Integrand size = 27, antiderivative size = 185 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {16493087661 \sqrt {3-x+2 x^2}}{29360128}+\frac {1572007407 x \sqrt {3-x+2 x^2}}{7340032}-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {2899366573 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8388608 \sqrt {2}} \] Output:
16493087661/29360128*(2*x^2-x+3)^(1/2)+1572007407/7340032*x*(2*x^2-x+3)^(1 /2)-15428243/131072*x^2*(2*x^2-x+3)^(1/2)-19750457/229376*x^3*(2*x^2-x+3)^ (1/2)+686531/6144*x^4*(2*x^2-x+3)^(1/2)+2116475/10752*x^5*(2*x^2-x+3)^(1/2 )+57375/448*x^6*(2*x^2-x+3)^(1/2)+625/16*x^7*(2*x^2-x+3)^(1/2)+2899366573/ 16777216*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)
Time = 0.92 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.46 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (49479262983+18864088884 x-10367779296 x^2-7584175488 x^3+9842108416 x^4+17338163200 x^5+11280384000 x^6+3440640000 x^7\right )+60886698033 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{352321536} \] Input:
Integrate[(2 + 3*x + 5*x^2)^4/Sqrt[3 - x + 2*x^2],x]
Output:
(4*Sqrt[3 - x + 2*x^2]*(49479262983 + 18864088884*x - 10367779296*x^2 - 75 84175488*x^3 + 9842108416*x^4 + 17338163200*x^5 + 11280384000*x^6 + 344064 0000*x^7) + 60886698033*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/35 2321536
Time = 0.76 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.19, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 1160, 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.
| \(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^4}{\sqrt {2 x^2-x+3}} \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{16} \int \frac {57375 x^7+48950 x^6+74880 x^5+56352 x^4+29952 x^3+12032 x^2+3072 x+512}{2 \sqrt {2 x^2-x+3}}dx+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \int \frac {57375 x^7+48950 x^6+74880 x^5+56352 x^4+29952 x^3+12032 x^2+3072 x+512}{\sqrt {2 x^2-x+3}}dx+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{14} \int \frac {2116475 x^6+31140 x^5+1577856 x^4+838656 x^3+336896 x^2+86016 x+14336}{2 \sqrt {2 x^2-x+3}}dx+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \int \frac {2116475 x^6+31140 x^5+1577856 x^4+838656 x^3+336896 x^2+86016 x+14336}{\sqrt {2 x^2-x+3}}dx+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{12} \int \frac {24028585 x^5-25625706 x^4+20127744 x^3+8085504 x^2+2064384 x+344064}{2 \sqrt {2 x^2-x+3}}dx+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \int \frac {24028585 x^5-25625706 x^4+20127744 x^3+8085504 x^2+2064384 x+344064}{\sqrt {2 x^2-x+3}}dx+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {1}{10} \int \frac {15 \left (-19750457 x^4-11608744 x^3+10780672 x^2+2752512 x+458752\right )}{2 \sqrt {2 x^2-x+3}}dx+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {3}{4} \int \frac {-19750457 x^4-11608744 x^3+10780672 x^2+2752512 x+458752}{\sqrt {2 x^2-x+3}}dx+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{8} \int \frac {-323993103 x^3+527998978 x^2+44040192 x+7340032}{2 \sqrt {2 x^2-x+3}}dx-\frac {19750457}{8} x^3 \sqrt {2 x^2-x+3}\right )+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \int \frac {-323993103 x^3+527998978 x^2+44040192 x+7340032}{\sqrt {2 x^2-x+3}}dx-\frac {19750457}{8} x^3 \sqrt {2 x^2-x+3}\right )+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{6} \int \frac {3 \left (1572007407 x^2+1472133180 x+29360128\right )}{2 \sqrt {2 x^2-x+3}}dx-\frac {107997701}{2} x^2 \sqrt {2 x^2-x+3}\right )-\frac {19750457}{8} x^3 \sqrt {2 x^2-x+3}\right )+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \int \frac {1572007407 x^2+1472133180 x+29360128}{\sqrt {2 x^2-x+3}}dx-\frac {107997701}{2} x^2 \sqrt {2 x^2-x+3}\right )-\frac {19750457}{8} x^3 \sqrt {2 x^2-x+3}\right )+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \left (\frac {1}{4} \int -\frac {9197163418-16493087661 x}{2 \sqrt {2 x^2-x+3}}dx+\frac {1572007407}{4} \sqrt {2 x^2-x+3} x\right )-\frac {107997701}{2} x^2 \sqrt {2 x^2-x+3}\right )-\frac {19750457}{8} x^3 \sqrt {2 x^2-x+3}\right )+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \left (\frac {1572007407}{4} x \sqrt {2 x^2-x+3}-\frac {1}{8} \int \frac {9197163418-16493087661 x}{\sqrt {2 x^2-x+3}}dx\right )-\frac {107997701}{2} x^2 \sqrt {2 x^2-x+3}\right )-\frac {19750457}{8} x^3 \sqrt {2 x^2-x+3}\right )+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {16493087661}{2} \sqrt {2 x^2-x+3}-\frac {20295566011}{4} \int \frac {1}{\sqrt {2 x^2-x+3}}dx\right )+\frac {1572007407}{4} \sqrt {2 x^2-x+3} x\right )-\frac {107997701}{2} x^2 \sqrt {2 x^2-x+3}\right )-\frac {19750457}{8} x^3 \sqrt {2 x^2-x+3}\right )+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {16493087661}{2} \sqrt {2 x^2-x+3}-\frac {20295566011 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{4 \sqrt {46}}\right )+\frac {1572007407}{4} \sqrt {2 x^2-x+3} x\right )-\frac {107997701}{2} x^2 \sqrt {2 x^2-x+3}\right )-\frac {19750457}{8} x^3 \sqrt {2 x^2-x+3}\right )+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{24} \left (\frac {3}{4} \left (\frac {1}{16} \left (\frac {1}{4} \left (\frac {1}{8} \left (\frac {16493087661}{2} \sqrt {2 x^2-x+3}-\frac {20295566011 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{4 \sqrt {2}}\right )+\frac {1572007407}{4} \sqrt {2 x^2-x+3} x\right )-\frac {107997701}{2} x^2 \sqrt {2 x^2-x+3}\right )-\frac {19750457}{8} x^3 \sqrt {2 x^2-x+3}\right )+\frac {4805717}{2} \sqrt {2 x^2-x+3} x^4\right )+\frac {2116475}{12} \sqrt {2 x^2-x+3} x^5\right )+\frac {57375}{14} \sqrt {2 x^2-x+3} x^6\right )+\frac {625}{16} \sqrt {2 x^2-x+3} x^7\) |
Input:
Int[(2 + 3*x + 5*x^2)^4/Sqrt[3 - x + 2*x^2],x]
Output:
(625*x^7*Sqrt[3 - x + 2*x^2])/16 + ((57375*x^6*Sqrt[3 - x + 2*x^2])/14 + ( (2116475*x^5*Sqrt[3 - x + 2*x^2])/12 + ((4805717*x^4*Sqrt[3 - x + 2*x^2])/ 2 + (3*((-19750457*x^3*Sqrt[3 - x + 2*x^2])/8 + ((-107997701*x^2*Sqrt[3 - x + 2*x^2])/2 + ((1572007407*x*Sqrt[3 - x + 2*x^2])/4 + ((16493087661*Sqrt [3 - x + 2*x^2])/2 - (20295566011*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(4*Sqrt[2] ))/8)/4)/16))/4)/24)/28)/32
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[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 = 2.39 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.35
| method | result | size |
| risch | \(\frac {\left (3440640000 x^{7}+11280384000 x^{6}+17338163200 x^{5}+9842108416 x^{4}-7584175488 x^{3}-10367779296 x^{2}+18864088884 x +49479262983\right ) \sqrt {2 x^{2}-x +3}}{88080384}-\frac {2899366573 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16777216}\) | \(65\) |
| trager | \(\left (\frac {625}{16} x^{7}+\frac {57375}{448} x^{6}+\frac {2116475}{10752} x^{5}+\frac {686531}{6144} x^{4}-\frac {19750457}{229376} x^{3}-\frac {15428243}{131072} x^{2}+\frac {1572007407}{7340032} x +\frac {16493087661}{29360128}\right ) \sqrt {2 x^{2}-x +3}+\frac {2899366573 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{16777216}\) | \(89\) |
| default | \(-\frac {2899366573 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16777216}+\frac {16493087661 \sqrt {2 x^{2}-x +3}}{29360128}+\frac {1572007407 x \sqrt {2 x^{2}-x +3}}{7340032}-\frac {15428243 x^{2} \sqrt {2 x^{2}-x +3}}{131072}-\frac {19750457 x^{3} \sqrt {2 x^{2}-x +3}}{229376}+\frac {686531 x^{4} \sqrt {2 x^{2}-x +3}}{6144}+\frac {2116475 x^{5} \sqrt {2 x^{2}-x +3}}{10752}+\frac {57375 x^{6} \sqrt {2 x^{2}-x +3}}{448}+\frac {625 x^{7} \sqrt {2 x^{2}-x +3}}{16}\) | \(147\) |
Input:
int((5*x^2+3*x+2)^4/(2*x^2-x+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/88080384*(3440640000*x^7+11280384000*x^6+17338163200*x^5+9842108416*x^4- 7584175488*x^3-10367779296*x^2+18864088884*x+49479262983)*(2*x^2-x+3)^(1/2 )-2899366573/16777216*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.48 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {1}{88080384} \, {\left (3440640000 \, x^{7} + 11280384000 \, x^{6} + 17338163200 \, x^{5} + 9842108416 \, x^{4} - 7584175488 \, x^{3} - 10367779296 \, x^{2} + 18864088884 \, x + 49479262983\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {2899366573}{33554432} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3)^(1/2),x, algorithm="fricas")
Output:
1/88080384*(3440640000*x^7 + 11280384000*x^6 + 17338163200*x^5 + 984210841 6*x^4 - 7584175488*x^3 - 10367779296*x^2 + 18864088884*x + 49479262983)*sq rt(2*x^2 - x + 3) + 2899366573/33554432*sqrt(2)*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)
Time = 0.48 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.45 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\sqrt {3-x+2 x^2}} \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {625 x^{7}}{16} + \frac {57375 x^{6}}{448} + \frac {2116475 x^{5}}{10752} + \frac {686531 x^{4}}{6144} - \frac {19750457 x^{3}}{229376} - \frac {15428243 x^{2}}{131072} + \frac {1572007407 x}{7340032} + \frac {16493087661}{29360128}\right ) - \frac {2899366573 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{16777216} \] Input:
integrate((5*x**2+3*x+2)**4/(2*x**2-x+3)**(1/2),x)
Output:
sqrt(2*x**2 - x + 3)*(625*x**7/16 + 57375*x**6/448 + 2116475*x**5/10752 + 686531*x**4/6144 - 19750457*x**3/229376 - 15428243*x**2/131072 + 157200740 7*x/7340032 + 16493087661/29360128) - 2899366573*sqrt(2)*asinh(4*sqrt(23)* (x - 1/4)/23)/16777216
Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.80 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {625}{16} \, \sqrt {2 \, x^{2} - x + 3} x^{7} + \frac {57375}{448} \, \sqrt {2 \, x^{2} - x + 3} x^{6} + \frac {2116475}{10752} \, \sqrt {2 \, x^{2} - x + 3} x^{5} + \frac {686531}{6144} \, \sqrt {2 \, x^{2} - x + 3} x^{4} - \frac {19750457}{229376} \, \sqrt {2 \, x^{2} - x + 3} x^{3} - \frac {15428243}{131072} \, \sqrt {2 \, x^{2} - x + 3} x^{2} + \frac {1572007407}{7340032} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {2899366573}{16777216} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {16493087661}{29360128} \, \sqrt {2 \, x^{2} - x + 3} \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3)^(1/2),x, algorithm="maxima")
Output:
625/16*sqrt(2*x^2 - x + 3)*x^7 + 57375/448*sqrt(2*x^2 - x + 3)*x^6 + 21164 75/10752*sqrt(2*x^2 - x + 3)*x^5 + 686531/6144*sqrt(2*x^2 - x + 3)*x^4 - 1 9750457/229376*sqrt(2*x^2 - x + 3)*x^3 - 15428243/131072*sqrt(2*x^2 - x + 3)*x^2 + 1572007407/7340032*sqrt(2*x^2 - x + 3)*x - 2899366573/16777216*sq rt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) + 16493087661/29360128*sqrt(2*x^2 - x + 3)
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.45 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {1}{88080384} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, {\left (120 \, {\left (140 \, x + 459\right )} x + 84659\right )} x + 4805717\right )} x - 59251371\right )} x - 323993103\right )} x + 4716022221\right )} x + 49479262983\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {2899366573}{16777216} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3)^(1/2),x, algorithm="giac")
Output:
1/88080384*(4*(8*(4*(16*(100*(120*(140*x + 459)*x + 84659)*x + 4805717)*x - 59251371)*x - 323993103)*x + 4716022221)*x + 49479262983)*sqrt(2*x^2 - x + 3) + 2899366573/16777216*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
Timed out. \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\sqrt {3-x+2 x^2}} \, dx=\int \frac {{\left (5\,x^2+3\,x+2\right )}^4}{\sqrt {2\,x^2-x+3}} \,d x \] Input:
int((3*x + 5*x^2 + 2)^4/(2*x^2 - x + 3)^(1/2),x)
Output:
int((3*x + 5*x^2 + 2)^4/(2*x^2 - x + 3)^(1/2), x)
Time = 0.22 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.83 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\sqrt {3-x+2 x^2}} \, dx=\frac {625 \sqrt {2 x^{2}-x +3}\, x^{7}}{16}+\frac {57375 \sqrt {2 x^{2}-x +3}\, x^{6}}{448}+\frac {2116475 \sqrt {2 x^{2}-x +3}\, x^{5}}{10752}+\frac {686531 \sqrt {2 x^{2}-x +3}\, x^{4}}{6144}-\frac {19750457 \sqrt {2 x^{2}-x +3}\, x^{3}}{229376}-\frac {15428243 \sqrt {2 x^{2}-x +3}\, x^{2}}{131072}+\frac {1572007407 \sqrt {2 x^{2}-x +3}\, x}{7340032}+\frac {16493087661 \sqrt {2 x^{2}-x +3}}{29360128}-\frac {2899366573 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )}{16777216} \] Input:
int((5*x^2+3*x+2)^4/(2*x^2-x+3)^(1/2),x)
Output:
(13762560000*sqrt(2*x**2 - x + 3)*x**7 + 45121536000*sqrt(2*x**2 - x + 3)* x**6 + 69352652800*sqrt(2*x**2 - x + 3)*x**5 + 39368433664*sqrt(2*x**2 - x + 3)*x**4 - 30336701952*sqrt(2*x**2 - x + 3)*x**3 - 41471117184*sqrt(2*x* *2 - x + 3)*x**2 + 75456355536*sqrt(2*x**2 - x + 3)*x + 197917051932*sqrt( 2*x**2 - x + 3) - 60886698033*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 4*x - 1)/sqrt(23)))/352321536