\(\int \frac {(3-x+2 x^2)^{5/2}}{(2+3 x+5 x^2)^2} \, dx\) [111]

Optimal result

Integrand size = 27, antiderivative size = 288 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {1123 \sqrt {3-x+2 x^2}}{7750}-\frac {604}{775} x \sqrt {3-x+2 x^2}+\frac {52}{155} x^2 \sqrt {3-x+2 x^2}-\frac {8}{31} x^3 \sqrt {3-x+2 x^2}+\frac {(3+10 x) \left (3-x+2 x^2\right )^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {4799 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2500 \sqrt {2}}+\frac {11 \sqrt {\frac {11}{31} \left (224510383+194487500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (224510383+194487500 \sqrt {2}\right )}} \left (21136+33287 \sqrt {2}+\left (87710+54423 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{38750}-\frac {11 \sqrt {\frac {11}{31} \left (-224510383+194487500 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (-224510383+194487500 \sqrt {2}\right )}} \left (21136-33287 \sqrt {2}+\left (87710-54423 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{38750} \] Output:

1123/7750*(2*x^2-x+3)^(1/2)-604/775*x*(2*x^2-x+3)^(1/2)+52/155*x^2*(2*x^2- 
x+3)^(1/2)-8/31*x^3*(2*x^2-x+3)^(1/2)+(3+10*x)*(2*x^2-x+3)^(5/2)/(155*x^2+ 
93*x+62)-4799/5000*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+11/1201250*(7655 
8040603+66320237500*2^(1/2))^(1/2)*arctan(11^(1/2)/(13919643746+1205822500 
0*2^(1/2))^(1/2)*(21136+33287*2^(1/2)+(87710+54423*2^(1/2))*x)/(2*x^2-x+3) 
^(1/2))-11/1201250*(-76558040603+66320237500*2^(1/2))^(1/2)*arctanh(11^(1/ 
2)/(-13919643746+12058225000*2^(1/2))^(1/2)*(21136-33287*2^(1/2)+(87710-54 
423*2^(1/2))*x)/(2*x^2-x+3)^(1/2))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.07 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.50 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {\frac {500 \sqrt {3-x+2 x^2} \left (8996+9289 x-12555 x^2+3100 x^3\right )}{2+3 x+5 x^2}-3719225 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )+30008 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {5237 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+2880 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2225 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-242 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {639994 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-22980 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+1175 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{3875000} \] Input:

Integrate[(3 - x + 2*x^2)^(5/2)/(2 + 3*x + 5*x^2)^2,x]
 

Output:

((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
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.96, 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

\(\displaystyle \frac {5}{62} \left (\frac {1}{25} \left (\frac {1}{50} \left (\frac {148769 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{5 \sqrt {2}}+\frac {242}{5} \left (\sqrt {\frac {1}{341} \left (224510383+194487500 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{62 \left (224510383+194487500 \sqrt {2}\right )}} \left (\left (87710+54423 \sqrt {2}\right ) x+33287 \sqrt {2}+21136\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (224510383-194487500 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{62 \left (194487500 \sqrt {2}-224510383\right )}} \left (\left (87710-54423 \sqrt {2}\right ) x-33287 \sqrt {2}+21136\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {341 \left (194487500 \sqrt {2}-224510383\right )}}\right )\right )-\frac {1}{25} (2240 x+1277) \sqrt {2 x^2-x+3}\right )+\frac {8}{25} (4-5 x) \left (2 x^2-x+3\right )^{3/2}\right )+\frac {(10 x+3) \left (2 x^2-x+3\right )^{5/2}}{31 \left (5 x^2+3 x+2\right )}\)

Input:

Int[(3 - x + 2*x^2)^(5/2)/(2 + 3*x + 5*x^2)^2,x]
 

Output:

((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
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 219

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])
 

rule 222

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]
 

rule 1090

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]
 

rule 1302

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]
 

rule 1362

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]
 

rule 1368

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]
 

rule 2138

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]
 

rule 2143

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]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 5.75 (sec) , antiderivative size = 613, normalized size of antiderivative = 2.13

method	result	size
trager	\(\text {Expression too large to display}\)	\(613\)
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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (1114345 \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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (-1+\sqrt {2}+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (-1+\sqrt {2}+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+1584599 \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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (-1+\sqrt {2}+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (-1+\sqrt {2}+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}[phone] \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {-1+\sqrt {2}+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {-1+\sqrt {2}+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\)	\(740\)
default	\(\text {Expression too large to display}\)	\(40028\)

Input:

int((2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
 

Output:

1/7750*(3100*x^3-12555*x^2+9289*x+8996)/(5*x^2+3*x+2)*(2*x^2-x+3)^(1/2)-1/ 
1201250*RootOf(_Z^2+96100*RootOf(96100*_Z^4+9263522912963*_Z^2+33504990790 
8469531250)^2+9263522912963)*ln(-(50635674288*RootOf(_Z^2+96100*RootOf(961 
00*_Z^4+9263522912963*_Z^2+335049907908469531250)^2+9263522912963)*RootOf( 
96100*_Z^4+9263522912963*_Z^2+335049907908469531250)^4*x+24839430743324904 
08*RootOf(96100*_Z^4+9263522912963*_Z^2+335049907908469531250)^2*RootOf(_Z 
^2+96100*RootOf(96100*_Z^4+9263522912963*_Z^2+335049907908469531250)^2+926 
3522912963)*x-5394974354905351392*RootOf(96100*_Z^4+9263522912963*_Z^2+335 
049907908469531250)^2*RootOf(_Z^2+96100*RootOf(96100*_Z^4+9263522912963*_Z 
^2+335049907908469531250)^2+9263522912963)+19722622249345627868333000*Root 
Of(96100*_Z^4+9263522912963*_Z^2+335049907908469531250)^2*(2*x^2-x+3)^(1/2 
)-176231528680762367884053125*RootOf(_Z^2+96100*RootOf(96100*_Z^4+92635229 
12963*_Z^2+335049907908469531250)^2+9263522912963)*x+212361413450122530324 
775000*RootOf(_Z^2+96100*RootOf(96100*_Z^4+9263522912963*_Z^2+335049907908 
469531250)^2+9263522912963)+916476229734634762910552969993750*(2*x^2-x+3)^ 
(1/2))/(3100*x*RootOf(96100*_Z^4+9263522912963*_Z^2+335049907908469531250) 
^2+92436758755*x-75966534842))+1/3875*RootOf(96100*_Z^4+9263522912963*_Z^2 
+335049907908469531250)*ln((7911824107500*x*RootOf(96100*_Z^4+926352291296 
3*_Z^2+335049907908469531250)^5+1137198439966646972200*RootOf(96100*_Z^4+9 
263522912963*_Z^2+335049907908469531250)^3*x+842964742953961155000*Root...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (218) = 436\).

Time = 0.11 (sec) , antiderivative size = 616, normalized size of antiderivative = 2.14 \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^2,x, algorithm="fricas")
 

Output:

-1/310000*(44*sqrt(11/31)*(5*x^2 + 3*x + 2)*sqrt(194487500*sqrt(2) + 22451 
0383)*arctan(-1/313911301*sqrt(11/31)*(sqrt(11/31)*(171*x^4 + 1212*x^3 - 1 
640*x^2 - 176*sqrt(2)*(6*x^4 + 5*x^3 + 5*x^2 + 12*x) - 3936*x)*sqrt(194487 
500*sqrt(2) - 224510383) + 88*(100035*x^3 - 255241*x^2 - sqrt(2)*(99487*x^ 
3 - 206202*x^2 - 17048*x + 111264) - 161712*x + 91224)*sqrt(2*x^2 - x + 3) 
)*sqrt(194487500*sqrt(2) + 224510383)/(343*x^4 - 400*x^3 + 1136*x^2 + 384* 
x - 576)) - 44*sqrt(11/31)*(5*x^2 + 3*x + 2)*sqrt(194487500*sqrt(2) + 2245 
10383)*arctan(-1/313911301*sqrt(11/31)*(sqrt(11/31)*(171*x^4 + 1212*x^3 - 
1640*x^2 - 176*sqrt(2)*(6*x^4 + 5*x^3 + 5*x^2 + 12*x) - 3936*x)*sqrt(19448 
7500*sqrt(2) - 224510383) - 88*(100035*x^3 - 255241*x^2 - sqrt(2)*(99487*x 
^3 - 206202*x^2 - 17048*x + 111264) - 161712*x + 91224)*sqrt(2*x^2 - x + 3 
))*sqrt(194487500*sqrt(2) + 224510383)/(343*x^4 - 400*x^3 + 1136*x^2 + 384 
*x - 576)) + 22*sqrt(11/31)*(5*x^2 + 3*x + 2)*sqrt(194487500*sqrt(2) - 224 
510383)*log(11*(2*sqrt(11/31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(86127*x - 2736 
98) + 187571*x - 359825)*sqrt(194487500*sqrt(2) - 224510383) + 1398332159* 
x^2 + 1255645204*sqrt(2)*(2*x^2 - x + 3) - 4309146041*x + 5707478200)/x^2) 
 - 22*sqrt(11/31)*(5*x^2 + 3*x + 2)*sqrt(194487500*sqrt(2) - 224510383)*lo 
g(-11*(2*sqrt(11/31)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(86127*x - 273698) + 187 
571*x - 359825)*sqrt(194487500*sqrt(2) - 224510383) - 1398332159*x^2 - 125 
5645204*sqrt(2)*(2*x^2 - x + 3) + 4309146041*x - 5707478200)/x^2) - 148...
 

Sympy [F]

\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \] Input:

integrate((2*x**2-x+3)**(5/2)/(5*x**2+3*x+2)**2,x)
 

Output:

Integral((2*x**2 - x + 3)**(5/2)/(5*x**2 + 3*x + 2)**2, x)
 

Maxima [F]

\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {{\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \,d x } \] Input:

integrate((2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^2,x, algorithm="maxima")
 

Output:

integrate((2*x^2 - x + 3)^(5/2)/(5*x^2 + 3*x + 2)^2, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^2,x, algorithm="giac")
 

Output:

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]%
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{5/2}}{{\left (5\,x^2+3\,x+2\right )}^2} \,d x \] Input:

int((2*x^2 - x + 3)^(5/2)/(3*x + 5*x^2 + 2)^2,x)
 

Output:

int((2*x^2 - x + 3)^(5/2)/(3*x + 5*x^2 + 2)^2, x)
 

Reduce [F]

\[ \int \frac {\left (3-x+2 x^2\right )^{5/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {1906000 \sqrt {2 x^{2}-x +3}\, x^{3}-7719300 \sqrt {2 x^{2}-x +3}\, x^{2}-36431580 \sqrt {2 x^{2}-x +3}\, x [phone] \sqrt {2 x^{2}-x +3}[phone] \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right ) x^{2}[phone] \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right ) x +9146894 \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}-4 x +1\right )[phone] \left (\int \frac {\sqrt {2 x^{2}-x +3}}{50 x^{6}+35 x^{5}+103 x^{4}+85 x^{3}+83 x^{2}+32 x +12}d x \right ) x^{2}[phone] \left (\int \frac {\sqrt {2 x^{2}-x +3}}{50 x^{6}+35 x^{5}+103 x^{4}+85 x^{3}+83 x^{2}+32 x +12}d x \right ) x [phone] \left (\int \frac {\sqrt {2 x^{2}-x +3}}{50 x^{6}+35 x^{5}+103 x^{4}+85 x^{3}+83 x^{2}+32 x +12}d x \right )-1781250680 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{50 x^{6}+35 x^{5}+103 x^{4}+85 x^{3}+83 x^{2}+32 x +12}d x \right ) x^{2}-1068750408 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{50 x^{6}+35 x^{5}+103 x^{4}+85 x^{3}+83 x^{2}+32 x +12}d x \right ) x -712500272 \left (\int \frac {\sqrt {2 x^{2}-x +3}\, x^{2}}{50 x^{6}+35 x^{5}+103 x^{4}+85 x^{3}+83 x^{2}+32 x +12}d x \right )}{23825000 x^{2}[phone] x +9530000} \] Input:

int((2*x^2-x+3)^(5/2)/(5*x^2+3*x+2)^2,x)
 

Output:

(1906000*sqrt(2*x**2 - x + 3)*x**3 - 7719300*sqrt(2*x**2 - x + 3)*x**2 - 3 
6431580*sqrt(2*x**2 - x + 3)*x + 10411464*sqrt(2*x**2 - x + 3) + 22867235* 
sqrt(2)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1)*x**2 + 13720341*s 
qrt(2)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1)*x + 9146894*sqrt(2 
)*log( - 2*sqrt(2*x**2 - x + 3)*sqrt(2) - 4*x + 1) + 2073857720*int(sqrt(2 
*x**2 - x + 3)/(50*x**6 + 35*x**5 + 103*x**4 + 85*x**3 + 83*x**2 + 32*x + 
12),x)*x**2 + 1244314632*int(sqrt(2*x**2 - x + 3)/(50*x**6 + 35*x**5 + 103 
*x**4 + 85*x**3 + 83*x**2 + 32*x + 12),x)*x + 829543088*int(sqrt(2*x**2 - 
x + 3)/(50*x**6 + 35*x**5 + 103*x**4 + 85*x**3 + 83*x**2 + 32*x + 12),x) - 
 1781250680*int((sqrt(2*x**2 - x + 3)*x**2)/(50*x**6 + 35*x**5 + 103*x**4 
+ 85*x**3 + 83*x**2 + 32*x + 12),x)*x**2 - 1068750408*int((sqrt(2*x**2 - x 
 + 3)*x**2)/(50*x**6 + 35*x**5 + 103*x**4 + 85*x**3 + 83*x**2 + 32*x + 12) 
,x)*x - 712500272*int((sqrt(2*x**2 - x + 3)*x**2)/(50*x**6 + 35*x**5 + 103 
*x**4 + 85*x**3 + 83*x**2 + 32*x + 12),x))/(4765000*(5*x**2 + 3*x + 2))