\(\int \frac {1}{(3-x+2 x^2)^3 (2+3 x+5 x^2)} \, dx\) [84]

Optimal result

Integrand size = 25, antiderivative size = 115 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )} \, dx=\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2}+\frac {3625-746 x}{256036 \left (3-x+2 x^2\right )}-\frac {53403 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{5632792 \sqrt {23}}+\frac {247 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{10648 \sqrt {31}}-\frac {119 \log \left (3-x+2 x^2\right )}{21296}+\frac {119 \log \left (2+3 x+5 x^2\right )}{21296} \] Output:

1/1012*(13-6*x)/(2*x^2-x+3)^2+(3625-746*x)/(512072*x^2-256036*x[phone] 
403/129554216*arctan(1/23*(1-4*x)*23^(1/2))*23^(1/2)+247/330088*arctan(1/3 
1*(3+10*x)*31^(1/2))*31^(1/2)-119/21296*ln(2*x^2-x+3)+119/21296*ln(5*x^2+3 
*x+2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )} \, dx=\frac {3310986 \sqrt {23} \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )+6010498 \sqrt {31} \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )+713 \left (-\frac {44 \left (-14164+7381 x-7996 x^2+1492 x^3\right )}{\left (-3+x-2 x^2\right )^2}-62951 \log \left (3-x+2 x^2\right )+62951 \log \left (2+3 x+5 x^2\right )\right )}{8032361392} \] Input:

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

Output:

(3310986*Sqrt[23]*ArcTan[(-1 + 4*x)/Sqrt[23]] + 6010498*Sqrt[31]*ArcTan[(3 
 + 10*x)/Sqrt[31]] + 713*((-44*(-14164 + 7381*x - 7996*x^2 + 1492*x^3))/(- 
3 + x - 2*x^2)^2 - 62951*Log[3 - x + 2*x^2] + 62951*Log[2 + 3*x + 5*x^2])) 
/8032361392
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1305, 27, 2135, 27, 2141, 27, 1142, 25, 1083, 217, 1103}

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.

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

\(\Big \downarrow \) 1305

\(\displaystyle \frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}-\frac {\int -\frac {22 \left (-45 x^2+88 x+166\right )}{\left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )}dx}{11132}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{506} \int \frac {-45 x^2+88 x+166}{\left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )}dx+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}\)

\(\Big \downarrow \) 2135

\(\displaystyle \frac {1}{506} \left (\frac {\int \frac {11 \left (-3730 x^2+32147 x+27074\right )}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}dx}{5566}+\frac {3625-746 x}{506 \left (2 x^2-x+3\right )}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{506} \left (\frac {1}{506} \int \frac {-3730 x^2+32147 x+27074}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}dx+\frac {3625-746 x}{506 \left (2 x^2-x+3\right )}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}\)

\(\Big \downarrow \) 2141

\(\displaystyle \frac {1}{506} \left (\frac {1}{506} \left (\frac {1}{242} \int \frac {77 (8311-17986 x)}{2 x^2-x+3}dx+\frac {1}{242} \int \frac {5819 (595 x+302)}{5 x^2+3 x+2}dx\right )+\frac {3625-746 x}{506 \left (2 x^2-x+3\right )}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{506} \left (\frac {1}{506} \left (\frac {7}{22} \int \frac {8311-17986 x}{2 x^2-x+3}dx+\frac {529}{22} \int \frac {595 x+302}{5 x^2+3 x+2}dx\right )+\frac {3625-746 x}{506 \left (2 x^2-x+3\right )}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{506} \left (\frac {1}{506} \left (\frac {7}{22} \left (\frac {7629}{2} \int \frac {1}{2 x^2-x+3}dx-\frac {8993}{2} \int -\frac {1-4 x}{2 x^2-x+3}dx\right )+\frac {529}{22} \left (\frac {247}{2} \int \frac {1}{5 x^2+3 x+2}dx+\frac {119}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx\right )\right )+\frac {3625-746 x}{506 \left (2 x^2-x+3\right )}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{506} \left (\frac {1}{506} \left (\frac {7}{22} \left (\frac {7629}{2} \int \frac {1}{2 x^2-x+3}dx+\frac {8993}{2} \int \frac {1-4 x}{2 x^2-x+3}dx\right )+\frac {529}{22} \left (\frac {247}{2} \int \frac {1}{5 x^2+3 x+2}dx+\frac {119}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx\right )\right )+\frac {3625-746 x}{506 \left (2 x^2-x+3\right )}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{506} \left (\frac {1}{506} \left (\frac {7}{22} \left (\frac {8993}{2} \int \frac {1-4 x}{2 x^2-x+3}dx-7629 \int \frac {1}{-(4 x-1)^2-23}d(4 x-1)\right )+\frac {529}{22} \left (\frac {119}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx-247 \int \frac {1}{-(10 x+3)^2-31}d(10 x+3)\right )\right )+\frac {3625-746 x}{506 \left (2 x^2-x+3\right )}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{506} \left (\frac {1}{506} \left (\frac {7}{22} \left (\frac {8993}{2} \int \frac {1-4 x}{2 x^2-x+3}dx+\frac {7629 \arctan \left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {23}}\right )+\frac {529}{22} \left (\frac {119}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx+\frac {247 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{\sqrt {31}}\right )\right )+\frac {3625-746 x}{506 \left (2 x^2-x+3\right )}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{506} \left (\frac {1}{506} \left (\frac {7}{22} \left (\frac {7629 \arctan \left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {23}}-\frac {8993}{2} \log \left (2 x^2-x+3\right )\right )+\frac {529}{22} \left (\frac {247 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{\sqrt {31}}+\frac {119}{2} \log \left (5 x^2+3 x+2\right )\right )\right )+\frac {3625-746 x}{506 \left (2 x^2-x+3\right )}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2}\)

Input:

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

Output:

(13 - 6*x)/(1012*(3 - x + 2*x^2)^2) + ((3625 - 746*x)/(506*(3 - x + 2*x^2) 
) + ((7*((7629*ArcTan[(-1 + 4*x)/Sqrt[23]])/Sqrt[23] - (8993*Log[3 - x + 2 
*x^2])/2))/22 + (529*((247*ArcTan[(3 + 10*x)/Sqrt[31]])/Sqrt[31] + (119*Lo 
g[2 + 3*x + 5*x^2])/2))/22)/506)/506
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

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 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]
 

rule 2141

Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x 
_)^2)), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Co 
eff[Px, x, 2], q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b* 
e*f + a^2*f^2}, Simp[1/q   Int[(A*c^2*d - a*c*C*d - A*b*c*e + a*B*c*e + A*b 
^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d - A*c*e + a*C*e + A*b 
*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Simp[1/q   Int[(c*C*d^2 - B*c*d* 
e + A*c*e^2 + b*B*d*f - A*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(B*c*d - 
b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(d + e*x + f*x^2), x], x] /; NeQ[ 
q, 0]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
 

Maple [A] (verified)

Time = 3.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77

method	result	size
default	\(-\frac {\frac {8206}{529} x^{3}-\frac {43978}{529} x^{2}+\frac {81191}{1058} x -\frac {77902}{529}}{2662 \left (2 x^{2}-x +3\right )^{2}}-\frac {119 \ln \left (2 x^{2}-x +3\right )}{21296}+\frac {53403 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{129554216}+\frac {119 \ln \left (5 x^{2}+3 x +2\right )}{21296}+\frac {247 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{330088}\)	\(89\)
risch	\(\frac {-\frac {373}{64009} x^{3}+\frac {1999}{64009} x^{2}-\frac {61}{2116} x +\frac {3541}{64009}}{\left (2 x^{2}-x +3\right )^{2}}+\frac {119 \ln \left (100 x^{2}+60 x +40\right )}{21296}+\frac {247 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{330088}-\frac {119 \ln \left (16 x^{2}-8 x +24\right )}{21296}+\frac {53403 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{129554216}\)	\(89\)

Input:

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

Output:

-1/2662*(8206/529*x^3-43978/529*x^2+81191/1058*x-77902/529)/(2*x^2-x+3)^2- 
119/21296*ln(2*x^2-x+3)+53403/129554216*23^(1/2)*arctan(1/23*(4*x-1)*23^(1 
/2))+119/21296*ln(5*x^2+3*x+2)+247/330088*arctan(1/31*(10*x+3)*31^(1/2))*3 
1^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )} \, dx=-\frac {46807024 \, x^{3} - 6010498 \, \sqrt {31} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - 3310986 \, \sqrt {23} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - 250850512 \, x^{2} - 44884063 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 44884063 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) + 231556732 \, x - 444353008}{8032361392 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \] Input:

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

Output:

-1/8032361392*(46807024*x^3 - 6010498*sqrt(31)*(4*x^4 - 4*x^3 + 13*x^2 - 6 
*x + 9)*arctan(1/31*sqrt(31)*(10*x + 3)) - 3310986*sqrt(23)*(4*x^4 - 4*x^3 
 + 13*x^2 - 6*x + 9)*arctan(1/23*sqrt(23)*(4*x - 1)) - 250850512*x^2 - 448 
84063*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(5*x^2 + 3*x + 2) + 44884063*( 
4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(2*x^2 - x + 3) + 231556732*x - 44435 
3008)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
 

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )} \, dx=\frac {- 1492 x^{3} + 7996 x^{2} - 7381 x + 14164}{1024144 x^{4} - 1024144 x^{3} + 3328468 x^{2} - 1536216 x + 2304324} - \frac {119 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{21296} + \frac {119 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{21296} + \frac {53403 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{129554216} + \frac {247 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{330088} \] Input:

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

Output:

(-1492*x**3 + 7996*x**2 - 7381*x + 14164)/(1024144*x**4 - 1024144*x**3 + 3 
328468*x**2 - 1536216*x + 2304324) - 119*log(x**2 - x/2 + 3/2)/21296 + 119 
*log(x**2 + 3*x/5 + 2/5)/21296 + 53403*sqrt(23)*atan(4*sqrt(23)*x/23 - sqr 
t(23)/23)/129554216 + 247*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/ 
330088
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )} \, dx=\frac {247}{330088} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {53403}{129554216} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {1492 \, x^{3} - 7996 \, x^{2} + 7381 \, x - 14164}{256036 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} + \frac {119}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac {119}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:

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

Output:

247/330088*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 53403/129554216*sqr 
t(23)*arctan(1/23*sqrt(23)*(4*x - 1)) - 1/256036*(1492*x^3 - 7996*x^2 + 73 
81*x - 14164)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9) + 119/21296*log(5*x^2 + 3 
*x + 2) - 119/21296*log(2*x^2 - x + 3)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )} \, dx=\frac {247}{330088} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {53403}{129554216} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {1492 \, x^{3} - 7996 \, x^{2} + 7381 \, x - 14164}{256036 \, {\left (2 \, x^{2} - x + 3\right )}^{2}} + \frac {119}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac {119}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:

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

Output:

247/330088*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 53403/129554216*sqr 
t(23)*arctan(1/23*sqrt(23)*(4*x - 1)) - 1/256036*(1492*x^3 - 7996*x^2 + 73 
81*x - 14164)/(2*x^2 - x + 3)^2 + 119/21296*log(5*x^2 + 3*x + 2) - 119/212 
96*log(2*x^2 - x + 3)
 

Mupad [B] (verification not implemented)

Time = 15.94 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )} \, dx=-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {119}{21296}+\frac {\sqrt {31}\,247{}\mathrm {i}}{660176}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {119}{21296}+\frac {\sqrt {31}\,247{}\mathrm {i}}{660176}\right )-\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {119}{21296}+\frac {\sqrt {23}\,53403{}\mathrm {i}}{259108432}\right )+\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {119}{21296}+\frac {\sqrt {23}\,53403{}\mathrm {i}}{259108432}\right )-\frac {\frac {373\,x^3}{256036}-\frac {1999\,x^2}{256036}+\frac {61\,x}{8464}-\frac {3541}{256036}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \] Input:

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

Output:

log(x + (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*247i)/660176 + 119/21296) - lo 
g(x - (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*247i)/660176 - 119/21296) - log( 
x - (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*53403i)/259108432 + 119/21296) + log 
(x + (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*53403i)/259108432 - 119/21296) - (( 
61*x)/8464 - (1999*x^2)/256036 + (373*x^3)/256036 - 3541/256036)/((13*x^2) 
/4 - (3*x)/2 - x^3 + x^4 + 9/4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 358, normalized size of antiderivative = 3.11 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )} \, dx=\frac {24041992 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{4}-24041992 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{3}[phone] \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{2}-36062988 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x [phone] \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right )[phone] \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{4}-13243944 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{3}[phone] \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{2}-19865916 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x [phone] \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right )[phone] \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{4}-179536252 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{3}[phone] \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{2}-269304378 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x [phone] \,\mathrm {log}\left (5 x^{2}+3 x +2\right )-179536252 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{4}[phone] \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{3}-583492819 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{2}[phone] \,\mathrm {log}\left (2 x^{2}-x +3\right ) x -403956567 \,\mathrm {log}\left (2 x^{2}-x +3\right )-46807024 x^{4}[phone] x^{2}-161346196 x [phone]}{32129445568 x^{4}-32129445568 x^{3}[phone] x^{2}-48194168352 x [phone]} \] Input:

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

Output:

(24041992*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**4 - 24041992*sqrt(31)*atan 
((10*x + 3)/sqrt(31))*x**3 + 78136474*sqrt(31)*atan((10*x + 3)/sqrt(31))*x 
**2 - 36062988*sqrt(31)*atan((10*x + 3)/sqrt(31))*x + 54094482*sqrt(31)*at 
an((10*x + 3)/sqrt(31)) + 13243944*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**4 
- 13243944*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**3 + 43042818*sqrt(23)*atan 
((4*x - 1)/sqrt(23))*x**2 - 19865916*sqrt(23)*atan((4*x - 1)/sqrt(23))*x + 
 29798874*sqrt(23)*atan((4*x - 1)/sqrt(23)) + 179536252*log(5*x**2 + 3*x + 
 2)*x**4 - 179536252*log(5*x**2 + 3*x + 2)*x**3 + 583492819*log(5*x**2 + 3 
*x + 2)*x**2 - 269304378*log(5*x**2 + 3*x + 2)*x + 403956567*log(5*x**2 + 
3*x + 2) - 179536252*log(2*x**2 - x + 3)*x**4 + 179536252*log(2*x**2 - x + 
 3)*x**3 - 583492819*log(2*x**2 - x + 3)*x**2 + 269304378*log(2*x**2 - x + 
 3)*x - 403956567*log(2*x**2 - x + 3) - 46807024*x**4 + 98727684*x**2 - 16 
1346196*x + 339037204)/(8032361392*(4*x**4 - 4*x**3 + 13*x**2 - 6*x + 9))