\(\int \frac {1}{(3-x+2 x^2)^{3/2} (2+3 x+5 x^2)^2} \, dx\) [125]

Optimal result

Integrand size = 27, antiderivative size = 211 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {6315-2306 x}{345092 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{682 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{682} \left (129694447+103775000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (129694447+103775000 \sqrt {2}\right )}} \left (12611+16454 \sqrt {2}+\left (45519+29065 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{30008}-\frac {\sqrt {\frac {1}{682} \left (-129694447+103775000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-129694447+103775000 \sqrt {2}\right )}} \left (12611-16454 \sqrt {2}+\left (45519-29065 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{30008} \] Output:

-1/345092*(6315-2306*x)/(2*x^2-x+3)^(1/2)+1/682*(4+65*x)/(2*x^2-x+3)^(1/2) 
/(5*x^2+3*x+2)+1/20465456*(88451612854+70774550000*2^(1/2))^(1/2)*arctan(1 
1^(1/2)/(4020527857+3217025000*2^(1/2))^(1/2)*(12611+16454*2^(1/2)+(45519+ 
29065*2^(1/2))*x)/(2*x^2-x+3)^(1/2))-1/20465456*(-88451612854+70774550000* 
2^(1/2))^(1/2)*arctanh(11^(1/2)/(-4020527857+3217025000*2^(1/2))^(1/2)*(12 
611-16454*2^(1/2)+(45519-29065*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 = 0.98 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.96 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {\sqrt {3-x+2 x^2} \left (-10606+18557 x-24657 x^2+11530 x^3\right )}{345092 \left (6+7 x+16 x^2+x^3+10 x^4\right )}-\frac {1}{484} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {225 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+8 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-15 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+\frac {\text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {8623 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+9624 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+1565 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{30008 \sqrt {2}} \] Input:

Integrate[1/((3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2),x]
 

Output:

(Sqrt[3 - x + 2*x^2]*(-10606 + 18557*x - 24657*x^2 + 11530*x^3))/(345092*( 
6 + 7*x + 16*x^2 + x^3 + 10*x^4)) - RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 
+ 6*Sqrt[2]*#1^3 - 5*#1^4 & , (225*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] 
- #1] + 8*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 15*Log 
[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1^2)/(-13*Sqrt[2] + 17*#1 + 9*S 
qrt[2]*#1^2 - 10*#1^3) & ]/484 + RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6 
*Sqrt[2]*#1^3 - 5*#1^4 & , (8623*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2 
*x^2] - #1] + 9624*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 + 1565* 
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) & ]/(30008*Sqrt[2])
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 222, 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 = {1305, 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.

\(\displaystyle \int \frac {1}{\left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2} \, dx\)

\(\Big \downarrow \) 1305

\(\displaystyle \frac {65 x+4}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}-\frac {\int -\frac {11 \left (520 x^2-303 x+324\right )}{2 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}dx}{7502}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {520 x^2-303 x+324}{\left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}dx}{1364}+\frac {65 x+4}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 2135

\(\displaystyle \frac {\frac {\int \frac {253 (2158-2495 x)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2783}-\frac {6315-2306 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {65 x+4}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{22} \int \frac {2158-2495 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx-\frac {6315-2306 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {65 x+4}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1368

\(\displaystyle \frac {\frac {1}{22} \left (\frac {\int -\frac {11 \left (\left (337+2495 \sqrt {2}\right ) x-2158 \sqrt {2}+4653\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (\left (337-2495 \sqrt {2}\right ) x+2158 \sqrt {2}+4653\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )-\frac {6315-2306 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {65 x+4}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{22} \left (\frac {\int \frac {\left (337-2495 \sqrt {2}\right ) x+2158 \sqrt {2}+4653}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {\left (337+2495 \sqrt {2}\right ) x-2158 \sqrt {2}+4653}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )-\frac {6315-2306 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {65 x+4}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1362

\(\displaystyle \frac {\frac {1}{22} \left (\frac {\left (129694447-103775000 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (45519-29065 \sqrt {2}\right ) x-16454 \sqrt {2}+12611\right )^2}{2 x^2-x+3}-31 \left (129694447-103775000 \sqrt {2}\right )}d\frac {\left (45519-29065 \sqrt {2}\right ) x-16454 \sqrt {2}+12611}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (129694447+103775000 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (45519+29065 \sqrt {2}\right ) x+16454 \sqrt {2}+12611\right )^2}{2 x^2-x+3}-31 \left (129694447+103775000 \sqrt {2}\right )}d\frac {\left (45519+29065 \sqrt {2}\right ) x+16454 \sqrt {2}+12611}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}\right )-\frac {6315-2306 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {65 x+4}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {1}{22} \left (\frac {\left (129694447-103775000 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (45519-29065 \sqrt {2}\right ) x-16454 \sqrt {2}+12611\right )^2}{2 x^2-x+3}-31 \left (129694447-103775000 \sqrt {2}\right )}d\frac {\left (45519-29065 \sqrt {2}\right ) x-16454 \sqrt {2}+12611}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (129694447+103775000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (129694447+103775000 \sqrt {2}\right )}} \left (\left (45519+29065 \sqrt {2}\right ) x+16454 \sqrt {2}+12611\right )}{\sqrt {2 x^2-x+3}}\right )\right )-\frac {6315-2306 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {65 x+4}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{22} \left (\sqrt {\frac {1}{682} \left (129694447+103775000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (129694447+103775000 \sqrt {2}\right )}} \left (\left (45519+29065 \sqrt {2}\right ) x+16454 \sqrt {2}+12611\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (129694447-103775000 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (103775000 \sqrt {2}-129694447\right )}} \left (\left (45519-29065 \sqrt {2}\right ) x-16454 \sqrt {2}+12611\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (103775000 \sqrt {2}-129694447\right )}}\right )-\frac {6315-2306 x}{253 \sqrt {2 x^2-x+3}}}{1364}+\frac {65 x+4}{682 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}\)

Input:

Int[1/((3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2),x]
 

Output:

(4 + 65*x)/(682*Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)) + (-1/253*(6315 - 2 
306*x)/Sqrt[3 - x + 2*x^2] + (Sqrt[(129694447 + 103775000*Sqrt[2])/682]*Ar 
cTan[(Sqrt[11/(31*(129694447 + 103775000*Sqrt[2]))]*(12611 + 16454*Sqrt[2] 
 + (45519 + 29065*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + ((129694447 - 103775 
000*Sqrt[2])*ArcTanh[(Sqrt[11/(31*(-129694447 + 103775000*Sqrt[2]))]*(1261 
1 - 16454*Sqrt[2] + (45519 - 29065*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt 
[682*(-129694447 + 103775000*Sqrt[2])])/22)/1364
 

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 1305

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]
 

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 2135

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]
 

Maple [C] (verified)

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

Time = 5.18 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.32

method	result	size
trager	\(\text {Expression too large to display}\)	\(490\)
risch	\(\frac {11530 x^{3}-24657 x^{2}+18557 x -10606}{345092 \left (5 x^{2}+3 x +2\right ) \sqrt {2 x^{2}-x +3}}+\frac {\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 (1173047 \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}}+1666070 \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}-2028849746 \,\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 )}{634429136 \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}}}\)	\(726\)
default	\(\text {Expression too large to display}\)	\(5942\)

Input:

int(1/(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
 

Output:

1/345092*(11530*x^3-24657*x^2+18557*x-10606)/(10*x^4+x^3+16*x^2+7*x+6)*(2* 
x^2-x+3)^(1/2)+9/7502*RootOf(24413428512*_Z^4+3582290320587*_Z^2+168269541 
015625)*ln((1593373465716747264*x*RootOf(24413428512*_Z^4+3582290320587*_Z 
^2+168269541015625)^5+356051747055336070464*RootOf(24413428512*_Z^4+358229 
0320587*_Z^2+168269541015625)^3*x+193384498382205292800*RootOf(24413428512 
*_Z^4+3582290320587*_Z^2+168269541015625)^3+3247545983585953896000*RootOf( 
24413428512*_Z^4+3582290320587*_Z^2+168269541015625)^2*(2*x^2-x+3)^(1/2)+8 
600066701843343049675*RootOf(24413428512*_Z^4+3582290320587*_Z^2+168269541 
015625)*x+37885378432347118747800*RootOf(24413428512*_Z^4+3582290320587*_Z 
^2+168269541015625)+244659713848830018593750*(2*x^2-x+3)^(1/2))/(883872*x* 
RootOf(24413428512*_Z^4+3582290320587*_Z^2+168269541015625)^2+83351945*x+2 
4672962))+1/20465456*RootOf(_Z^2+602800704*RootOf(24413428512*_Z^4+3582290 
320587*_Z^2+168269541015625)^2+88451612854)*ln((691568344495116*x*RootOf(2 
4413428512*_Z^4+3582290320587*_Z^2+168269541015625)^4*RootOf(_Z^2+60280070 
4*RootOf(24413428512*_Z^4+3582290320587*_Z^2+168269541015625)^2+8845161285 
4)+48417413991953391*RootOf(24413428512*_Z^4+3582290320587*_Z^2+1682695410 
15625)^2*RootOf(_Z^2+602800704*RootOf(24413428512*_Z^4+3582290320587*_Z^2+ 
168269541015625)^2+88451612854)*x-34606661887587821204250*RootOf(244134285 
12*_Z^4+3582290320587*_Z^2+168269541015625)^2*(2*x^2-x+3)^(1/2)-8393424408 
9498825*RootOf(24413428512*_Z^4+3582290320587*_Z^2+168269541015625)^2*R...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (158) = 316\).

Time = 0.09 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.73 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^2,x, algorithm="fricas")
 

Output:

-1/2760736*(46*(10*x^4 + x^3 + 16*x^2 + 7*x + 6)*sqrt(51887500/341*sqrt(2) 
 + 129694447/682)*arctan(-22/12336481*(4*(109922*x^3 - 272712*x^2 - sqrt(2 
)*(101355*x^3 - 213973*x^2 - 29136*x + 111672) - 154288*x + 103584)*sqrt(2 
*x^2 - x + 3) + (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(51887500/341*sqrt(2) - 129694447/682))*sq 
rt(51887500/341*sqrt(2) + 129694447/682)/(343*x^4 - 400*x^3 + 1136*x^2 + 3 
84*x - 576)) - 46*(10*x^4 + x^3 + 16*x^2 + 7*x + 6)*sqrt(51887500/341*sqrt 
(2) + 129694447/682)*arctan(22/12336481*(4*(109922*x^3 - 272712*x^2 - sqrt 
(2)*(101355*x^3 - 213973*x^2 - 29136*x + 111672) - 154288*x + 103584)*sqrt 
(2*x^2 - x + 3) - (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(51887500/341*sqrt(2) - 129694447/682))* 
sqrt(51887500/341*sqrt(2) + 129694447/682)/(343*x^4 - 400*x^3 + 1136*x^2 + 
 384*x - 576)) + 23*(10*x^4 + x^3 + 16*x^2 + 7*x + 6)*sqrt(51887500/341*sq 
rt(2) - 129694447/682)*log((604487569*x^2 + 22*sqrt(2*x^2 - x + 3)*(sqrt(2 
)*(95963*x - 285725) + 189762*x - 381688)*sqrt(51887500/341*sqrt(2) - 1296 
94447/682) + 542805164*sqrt(2)*(2*x^2 - x + 3) - 1862808631*x + 2467296200 
)/x^2) - 23*(10*x^4 + x^3 + 16*x^2 + 7*x + 6)*sqrt(51887500/341*sqrt(2) - 
129694447/682)*log((604487569*x^2 - 22*sqrt(2*x^2 - x + 3)*(sqrt(2)*(95963 
*x - 285725) + 189762*x - 381688)*sqrt(51887500/341*sqrt(2) - 129694447/68 
2) + 542805164*sqrt(2)*(2*x^2 - x + 3) - 1862808631*x + 2467296200)/x^2...
 

Sympy [F]

\[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {1}{\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \] Input:

integrate(1/(2*x**2-x+3)**(3/2)/(5*x**2+3*x+2)**2,x)
 

Output:

Integral(1/((2*x**2 - x + 3)**(3/2)*(5*x**2 + 3*x + 2)**2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^2,x, algorithm="maxima")
 

Output:

integrate(1/((5*x^2 + 3*x + 2)^2*(2*x^2 - x + 3)^(3/2)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(2*x^2-x+3)^(3/2)/(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:Francis algorithm failure for[-1.0, 
infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,inf 
inity,inf
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {1}{{\left (2\,x^2-x+3\right )}^{3/2}\,{\left (5\,x^2+3\,x+2\right )}^2} \,d x \] Input:

int(1/((2*x^2 - x + 3)^(3/2)*(3*x + 5*x^2 + 2)^2),x)
                                                                                    
                                                                                    
 

Output:

int(1/((2*x^2 - x + 3)^(3/2)*(3*x + 5*x^2 + 2)^2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\sqrt {2 x^{2}-x +3}}{100 x^{8}+20 x^{7}+321 x^{6}+172 x^{5}+390 x^{4}+236 x^{3}+241 x^{2}+84 x +36}d x \] Input:

int(1/(2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^2,x)
 

Output:

int(sqrt(2*x**2 - x + 3)/(100*x**8 + 20*x**7 + 321*x**6 + 172*x**5 + 390*x 
**4 + 236*x**3 + 241*x**2 + 84*x + 36),x)