Integrand size = 27, antiderivative size = 189 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3 \, dx=-\frac {46077855 (1-4 x) \sqrt {3-x+2 x^2}}{33554432}-\frac {667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac {4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac {81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac {384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac {27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac {725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}-\frac {1059790665 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{67108864 \sqrt {2}} \]
-667795/2097152*(1-4*x)*(2*x^2-x+3)^(3/2)-4625907/2293760*(2*x^2-x+3)^(5/2 )-81685/114688*x*(2*x^2-x+3)^(5/2)+384739/43008*x^2*(2*x^2-x+3)^(5/2)+2778 5/1536*x^3*(2*x^2-x+3)^(5/2)+725/48*x^4*(2*x^2-x+3)^(5/2)+25/4*x^5*(2*x^2- x+3)^(5/2)-1059790665/134217728*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-460 77855/33554432*(1-4*x)*(2*x^2-x+3)^(1/2)
Time = 0.79 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.50 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3 \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (-72152399943+53985432012 x+199615064544 x^2+389257196928 x^3+487891884032 x^4+571298324480 x^5+430820229120 x^6+328328806400 x^7+124780544000 x^8+88080384000 x^9\right )-111278019825 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{14092861440} \]
(4*Sqrt[3 - x + 2*x^2]*(-72152399943 + 53985432012*x + 199615064544*x^2 + 389257196928*x^3 + 487891884032*x^4 + 571298324480*x^5 + 430820229120*x^6 + 328328806400*x^7 + 124780544000*x^8 + 88080384000*x^9) - 111278019825*Sq rt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/14092861440
Time = 0.57 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.19, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 1160, 1087, 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.
| \(\displaystyle \int \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^3 \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{20} \int \frac {5}{2} \left (2 x^2-x+3\right )^{3/2} \left (2175 x^5+1530 x^4+1656 x^3+912 x^2+288 x+64\right )dx+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \left (2 x^2-x+3\right )^{3/2} \left (2175 x^5+1530 x^4+1656 x^3+912 x^2+288 x+64\right )dx+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{18} \int \frac {3}{2} \left (2 x^2-x+3\right )^{3/2} \left (27785 x^4+2472 x^3+10944 x^2+3456 x+768\right )dx+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \int \left (2 x^2-x+3\right )^{3/2} \left (27785 x^4+2472 x^3+10944 x^2+3456 x+768\right )dx+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{16} \int \frac {1}{2} \left (2 x^2-x+3\right )^{3/2} \left (384739 x^3-149922 x^2+110592 x+24576\right )dx+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \int \left (2 x^2-x+3\right )^{3/2} \left (384739 x^3-149922 x^2+110592 x+24576\right )dx+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {1}{14} \int \frac {3}{2} \left (-245055 x^2-506764 x+229376\right ) \left (2 x^2-x+3\right )^{3/2}dx+\frac {384739}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {3}{28} \int \left (-245055 x^2-506764 x+229376\right ) \left (2 x^2-x+3\right )^{3/2}dx+\frac {384739}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {1}{12} \int \frac {3}{2} (2325118-4625907 x) \left (2 x^2-x+3\right )^{3/2}dx-\frac {81685}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {384739}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {1}{8} \int (2325118-4625907 x) \left (2 x^2-x+3\right )^{3/2}dx-\frac {81685}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {384739}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {1}{8} \left (\frac {4674565}{4} \int \left (2 x^2-x+3\right )^{3/2}dx-\frac {4625907}{10} \left (2 x^2-x+3\right )^{5/2}\right )-\frac {81685}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {384739}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {1}{8} \left (\frac {4674565}{4} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {4625907}{10} \left (2 x^2-x+3\right )^{5/2}\right )-\frac {81685}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {384739}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {1}{8} \left (\frac {4674565}{4} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {4625907}{10} \left (2 x^2-x+3\right )^{5/2}\right )-\frac {81685}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {384739}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {1}{8} \left (\frac {4674565}{4} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {4625907}{10} \left (2 x^2-x+3\right )^{5/2}\right )-\frac {81685}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {384739}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{12} \left (\frac {1}{32} \left (\frac {3}{28} \left (\frac {1}{8} \left (\frac {4674565}{4} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {4625907}{10} \left (2 x^2-x+3\right )^{5/2}\right )-\frac {81685}{4} x \left (2 x^2-x+3\right )^{5/2}\right )+\frac {384739}{14} x^2 \left (2 x^2-x+3\right )^{5/2}\right )+\frac {27785}{16} \left (2 x^2-x+3\right )^{5/2} x^3\right )+\frac {725}{6} \left (2 x^2-x+3\right )^{5/2} x^4\right )+\frac {25}{4} \left (2 x^2-x+3\right )^{5/2} x^5\) |
(25*x^5*(3 - x + 2*x^2)^(5/2))/4 + ((725*x^4*(3 - x + 2*x^2)^(5/2))/6 + (( 27785*x^3*(3 - x + 2*x^2)^(5/2))/16 + ((384739*x^2*(3 - x + 2*x^2)^(5/2))/ 14 + (3*((-81685*x*(3 - x + 2*x^2)^(5/2))/4 + ((-4625907*(3 - x + 2*x^2)^( 5/2))/10 + (4674565*(-1/16*((1 - 4*x)*(3 - x + 2*x^2)^(3/2)) + (69*(-1/8*( (1 - 4*x)*Sqrt[3 - x + 2*x^2]) + (23*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(16*Sqr t[2])))/32))/4)/8))/28)/32)/12)/8
3.1.66.3.1 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_))*((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[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Time = 0.76 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(\frac {\left (88080384000 x^{9}+124780544000 x^{8}+328328806400 x^{7}+430820229120 x^{6}+571298324480 x^{5}+487891884032 x^{4}+389257196928 x^{3}+199615064544 x^{2}+53985432012 x -72152399943\right ) \sqrt {2 x^{2}-x +3}}{3523215360}+\frac {1059790665 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{134217728}\) | \(75\) |
| trager | \(\left (25 x^{9}+\frac {425}{12} x^{8}+\frac {35785}{384} x^{7}+\frac {438253}{3584} x^{6}+\frac {13947713}{86016} x^{5}+\frac {34032637}{245760} x^{4}+\frac {1013690617}{9175040} x^{3}+\frac {297046227}{5242880} x^{2}+\frac {4498786001}{293601280} x -\frac {24050799981}{1174405120}\right ) \sqrt {2 x^{2}-x +3}-\frac {1059790665 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )\right )}{134217728}\) | \(99\) |
| default | \(\frac {667795 \left (-1+4 x \right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{2097152}+\frac {46077855 \left (-1+4 x \right ) \sqrt {2 x^{2}-x +3}}{33554432}+\frac {1059790665 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{134217728}-\frac {4625907 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{2293760}+\frac {25 x^{5} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{4}+\frac {725 x^{4} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{48}+\frac {27785 x^{3} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{1536}+\frac {384739 x^{2} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{43008}-\frac {81685 x \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{114688}\) | \(151\) |
1/3523215360*(88080384000*x^9+124780544000*x^8+328328806400*x^7+4308202291 20*x^6+571298324480*x^5+487891884032*x^4+389257196928*x^3+199615064544*x^2 +53985432012*x-72152399943)*(2*x^2-x+3)^(1/2)+1059790665/134217728*2^(1/2) *arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3 \, dx=\frac {1}{3523215360} \, {\left (88080384000 \, x^{9} + 124780544000 \, x^{8} + 328328806400 \, x^{7} + 430820229120 \, x^{6} + 571298324480 \, x^{5} + 487891884032 \, x^{4} + 389257196928 \, x^{3} + 199615064544 \, x^{2} + 53985432012 \, x - 72152399943\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {1059790665}{268435456} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
1/3523215360*(88080384000*x^9 + 124780544000*x^8 + 328328806400*x^7 + 4308 20229120*x^6 + 571298324480*x^5 + 487891884032*x^4 + 389257196928*x^3 + 19 9615064544*x^2 + 53985432012*x - 72152399943)*sqrt(2*x^2 - x + 3) + 105979 0665/268435456*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x ^2 + 16*x - 25)
Time = 0.49 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.50 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3 \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (25 x^{9} + \frac {425 x^{8}}{12} + \frac {35785 x^{7}}{384} + \frac {438253 x^{6}}{3584} + \frac {13947713 x^{5}}{86016} + \frac {34032637 x^{4}}{245760} + \frac {1013690617 x^{3}}{9175040} + \frac {297046227 x^{2}}{5242880} + \frac {4498786001 x}{293601280} - \frac {24050799981}{1174405120}\right ) + \frac {1059790665 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{134217728} \]
sqrt(2*x**2 - x + 3)*(25*x**9 + 425*x**8/12 + 35785*x**7/384 + 438253*x**6 /3584 + 13947713*x**5/86016 + 34032637*x**4/245760 + 1013690617*x**3/91750 40 + 297046227*x**2/5242880 + 4498786001*x/293601280 - 24050799981/1174405 120) + 1059790665*sqrt(2)*asinh(4*sqrt(23)*(x - 1/4)/23)/134217728
Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.91 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3 \, dx=\frac {25}{4} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{5} + \frac {725}{48} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{4} + \frac {27785}{1536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {384739}{43008} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} - \frac {81685}{114688} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {4625907}{2293760} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {667795}{524288} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {667795}{2097152} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {46077855}{8388608} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {1059790665}{134217728} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {46077855}{33554432} \, \sqrt {2 \, x^{2} - x + 3} \]
25/4*(2*x^2 - x + 3)^(5/2)*x^5 + 725/48*(2*x^2 - x + 3)^(5/2)*x^4 + 27785/ 1536*(2*x^2 - x + 3)^(5/2)*x^3 + 384739/43008*(2*x^2 - x + 3)^(5/2)*x^2 - 81685/114688*(2*x^2 - x + 3)^(5/2)*x - 4625907/2293760*(2*x^2 - x + 3)^(5/ 2) + 667795/524288*(2*x^2 - x + 3)^(3/2)*x - 667795/2097152*(2*x^2 - x + 3 )^(3/2) + 46077855/8388608*sqrt(2*x^2 - x + 3)*x + 1059790665/134217728*sq rt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 46077855/33554432*sqrt(2*x^2 - x + 3)
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.49 \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3 \, dx=\frac {1}{3523215360} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (8 \, {\left (140 \, {\left (160 \, {\left (12 \, x + 17\right )} x + 7157\right )} x + 1314759\right )} x + 13947713\right )} x + 238228459\right )} x + 3041071851\right )} x + 6237970767\right )} x + 13496358003\right )} x - 72152399943\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {1059790665}{134217728} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
1/3523215360*(4*(8*(4*(16*(20*(8*(140*(160*(12*x + 17)*x + 7157)*x + 13147 59)*x + 13947713)*x + 238228459)*x + 3041071851)*x + 6237970767)*x + 13496 358003)*x - 72152399943)*sqrt(2*x^2 - x + 3) - 1059790665/134217728*sqrt(2 )*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3 \, dx=\int {\left (2\,x^2-x+3\right )}^{3/2}\,{\left (5\,x^2+3\,x+2\right )}^3 \,d x \]