Integrand size = 27, antiderivative size = 223 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {(3+10 x) \left (3-x+2 x^2\right )^{3/2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {3 (277+696 x) \sqrt {3-x+2 x^2}}{3844 \left (2+3 x+5 x^2\right )}+\frac {3 \sqrt {\frac {1}{682} \left (366990269+259509026 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (366990269+259509026 \sqrt {2}\right )}} \left (29367+20575 \sqrt {2}+\left (70517+49942 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{7688}-\frac {3 \sqrt {\frac {1}{682} \left (-366990269+259509026 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-366990269+259509026 \sqrt {2}\right )}} \left (29367-20575 \sqrt {2}+\left (70517-49942 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{7688} \]
1/62*(3+10*x)*(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^2+3/3844*(277+696*x)*(2*x^2- x+3)^(1/2)/(5*x^2+3*x+2)-3/5243216*arctanh(1/31*(29367+x*(70517-49942*2^(1 /2))-20575*2^(1/2))*341^(1/2)/(-366990269+259509026*2^(1/2))^(1/2)/(2*x^2- x+3)^(1/2))*(-250287363458+176985155732*2^(1/2))^(1/2)+3/5243216*arctan(1/ 31*(29367+20575*2^(1/2)+x*(70517+49942*2^(1/2)))*341^(1/2)/(366990269+2595 09026*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(250287363458+176985155732*2^(1/2) )^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.83 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.57 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {3306250 \sqrt {3-x+2 x^2} \left (2220+8343 x+10171 x^2+11680 x^3\right )}{\left (2+3 x+5 x^2\right )^2}-42578694225 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+406695200 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {93 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+10 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+14 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {4926449381 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-2660991465 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-186 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {155209944 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-248390285 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{12709225000} \]
((3306250*Sqrt[3 - x + 2*x^2]*(2220 + 8343*x + 10171*x^2 + 11680*x^3))/(2 + 3*x + 5*x^2)^2 - 42578694225*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*S qrt[2]*#1^3 - 5*#1^4 & , Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]/(-13 *Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ] + 406695200*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (93*Log[-(Sqrt[2]*x ) + Sqrt[3 - x + 2*x^2] - #1] + 10*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ] + 1 4*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (492 6449381*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 26609914 65*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) & ] - 186*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1 ^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (155209944*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqr t[3 - x + 2*x^2] - #1]*#1 - 248390285*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) & ])/12709 225000
Time = 0.57 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1302, 27, 1346, 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.
| \(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{3/2}}{\left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1302 |
| \(\displaystyle \frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac {1}{62} \int -\frac {3 (63-22 x) \sqrt {2 x^2-x+3}}{2 \left (5 x^2+3 x+2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{124} \int \frac {(63-22 x) \sqrt {2 x^2-x+3}}{\left (5 x^2+3 x+2\right )^2}dx+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1346 |
| \(\displaystyle \frac {3}{124} \left (\frac {(696 x+277) \sqrt {2 x^2-x+3}}{31 \left (5 x^2+3 x+2\right )}-\frac {1}{31} \int -\frac {4453-1804 x}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \int \frac {4453-1804 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \left (\frac {\int -\frac {11 \left (-\left (\left (2649-1804 \sqrt {2}\right ) x\right )-4453 \sqrt {2}+6257\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (2649+1804 \sqrt {2}\right ) x\right )+4453 \sqrt {2}+6257\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \left (\frac {\int \frac {-\left (\left (2649+1804 \sqrt {2}\right ) x\right )+4453 \sqrt {2}+6257}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (2649-1804 \sqrt {2}\right ) x\right )-4453 \sqrt {2}+6257}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \left (\frac {\left (366990269-259509026 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367\right )^2}{2 x^2-x+3}-31 \left (366990269-259509026 \sqrt {2}\right )}d\frac {\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (366990269+259509026 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (70517+49942 \sqrt {2}\right ) x+20575 \sqrt {2}+29367\right )^2}{2 x^2-x+3}-31 \left (366990269+259509026 \sqrt {2}\right )}d\frac {\left (70517+49942 \sqrt {2}\right ) x+20575 \sqrt {2}+29367}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \left (\frac {\left (366990269-259509026 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367\right )^2}{2 x^2-x+3}-31 \left (366990269-259509026 \sqrt {2}\right )}d\frac {\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (366990269+259509026 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (366990269+259509026 \sqrt {2}\right )}} \left (\left (70517+49942 \sqrt {2}\right ) x+20575 \sqrt {2}+29367\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{124} \left (\frac {1}{62} \left (\sqrt {\frac {1}{682} \left (366990269+259509026 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (366990269+259509026 \sqrt {2}\right )}} \left (\left (70517+49942 \sqrt {2}\right ) x+20575 \sqrt {2}+29367\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (366990269-259509026 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (259509026 \sqrt {2}-366990269\right )}} \left (\left (70517-49942 \sqrt {2}\right ) x-20575 \sqrt {2}+29367\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (259509026 \sqrt {2}-366990269\right )}}\right )+\frac {\sqrt {2 x^2-x+3} (696 x+277)}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}\) |
((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(62*(2 + 3*x + 5*x^2)^2) + (3*(((277 + 696*x)*Sqrt[3 - x + 2*x^2])/(31*(2 + 3*x + 5*x^2)) + (Sqrt[(366990269 + 25 9509026*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(366990269 + 259509026*Sqrt[2])) ]*(29367 + 20575*Sqrt[2] + (70517 + 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2] ] + ((366990269 - 259509026*Sqrt[2])*ArcTanh[(Sqrt[11/(31*(-366990269 + 25 9509026*Sqrt[2]))]*(29367 - 20575*Sqrt[2] + (70517 - 49942*Sqrt[2])*x))/Sq rt[3 - x + 2*x^2]])/Sqrt[682*(-366990269 + 259509026*Sqrt[2])])/62))/124
3.1.71.3.1 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[(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e *x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e _.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(g*b - 2*a*h - (b*h - 2*g* c)*x)*(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1) )), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[e*q*(g*b - 2*a*h) - d*(b*h - 2*g*c)*(2*p + 3) + (2*f*q*(g*b - 2*a*h) - e*(b*h - 2*g*c)*(2*p + q + 3))*x - f*(b*h - 2*g*c)* (2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Ne Q[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.81 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.17
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(483\) |
| risch | \(\frac {\left (11680 x^{3}+10171 x^{2}+8343 x +2220\right ) \sqrt {2 x^{2}-x +3}}{3844 \left (5 x^{2}+3 x +2\right )^{2}}+\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 (1915561 \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 )+2708832 \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 )+2795860364 \,\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}-3974378870 \,\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 )}{162539696 \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}}}\) | \(726\) |
| default | \(\text {Expression too large to display}\) | \(81552\) |
1/3844*(11680*x^3+10171*x^2+8343*x+2220)/(5*x^2+3*x+2)^2*(2*x^2-x+3)^(1/2) +3/5243216*RootOf(_Z^2+29767936*RootOf(952573952*_Z^4+8009195630656*_Z^2+1 6836233643867169)^2+250287363458)*ln((36607893262336*RootOf(_Z^2+29767936* RootOf(952573952*_Z^4+8009195630656*_Z^2+16836233643867169)^2+250287363458 )*RootOf(952573952*_Z^4+8009195630656*_Z^2+16836233643867169)^4*x+32626452 3201744512*RootOf(952573952*_Z^4+8009195630656*_Z^2+16836233643867169)^2*R ootOf(_Z^2+29767936*RootOf(952573952*_Z^4+8009195630656*_Z^2+1683623364386 7169)^2+250287363458)*x+3707589189779200*RootOf(952573952*_Z^4+80091956306 56*_Z^2+16836233643867169)^2*RootOf(_Z^2+29767936*RootOf(952573952*_Z^4+80 09195630656*_Z^2+16836233643867169)^2+250287363458)+1225107842671457930662 4*(2*x^2-x+3)^(1/2)*RootOf(952573952*_Z^4+8009195630656*_Z^2+1683623364386 7169)^2+723862202733749385201*RootOf(_Z^2+29767936*RootOf(952573952*_Z^4+8 009195630656*_Z^2+16836233643867169)^2+250287363458)*x+1759868749834835570 0*RootOf(_Z^2+29767936*RootOf(952573952*_Z^4+8009195630656*_Z^2+1683623364 3867169)^2+250287363458)+51523372375740505057718054*(2*x^2-x+3)^(1/2))/(21 824*x*RootOf(952573952*_Z^4+8009195630656*_Z^2+16836233643867169)^2+921288 44*x+508369))+3/961*RootOf(952573952*_Z^4+8009195630656*_Z^2+1683623364386 7169)*ln(-(-585726292197376*x*RootOf(952573952*_Z^4+8009195630656*_Z^2+168 36233643867169)^5-4629284194657700864*RootOf(952573952*_Z^4+8009195630656* _Z^2+16836233643867169)^3*x+35926916207374132864*(2*x^2-x+3)^(1/2)*Root...
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.78 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {\sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {4575321 i \, \sqrt {31} - 3302912421} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {4575321 i \, \sqrt {31} - 3302912421} {\left (29367 i \, \sqrt {31} + 193967\right )} - 12067169709 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 229276224471 \, x - 265477733598}{x}\right ) - \sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {4575321 i \, \sqrt {31} - 3302912421} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {4575321 i \, \sqrt {31} - 3302912421} {\left (-29367 i \, \sqrt {31} - 193967\right )} - 12067169709 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 229276224471 \, x - 265477733598}{x}\right ) - \sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-4575321 i \, \sqrt {31} - 3302912421} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (29367 i \, \sqrt {31} - 193967\right )} \sqrt {-4575321 i \, \sqrt {31} - 3302912421} - 12067169709 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 229276224471 \, x - 265477733598}{x}\right ) + \sqrt {341} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {-4575321 i \, \sqrt {31} - 3302912421} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (-29367 i \, \sqrt {31} + 193967\right )} \sqrt {-4575321 i \, \sqrt {31} - 3302912421} - 12067169709 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 229276224471 \, x - 265477733598}{x}\right ) - 2728 \, {\left (11680 \, x^{3} + 10171 \, x^{2} + 8343 \, x + 2220\right )} \sqrt {2 \, x^{2} - x + 3}}{10486432 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \]
-1/10486432*(sqrt(341)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(4575321* I*sqrt(31) - 3302912421)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(4575321*I *sqrt(31) - 3302912421)*(29367*I*sqrt(31) + 193967) - 12067169709*sqrt(31) *(I*x - 6*I) + 229276224471*x - 265477733598)/x) - sqrt(341)*(25*x^4 + 30* x^3 + 29*x^2 + 12*x + 4)*sqrt(4575321*I*sqrt(31) - 3302912421)*log((sqrt(3 41)*sqrt(2*x^2 - x + 3)*sqrt(4575321*I*sqrt(31) - 3302912421)*(-29367*I*sq rt(31) - 193967) - 12067169709*sqrt(31)*(I*x - 6*I) + 229276224471*x - 265 477733598)/x) - sqrt(341)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-4575 321*I*sqrt(31) - 3302912421)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*(29367*I*s qrt(31) - 193967)*sqrt(-4575321*I*sqrt(31) - 3302912421) - 12067169709*sqr t(31)*(-I*x + 6*I) + 229276224471*x - 265477733598)/x) + sqrt(341)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(-4575321*I*sqrt(31) - 3302912421)*log( (sqrt(341)*sqrt(2*x^2 - x + 3)*(-29367*I*sqrt(31) + 193967)*sqrt(-4575321* I*sqrt(31) - 3302912421) - 12067169709*sqrt(31)*(-I*x + 6*I) + 22927622447 1*x - 265477733598)/x) - 2728*(11680*x^3 + 10171*x^2 + 8343*x + 2220)*sqrt (2*x^2 - x + 3))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} \,d x } \]
Exception generated. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{3/2}}{{\left (5\,x^2+3\,x+2\right )}^3} \,d x \]