +3}}{3440640}+\frac {1183005 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{131072}\) | \(60\) |
| trager | \(\left (\frac {10}{7} x^{6}+\frac {317}{84} x^{5}+\frac {97}{240} x^{4}+\frac {52877}{8960} x^{3}+\frac {50701}{15360} x^{2}+\frac {3685583}{860160} x +\frac {2077039}{1146880}\right ) \sqrt {2 x^{2}-x +3}-\frac {1183005 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{131072}\) | \(84\) |
| default | \(\frac {51435 \left (-1+4 x \right ) \sqrt {2 x^{2}-x +3}}{32768}+\frac {1183005 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{131072}+\frac {283 x^{2} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{1120}-\frac {5179 x \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{17920}+\frac {242329 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{215040}+\frac {5 x^{4} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{7}+\frac {377 x^{3} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{168}\) | \(115\) |
1/3440640*(4915200*x^6+12984320*x^5+1390592*x^4+20304768*x^3+11357024*x^2+ 14742332*x+6231117)*(2*x^2-x+3)^(1/2)+1183005/131072*2^(1/2)*arcsinh(4/23* 23^(1/2)*(x-1/4))
Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.58 \[ \int (5+2 x) \sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {1}{3440640} \, {\left (4915200 \, x^{6} + 12984320 \, x^{5} + 1390592 \, x^{4} + 20304768 \, x^{3} + 11357024 \, x^{2} + 14742332 \, x + 6231117\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {1183005}{262144} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/3440640*(4915200*x^6 + 12984320*x^5 + 1390592*x^4 + 20304768*x^3 + 11357 024*x^2 + 14742332*x + 6231117)*sqrt(2*x^2 - x + 3) + 1183005/262144*sqrt( 2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
Time = 0.63 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.53 \[ \int (5+2 x) \sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {10 x^{6}}{7} + \frac {317 x^{5}}{84} + \frac {97 x^{4}}{240} + \frac {52877 x^{3}}{8960} + \frac {50701 x^{2}}{15360} + \frac {3685583 x}{860160} + \frac {2077039}{1146880}\right ) + \frac {1183005 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{131072} \]
sqrt(2*x**2 - x + 3)*(10*x**6/7 + 317*x**5/84 + 97*x**4/240 + 52877*x**3/8 960 + 50701*x**2/15360 + 3685583*x/860160 + 2077039/1146880) + 1183005*sqr t(2)*asinh(4*sqrt(23)*(x - 1/4)/23)/131072
Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.88 \[ \int (5+2 x) \sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {5}{7} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + \frac {377}{168} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {283}{1120} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {5179}{17920} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {242329}{215040} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {51435}{8192} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {1183005}{131072} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {51435}{32768} \, \sqrt {2 \, x^{2} - x + 3} \]
5/7*(2*x^2 - x + 3)^(3/2)*x^4 + 377/168*(2*x^2 - x + 3)^(3/2)*x^3 + 283/11 20*(2*x^2 - x + 3)^(3/2)*x^2 - 5179/17920*(2*x^2 - x + 3)^(3/2)*x + 242329 /215040*(2*x^2 - x + 3)^(3/2) + 51435/8192*sqrt(2*x^2 - x + 3)*x + 1183005 /131072*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 51435/32768*sqrt(2*x^2 - x + 3)
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.55 \[ \int (5+2 x) \sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {1}{3440640} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x + 317\right )} x + 679\right )} x + 158631\right )} x + 354907\right )} x + 3685583\right )} x + 6231117\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {1183005}{131072} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
1/3440640*(4*(8*(4*(16*(20*(120*x + 317)*x + 679)*x + 158631)*x + 354907)* x + 3685583)*x + 6231117)*sqrt(2*x^2 - x + 3) - 1183005/131072*sqrt(2)*log (-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
Time = 1.75 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.19 \[ \int (5+2 x) \sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right ) \, dx=\frac {283\,x^2\,{\left (2\,x^2-x+3\right )}^{3/2}}{1120}+\frac {377\,x^3\,{\left (2\,x^2-x+3\right )}^{3/2}}{168}+\frac {5\,x^4\,{\left (2\,x^2-x+3\right )}^{3/2}}{7}+\frac {4478951\,\sqrt {2}\,\ln \left (\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (2\,x-\frac {1}{2}\right )}{2}\right )}{573440}+\frac {194737\,\left (\frac {x}{2}-\frac {1}{8}\right )\,\sqrt {2\,x^2-x+3}}{17920}+\frac {242329\,\sqrt {2\,x^2-x+3}\,\left (32\,x^2-4\,x+45\right )}{3440640}-\frac {5179\,x\,{\left (2\,x^2-x+3\right )}^{3/2}}{17920}+\frac {5573567\,\sqrt {2}\,\ln \left (2\,\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (4\,x-1\right )}{2}\right )}{4587520} \]
(283*x^2*(2*x^2 - x + 3)^(3/2))/1120 + (377*x^3*(2*x^2 - x + 3)^(3/2))/168 + (5*x^4*(2*x^2 - x + 3)^(3/2))/7 + (4478951*2^(1/2)*log((2*x^2 - x + 3)^ (1/2) + (2^(1/2)*(2*x - 1/2))/2))/573440 + (194737*(x/2 - 1/8)*(2*x^2 - x + 3)^(1/2))/17920 + (242329*(2*x^2 - x + 3)^(1/2)*(32*x^2 - 4*x + 45))/344 0640 - (5179*x*(2*x^2 - x + 3)^(3/2))/17920 + (5573567*2^(1/2)*log(2*(2*x^ 2 - x + 3)^(1/2) + (2^(1/2)*(4*x - 1))/2))/4587520