Integrand size = 27, antiderivative size = 246 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {4353943-6508666 x}{941410976 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac {5 (7318+17315 x)}{1860496 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )}+\frac {3 \sqrt {\frac {1}{682} \left (13874275807943+9819738650000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (13874275807943+9819738650000 \sqrt {2}\right )}} \left (5538393+4123702 \sqrt {2}+\left (13785797+9662095 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{81861824}-\frac {3 \sqrt {\frac {1}{682} \left (-13874275807943+9819738650000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-13874275807943+9819738650000 \sqrt {2}\right )}} \left (5538393-4123702 \sqrt {2}+\left (13785797-9662095 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{81861824} \]
1/941410976*(-4353943+6508666*x)/(2*x^2-x+3)^(1/2)+1/1364*(4+65*x)/(5*x^2+ 3*x+2)^2/(2*x^2-x+3)^(1/2)+5/1860496*(7318+17315*x)/(5*x^2+3*x+2)/(2*x^2-x +3)^(1/2)-3/55829763968*arctanh(1/31*(5538393+x*(13785797-9662095*2^(1/2)) -4123702*2^(1/2))*341^(1/2)/(-13874275807943+9819738650000*2^(1/2))^(1/2)/ (2*x^2-x+3)^(1/2))*(-9462256101017126+6697061759300000*2^(1/2))^(1/2)+3/55 829763968*arctan(1/31*(5538393+4123702*2^(1/2)+x*(13785797+9662095*2^(1/2) ))*341^(1/2)/(13874275807943+9819738650000*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2 ))*(9462256101017126+6697061759300000*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.06 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.47 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {4 \left (22374044+161806828 x+175833195 x^2+277167774 x^3+86411405 x^4+162716650 x^5\right )}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}-176824 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-491 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+208 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+5 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+124 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {7194481 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-3798456 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+575915 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-7 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {143178771 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-105962920 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+6180225 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{3765643904} \]
((4*(22374044 + 161806828*x + 175833195*x^2 + 277167774*x^3 + 86411405*x^4 + 162716650*x^5))/(Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)^2) - 176824*Sqrt [2]*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (- 491*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 208*Log[-(Sqrt[ 2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 5*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ] + 124*Sqrt[2]*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (7194481*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] - 3798456*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 575915*Sqrt[2 ]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ] - 7*Sqrt[2]*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (143178771*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] - 105962920*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2* x^2] - #1]*#1 + 6180225*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - # 1]*#1^2)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ])/3765643904
Time = 0.76 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1305, 27, 2135, 27, 2135, 27, 1368, 27, 1362, 217, 219}
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.
| \(\displaystyle \int \frac {1}{\left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}-\frac {\int -\frac {11 \left (1040 x^2-687 x+1042\right )}{2 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}dx}{15004}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {1040 x^2-687 x+1042}{\left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}dx}{2728}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {\frac {\int \frac {11 \left (692600 x^2-28425 x+483914\right )}{2 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}dx}{7502}+\frac {5 (17315 x+7318)}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {692600 x^2-28425 x+483914}{\left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}dx}{1364}+\frac {5 (17315 x+7318)}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {759 (847654-395185 x)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2783}-\frac {4353943-6508666 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {5 (17315 x+7318)}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{22} \int \frac {847654-395185 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {4353943-6508666 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {5 (17315 x+7318)}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {\frac {\frac {3}{22} \left (\frac {\int -\frac {11 \left (-\left (\left (452469-395185 \sqrt {2}\right ) x\right )-847654 \sqrt {2}+1242839\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (452469+395185 \sqrt {2}\right ) x\right )+847654 \sqrt {2}+1242839\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )-\frac {4353943-6508666 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {5 (17315 x+7318)}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{22} \left (\frac {\int \frac {-\left (\left (452469+395185 \sqrt {2}\right ) x\right )+847654 \sqrt {2}+1242839}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (452469-395185 \sqrt {2}\right ) x\right )-847654 \sqrt {2}+1242839}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )-\frac {4353943-6508666 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {5 (17315 x+7318)}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {\frac {\frac {3}{22} \left (\frac {\left (13874275807943-9819738650000 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (13785797-9662095 \sqrt {2}\right ) x-4123702 \sqrt {2}+5538393\right )^2}{2 x^2-x+3}-31 \left (13874275807943-9819738650000 \sqrt {2}\right )}d\frac {\left (13785797-9662095 \sqrt {2}\right ) x-4123702 \sqrt {2}+5538393}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (13874275807943+9819738650000 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (13785797+9662095 \sqrt {2}\right ) x+4123702 \sqrt {2}+5538393\right )^2}{2 x^2-x+3}-31 \left (13874275807943+9819738650000 \sqrt {2}\right )}d\frac {\left (13785797+9662095 \sqrt {2}\right ) x+4123702 \sqrt {2}+5538393}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}\right )-\frac {4353943-6508666 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {5 (17315 x+7318)}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {3}{22} \left (\frac {\left (13874275807943-9819738650000 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (13785797-9662095 \sqrt {2}\right ) x-4123702 \sqrt {2}+5538393\right )^2}{2 x^2-x+3}-31 \left (13874275807943-9819738650000 \sqrt {2}\right )}d\frac {\left (13785797-9662095 \sqrt {2}\right ) x-4123702 \sqrt {2}+5538393}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (13874275807943+9819738650000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (13874275807943+9819738650000 \sqrt {2}\right )}} \left (\left (13785797+9662095 \sqrt {2}\right ) x+4123702 \sqrt {2}+5538393\right )}{\sqrt {2 x^2-x+3}}\right )\right )-\frac {4353943-6508666 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {5 (17315 x+7318)}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {3}{22} \left (\sqrt {\frac {1}{682} \left (13874275807943+9819738650000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (13874275807943+9819738650000 \sqrt {2}\right )}} \left (\left (13785797+9662095 \sqrt {2}\right ) x+4123702 \sqrt {2}+5538393\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (13874275807943-9819738650000 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (9819738650000 \sqrt {2}-13874275807943\right )}} \left (\left (13785797-9662095 \sqrt {2}\right ) x-4123702 \sqrt {2}+5538393\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (9819738650000 \sqrt {2}-13874275807943\right )}}\right )-\frac {4353943-6508666 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {5 (17315 x+7318)}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}}{2728}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}\) |
(4 + 65*x)/(1364*Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)^2) + ((5*(7318 + 17 315*x))/(682*Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)) + (-1/253*(4353943 - 6 508666*x)/Sqrt[3 - x + 2*x^2] + (3*(Sqrt[(13874275807943 + 9819738650000*S qrt[2])/682]*ArcTan[(Sqrt[11/(31*(13874275807943 + 9819738650000*Sqrt[2])) ]*(5538393 + 4123702*Sqrt[2] + (13785797 + 9662095*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + ((13874275807943 - 9819738650000*Sqrt[2])*ArcTanh[(Sqrt[11/(3 1*(-13874275807943 + 9819738650000*Sqrt[2]))]*(5538393 - 4123702*Sqrt[2] + (13785797 - 9662095*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt[682*(-1387427 5807943 + 9819738650000*Sqrt[2])]))/22)/1364)/2728
3.1.92.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_), x_Symbol] :> Simp[(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a *f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(( d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*( c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si mp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f *(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))* (2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b ^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q , 0]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g*(g*b - 2*a*h) Subst[I nt[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[ g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b , c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && Ne Q[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(c*e - b*f ), 0]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Simp[1/(2*q) Int[Simp[h*(b*d - a*e) - g*(c*d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqr t[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[Simp[h*(b*d - a*e) - g*(c* d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x + c*x^2) *Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^( q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*( (A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b *e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 - 4 *a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)) Int[(a + b*x + c *x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B )*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a *c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) *x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.56 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.00
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(493\) |
| risch | \(\frac {162716650 x^{5}+86411405 x^{4}+277167774 x^{3}+175833195 x^{2}+161806828 x +22374044}{941410976 \left (5 x^{2}+3 x +2\right )^{2} \sqrt {2 x^{2}-x +3}}+\frac {3 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (372457261 \sqrt {-775687+549362 \sqrt {2}}\, \sqrt {2}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right )+526930410 \sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right )+553009243226 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right ) \sqrt {2}-759830139398 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )\right )}{1730722683008 \sqrt {\frac {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\) | \(736\) |
| default | \(\text {Expression too large to display}\) | \(18981\) |
1/941410976*(162716650*x^5+86411405*x^4+277167774*x^3+175833195*x^2+161806 828*x+22374044)/(5*x^2+3*x+2)^2/(2*x^2-x+3)^(1/2)+27/10232728*RootOf(97653 714048*_Z^4+383221372091193603*_Z^2+376669012321499306640625)*ln((22056286 34920723955712*x*RootOf(97653714048*_Z^4+383221372091193603*_Z^2+376669012 321499306640625)^5+10625753964685160355671308032*RootOf(97653714048*_Z^4+3 83221372091193603*_Z^2+376669012321499306640625)^3*x+583787314976741656402 406400*RootOf(97653714048*_Z^4+383221372091193603*_Z^2+3766690123214993066 40625)^3+5361306898641920835078986208000*RootOf(97653714048*_Z^4+383221372 091193603*_Z^2+376669012321499306640625)^2*(2*x^2-x+3)^(1/2)+1213869077110 2960837254737279811025*RootOf(97653714048*_Z^4+383221372091193603*_Z^2+376 669012321499306640625)*x+1725287176493454792258505057955400*RootOf(9765371 4048*_Z^4+383221372091193603*_Z^2+376669012321499306640625)+10542686598921 071198677006035680781250*(2*x^2-x+3)^(1/2))/(3535488*x*RootOf(97653714048* _Z^4+383221372091193603*_Z^2+376669012321499306640625)^2+7098559162705*x+2 15228344978))-3/55829763968*RootOf(_Z^2+2411202816*RootOf(97653714048*_Z^4 +383221372091193603*_Z^2+376669012321499306640625)^2+9462256101017126)*ln( -(239326023754418832*RootOf(_Z^2+2411202816*RootOf(97653714048*_Z^4+383221 372091193603*_Z^2+376669012321499306640625)^2+9462256101017126)*RootOf(976 53714048*_Z^4+383221372091193603*_Z^2+376669012321499306640625)^4*x+725400 753296830666567677*RootOf(97653714048*_Z^4+383221372091193603*_Z^2+3766...
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.86 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {23 \, \sqrt {341} {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \sqrt {968527552401 i \, \sqrt {31} - 124868482271487} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {968527552401 i \, \sqrt {31} - 124868482271487} {\left (5538393 i \, \sqrt {31} - 38528009\right )} - 456617847225000 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 8675739097275000 \, x - 10045592638950000}{x}\right ) - 23 \, \sqrt {341} {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \sqrt {968527552401 i \, \sqrt {31} - 124868482271487} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {968527552401 i \, \sqrt {31} - 124868482271487} {\left (-5538393 i \, \sqrt {31} + 38528009\right )} - 456617847225000 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 8675739097275000 \, x - 10045592638950000}{x}\right ) - 23 \, \sqrt {341} {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \sqrt {-968527552401 i \, \sqrt {31} - 124868482271487} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (5538393 i \, \sqrt {31} + 38528009\right )} \sqrt {-968527552401 i \, \sqrt {31} - 124868482271487} - 456617847225000 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 8675739097275000 \, x - 10045592638950000}{x}\right ) + 23 \, \sqrt {341} {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \sqrt {-968527552401 i \, \sqrt {31} - 124868482271487} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (-5538393 i \, \sqrt {31} - 38528009\right )} \sqrt {-968527552401 i \, \sqrt {31} - 124868482271487} - 456617847225000 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 8675739097275000 \, x - 10045592638950000}{x}\right ) + 2728 \, {\left (162716650 \, x^{5} + 86411405 \, x^{4} + 277167774 \, x^{3} + 175833195 \, x^{2} + 161806828 \, x + 22374044\right )} \sqrt {2 \, x^{2} - x + 3}}{2568169142528 \, {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )}} \]
1/2568169142528*(23*sqrt(341)*(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*sqrt(968527552401*I*sqrt(31) - 124868482271487)*log((sqrt(34 1)*sqrt(2*x^2 - x + 3)*sqrt(968527552401*I*sqrt(31) - 124868482271487)*(55 38393*I*sqrt(31) - 38528009) - 456617847225000*sqrt(31)*(-I*x + 6*I) + 867 5739097275000*x - 10045592638950000)/x) - 23*sqrt(341)*(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*sqrt(968527552401*I*sqrt(31) - 1248 68482271487)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(968527552401*I*sqrt(3 1) - 124868482271487)*(-5538393*I*sqrt(31) + 38528009) - 456617847225000*s qrt(31)*(-I*x + 6*I) + 8675739097275000*x - 10045592638950000)/x) - 23*sqr t(341)*(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*sqrt(-968 527552401*I*sqrt(31) - 124868482271487)*log((sqrt(341)*sqrt(2*x^2 - x + 3) *(5538393*I*sqrt(31) + 38528009)*sqrt(-968527552401*I*sqrt(31) - 124868482 271487) - 456617847225000*sqrt(31)*(I*x - 6*I) + 8675739097275000*x - 1004 5592638950000)/x) + 23*sqrt(341)*(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83* x^2 + 32*x + 12)*sqrt(-968527552401*I*sqrt(31) - 124868482271487)*log((sqr t(341)*sqrt(2*x^2 - x + 3)*(-5538393*I*sqrt(31) - 38528009)*sqrt(-96852755 2401*I*sqrt(31) - 124868482271487) - 456617847225000*sqrt(31)*(I*x - 6*I) + 8675739097275000*x - 10045592638950000)/x) + 2728*(162716650*x^5 + 86411 405*x^4 + 277167774*x^3 + 175833195*x^2 + 161806828*x + 22374044)*sqrt(2*x ^2 - x + 3))/(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {1}{\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[-1.0, infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,inf inity,inf
Timed out. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {1}{{\left (2\,x^2-x+3\right )}^{3/2}\,{\left (5\,x^2+3\,x+2\right )}^3} \,d x \]