Integrand size = 27, antiderivative size = 255 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {(1277+2240 x) \sqrt {3-x+2 x^2}}{7750}+\frac {4}{155} (4-5 x) \left (3-x+2 x^2\right )^{3/2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {4799 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2500 \sqrt {2}}+\frac {11 \sqrt {\frac {11}{31} \left (224510383+194487500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (224510383+194487500 \sqrt {2}\right )}} \left (21136+33287 \sqrt {2}+\left (87710+54423 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{38750}-\frac {11 \sqrt {\frac {11}{31} \left (-224510383+194487500 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (-224510383+194487500 \sqrt {2}\right )}} \left (21136-33287 \sqrt {2}+\left (87710-54423 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{38750} \]
4/155*(4-5*x)*(2*x^2-x+3)^(3/2)+1/31*(3+10*x)*(2*x^2-x+3)^(5/2)/(5*x^2+3*x +2)-4799/5000*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-1/7750*(1277+2240*x)* (2*x^2-x+3)^(1/2)-11/1201250*arctanh(1/62*(21136+x*(87710-54423*2^(1/2))-3 3287*2^(1/2))*682^(1/2)/(-224510383+194487500*2^(1/2))^(1/2)/(2*x^2-x+3)^( 1/2))*(-76558040603+66320237500*2^(1/2))^(1/2)+11/1201250*arctan(1/62*(211 36+33287*2^(1/2)+x*(87710+54423*2^(1/2)))*682^(1/2)/(224510383+194487500*2 ^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(76558040603+66320237500*2^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.69 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.70 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {\frac {500 \sqrt {3-x+2 x^2} \left (8996+9289 x-12555 x^2+3100 x^3\right )}{2+3 x+5 x^2}-3719225 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )+30008 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {5237 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+2880 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2225 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-242 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {639994 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-22980 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+1175 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{3875000} \]
((500*Sqrt[3 - x + 2*x^2]*(8996 + 9289*x - 12555*x^2 + 3100*x^3))/(2 + 3*x + 5*x^2) - 3719225*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]] + 30008 *RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (5237 *Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 2880*Sqrt[2]*Log[-(Sqrt[2] *x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 2225*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) & ] - 242*Sqrt[2]*RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^ 4 & , (639994*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] - 22980 *Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 1175*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) & ])/3875000
Time = 0.86 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.08, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {1302, 27, 2138, 27, 2138, 27, 2143, 27, 1090, 222, 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.
| \(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{5/2}}{\left (5 x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1302 |
| \(\displaystyle \frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac {1}{31} \int -\frac {5 \left (-32 x^2-6 x+15\right ) \left (2 x^2-x+3\right )^{3/2}}{2 \left (5 x^2+3 x+2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{62} \int \frac {\left (-32 x^2-6 x+15\right ) \left (2 x^2-x+3\right )^{3/2}}{5 x^2+3 x+2}dx+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 2138 |
| \(\displaystyle \frac {5}{62} \left (\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}-\frac {1}{600} \int -\frac {24 \left (-896 x^2-905 x+1461\right ) \sqrt {2 x^2-x+3}}{5 x^2+3 x+2}dx\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \int \frac {\left (-896 x^2-905 x+1461\right ) \sqrt {2 x^2-x+3}}{5 x^2+3 x+2}dx+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 2138 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (-\frac {1}{100} \int -\frac {2 \left (148769 x^2-175535 x+243476\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {1}{25} \sqrt {2 x^2-x+3} (2240 x+1277)\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \int \frac {148769 x^2-175535 x+243476}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \left (\frac {148769}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {1}{5} \int \frac {242 (3801-5471 x)}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \left (\frac {148769}{5} \int \frac {1}{\sqrt {2 x^2-x+3}}dx+\frac {242}{5} \int \frac {3801-5471 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx\right )-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \left (\frac {242}{5} \int \frac {3801-5471 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {148769 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{5 \sqrt {46}}\right )-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \left (\frac {242}{5} \int \frac {3801-5471 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {148769 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}\right )-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1368 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \left (\frac {242}{5} \left (\frac {\int -\frac {11 \left (\left (1670+5471 \sqrt {2}\right ) x-3801 \sqrt {2}+9272\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (\left (1670-5471 \sqrt {2}\right ) x+3801 \sqrt {2}+9272\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {148769 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}\right )-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \left (\frac {242}{5} \left (\frac {\int \frac {\left (1670-5471 \sqrt {2}\right ) x+3801 \sqrt {2}+9272}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {\left (1670+5471 \sqrt {2}\right ) x-3801 \sqrt {2}+9272}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {148769 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}\right )-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1362 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \left (\frac {242}{5} \left (\sqrt {2} \left (224510383-194487500 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (87710-54423 \sqrt {2}\right ) x-33287 \sqrt {2}+21136\right )^2}{2 x^2-x+3}-62 \left (224510383-194487500 \sqrt {2}\right )}d\frac {\left (87710-54423 \sqrt {2}\right ) x-33287 \sqrt {2}+21136}{\sqrt {2 x^2-x+3}}-\sqrt {2} \left (224510383+194487500 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (87710+54423 \sqrt {2}\right ) x+33287 \sqrt {2}+21136\right )^2}{2 x^2-x+3}-62 \left (224510383+194487500 \sqrt {2}\right )}d\frac {\left (87710+54423 \sqrt {2}\right ) x+33287 \sqrt {2}+21136}{\sqrt {2 x^2-x+3}}\right )+\frac {148769 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}\right )-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \left (\frac {242}{5} \left (\sqrt {2} \left (224510383-194487500 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (87710-54423 \sqrt {2}\right ) x-33287 \sqrt {2}+21136\right )^2}{2 x^2-x+3}-62 \left (224510383-194487500 \sqrt {2}\right )}d\frac {\left (87710-54423 \sqrt {2}\right ) x-33287 \sqrt {2}+21136}{\sqrt {2 x^2-x+3}}+\sqrt {\frac {1}{341} \left (224510383+194487500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (224510383+194487500 \sqrt {2}\right )}} \left (\left (87710+54423 \sqrt {2}\right ) x+33287 \sqrt {2}+21136\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {148769 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}\right )-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \left (\frac {148769 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}+\frac {242}{5} \left (\sqrt {\frac {1}{341} \left (224510383+194487500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (224510383+194487500 \sqrt {2}\right )}} \left (\left (87710+54423 \sqrt {2}\right ) x+33287 \sqrt {2}+21136\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (224510383-194487500 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (194487500 \sqrt {2}-224510383\right )}} \left (\left (87710-54423 \sqrt {2}\right ) x-33287 \sqrt {2}+21136\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (194487500 \sqrt {2}-224510383\right )}}\right )\right )-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
((3 + 10*x)*(3 - x + 2*x^2)^(5/2))/(31*(2 + 3*x + 5*x^2)) + (5*((8*(4 - 5* x)*(3 - x + 2*x^2)^(3/2))/25 + (-1/25*((1277 + 2240*x)*Sqrt[3 - x + 2*x^2] ) + ((148769*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(5*Sqrt[2]) + (242*(Sqrt[(22451 0383 + 194487500*Sqrt[2])/341]*ArcTan[(Sqrt[11/(62*(224510383 + 194487500* Sqrt[2]))]*(21136 + 33287*Sqrt[2] + (87710 + 54423*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + ((224510383 - 194487500*Sqrt[2])*ArcTanh[(Sqrt[11/(62*(-22451 0383 + 194487500*Sqrt[2]))]*(21136 - 33287*Sqrt[2] + (87710 - 54423*Sqrt[2 ])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt[341*(-224510383 + 194487500*Sqrt[2])]))/ 5)/50)/25))/62
3.1.77.3.1 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[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[((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)*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[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Si mp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2* A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x + (p*( c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C* d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2* p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ .)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 1/c Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x ^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.10 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.05
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(522\) |
| risch | \(\frac {\left (3100 x^{3}-12555 x^{2}+9289 x +8996\right ) \sqrt {2 x^{2}-x +3}}{38750 x^{2}+23250 x +15500}+\frac {4799 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{5000}+\frac {11 \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 (1114345 \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 )+1584599 \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 )+1982813041 \,\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}-1820897034 \,\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 )}{37238750 \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}}}\) | \(740\) |
| default | \(\text {Expression too large to display}\) | \(40028\) |
1/7750*(3100*x^3-12555*x^2+9289*x+8996)/(5*x^2+3*x+2)*(2*x^2-x+3)^(1/2)-47 99/5000*RootOf(_Z^2-2)*ln(-4*RootOf(_Z^2-2)*x+4*(2*x^2-x+3)^(1/2)+RootOf(_ Z^2-2))+1/1201250*RootOf(_Z^2+96100*RootOf(96100*_Z^4+9263522912963*_Z^2+3 35049907908469531250)^2+9263522912963)*ln(-(-50635674288*RootOf(_Z^2+96100 *RootOf(96100*_Z^4+9263522912963*_Z^2+335049907908469531250)^2+92635229129 63)*RootOf(96100*_Z^4+9263522912963*_Z^2+335049907908469531250)^4*x-248394 3074332490408*RootOf(96100*_Z^4+9263522912963*_Z^2+335049907908469531250)^ 2*RootOf(_Z^2+96100*RootOf(96100*_Z^4+9263522912963*_Z^2+33504990790846953 1250)^2+9263522912963)*x+19722622249345627868333000*RootOf(96100*_Z^4+9263 522912963*_Z^2+335049907908469531250)^2*(2*x^2-x+3)^(1/2)+5394974354905351 392*RootOf(96100*_Z^4+9263522912963*_Z^2+335049907908469531250)^2*RootOf(_ Z^2+96100*RootOf(96100*_Z^4+9263522912963*_Z^2+335049907908469531250)^2+92 63522912963)+176231528680762367884053125*RootOf(_Z^2+96100*RootOf(96100*_Z ^4+9263522912963*_Z^2+335049907908469531250)^2+9263522912963)*x+9164762297 34634762910552969993750*(2*x^2-x+3)^(1/2)-212361413450122530324775000*Root Of(_Z^2+96100*RootOf(96100*_Z^4+9263522912963*_Z^2+335049907908469531250)^ 2+9263522912963))/(3100*x*RootOf(96100*_Z^4+9263522912963*_Z^2+33504990790 8469531250)^2+92436758755*x-75966534842))-1/3875*RootOf(96100*_Z^4+9263522 912963*_Z^2+335049907908469531250)*ln((-7911824107500*x*RootOf(96100*_Z^4+ 9263522912963*_Z^2+335049907908469531250)^5-1137198439966646972200*Root...
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.58 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {2 \, \sqrt {62} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {37983267421 i \, \sqrt {31} - 298823319773} \log \left (-\frac {2 \, \sqrt {62} \sqrt {2 \, x^{2} - x + 3} \sqrt {37983267421 i \, \sqrt {31} - 298823319773} {\left (2642 i \, \sqrt {31} - 35929\right )} + 16580059375 \, \sqrt {31} {\left (-i \, x + 6 i\right )} - 315021128125 \, x + 364761306250}{x}\right ) - 2 \, \sqrt {62} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {37983267421 i \, \sqrt {31} - 298823319773} \log \left (-\frac {2 \, \sqrt {62} \sqrt {2 \, x^{2} - x + 3} \sqrt {37983267421 i \, \sqrt {31} - 298823319773} {\left (-2642 i \, \sqrt {31} + 35929\right )} + 16580059375 \, \sqrt {31} {\left (-i \, x + 6 i\right )} - 315021128125 \, x + 364761306250}{x}\right ) - 2 \, \sqrt {62} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-37983267421 i \, \sqrt {31} - 298823319773} \log \left (-\frac {2 \, \sqrt {62} \sqrt {2 \, x^{2} - x + 3} {\left (2642 i \, \sqrt {31} + 35929\right )} \sqrt {-37983267421 i \, \sqrt {31} - 298823319773} + 16580059375 \, \sqrt {31} {\left (i \, x - 6 i\right )} - 315021128125 \, x + 364761306250}{x}\right ) + 2 \, \sqrt {62} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-37983267421 i \, \sqrt {31} - 298823319773} \log \left (-\frac {2 \, \sqrt {62} \sqrt {2 \, x^{2} - x + 3} {\left (-2642 i \, \sqrt {31} - 35929\right )} \sqrt {-37983267421 i \, \sqrt {31} - 298823319773} + 16580059375 \, \sqrt {31} {\left (i \, x - 6 i\right )} - 315021128125 \, x + 364761306250}{x}\right ) - 4611839 \, \sqrt {2} {\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) - 1240 \, {\left (3100 \, x^{3} - 12555 \, x^{2} + 9289 \, x + 8996\right )} \sqrt {2 \, x^{2} - x + 3}}{9610000 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]
-1/9610000*(2*sqrt(62)*(5*x^2 + 3*x + 2)*sqrt(37983267421*I*sqrt(31) - 298 823319773)*log(-(2*sqrt(62)*sqrt(2*x^2 - x + 3)*sqrt(37983267421*I*sqrt(31 ) - 298823319773)*(2642*I*sqrt(31) - 35929) + 16580059375*sqrt(31)*(-I*x + 6*I) - 315021128125*x + 364761306250)/x) - 2*sqrt(62)*(5*x^2 + 3*x + 2)*s qrt(37983267421*I*sqrt(31) - 298823319773)*log(-(2*sqrt(62)*sqrt(2*x^2 - x + 3)*sqrt(37983267421*I*sqrt(31) - 298823319773)*(-2642*I*sqrt(31) + 3592 9) + 16580059375*sqrt(31)*(-I*x + 6*I) - 315021128125*x + 364761306250)/x) - 2*sqrt(62)*(5*x^2 + 3*x + 2)*sqrt(-37983267421*I*sqrt(31) - 29882331977 3)*log(-(2*sqrt(62)*sqrt(2*x^2 - x + 3)*(2642*I*sqrt(31) + 35929)*sqrt(-37 983267421*I*sqrt(31) - 298823319773) + 16580059375*sqrt(31)*(I*x - 6*I) - 315021128125*x + 364761306250)/x) + 2*sqrt(62)*(5*x^2 + 3*x + 2)*sqrt(-379 83267421*I*sqrt(31) - 298823319773)*log(-(2*sqrt(62)*sqrt(2*x^2 - x + 3)*( -2642*I*sqrt(31) - 35929)*sqrt(-37983267421*I*sqrt(31) - 298823319773) + 1 6580059375*sqrt(31)*(I*x - 6*I) - 315021128125*x + 364761306250)/x) - 4611 839*sqrt(2)*(5*x^2 + 3*x + 2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) - 1240*(3100*x^3 - 12555*x^2 + 9289*x + 8996)*sqrt( 2*x^2 - x + 3))/(5*x^2 + 3*x + 2)
\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \]
\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {{\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \,d x } \]
Exception generated. \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{15625,[8]%%%}+%%%{%%{[-37500,0]:[1,0,-2]%%},[7]%%%}+%%%{-6 1250,[6]%
Timed out. \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{5/2}}{{\left (5\,x^2+3\,x+2\right )}^2} \,d x \]