\(\int \frac {1}{(3-x+2 x^2) (2+3 x+5 x^2)^3} \, dx\) [72]

Optimal result

Integrand size = 25, antiderivative size = 115 \[ \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac {45 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{10648 \sqrt {23}}+\frac {847793 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{10232728 \sqrt {31}}-\frac {\log \left (3-x+2 x^2\right )}{21296}+\frac {\log \left (2+3 x+5 x^2\right )}{21296} \] Output:

1/1364*(4+65*x)/(5*x^2+3*x+2)^2+(7923+21605*x)/(2325620*x^2+1395372*x+9302 
48)-45/244904*arctan(1/23*(1-4*x)*23^(1/2))*23^(1/2)+847793/317214568*arct 
an(1/31*(3+10*x)*31^(1/2))*31^(1/2)-1/21296*ln(2*x^2-x+3)+1/21296*ln(5*x^2 
+3*x+2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {45 \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{10648 \sqrt {23}}+\frac {1695586 \sqrt {31} \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )+31 \left (\frac {44 \left (17210+89144 x+104430 x^2+108025 x^3\right )}{\left (2+3 x+5 x^2\right )^2}-961 \log \left (3-x+2 x^2\right )+961 \log \left (2+3 x+5 x^2\right )\right )}{634429136} \] Input:

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

Output:

(45*ArcTan[(-1 + 4*x)/Sqrt[23]])/(10648*Sqrt[23]) + (1695586*Sqrt[31]*ArcT 
an[(3 + 10*x)/Sqrt[31]] + 31*((44*(17210 + 89144*x + 104430*x^2 + 108025*x 
^3))/(2 + 3*x + 5*x^2)^2 - 961*Log[3 - x + 2*x^2] + 961*Log[2 + 3*x + 5*x^ 
2]))/634429136
 

Rubi [A] (verified)

Time = 0.44 (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.

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

\(\Big \downarrow \) 1305

\(\displaystyle \frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}-\frac {\int -\frac {11 \left (390 x^2-319 x+523\right )}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}dx}{15004}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {390 x^2-319 x+523}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}dx}{1364}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 2135

\(\displaystyle \frac {\frac {\int \frac {22 \left (43210 x^2-15839 x+60010\right )}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}dx}{7502}+\frac {21605 x+7923}{341 \left (5 x^2+3 x+2\right )}}{1364}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{341} \int \frac {43210 x^2-15839 x+60010}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}dx+\frac {21605 x+7923}{341 \left (5 x^2+3 x+2\right )}}{1364}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 2141

\(\displaystyle \frac {\frac {1}{341} \left (\frac {1}{242} \int \frac {10571 (23-2 x)}{2 x^2-x+3}dx+\frac {1}{242} \int \frac {11 (4805 x+425338)}{5 x^2+3 x+2}dx\right )+\frac {21605 x+7923}{341 \left (5 x^2+3 x+2\right )}}{1364}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{341} \left (\frac {961}{22} \int \frac {23-2 x}{2 x^2-x+3}dx+\frac {1}{22} \int \frac {4805 x+425338}{5 x^2+3 x+2}dx\right )+\frac {21605 x+7923}{341 \left (5 x^2+3 x+2\right )}}{1364}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {1}{341} \left (\frac {961}{22} \left (\frac {45}{2} \int \frac {1}{2 x^2-x+3}dx-\frac {1}{2} \int -\frac {1-4 x}{2 x^2-x+3}dx\right )+\frac {1}{22} \left (\frac {847793}{2} \int \frac {1}{5 x^2+3 x+2}dx+\frac {961}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx\right )\right )+\frac {21605 x+7923}{341 \left (5 x^2+3 x+2\right )}}{1364}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{341} \left (\frac {961}{22} \left (\frac {45}{2} \int \frac {1}{2 x^2-x+3}dx+\frac {1}{2} \int \frac {1-4 x}{2 x^2-x+3}dx\right )+\frac {1}{22} \left (\frac {847793}{2} \int \frac {1}{5 x^2+3 x+2}dx+\frac {961}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx\right )\right )+\frac {21605 x+7923}{341 \left (5 x^2+3 x+2\right )}}{1364}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {1}{341} \left (\frac {961}{22} \left (\frac {1}{2} \int \frac {1-4 x}{2 x^2-x+3}dx-45 \int \frac {1}{-(4 x-1)^2-23}d(4 x-1)\right )+\frac {1}{22} \left (\frac {961}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx-847793 \int \frac {1}{-(10 x+3)^2-31}d(10 x+3)\right )\right )+\frac {21605 x+7923}{341 \left (5 x^2+3 x+2\right )}}{1364}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {1}{341} \left (\frac {961}{22} \left (\frac {1}{2} \int \frac {1-4 x}{2 x^2-x+3}dx+\frac {45 \arctan \left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {23}}\right )+\frac {1}{22} \left (\frac {961}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx+\frac {847793 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{\sqrt {31}}\right )\right )+\frac {21605 x+7923}{341 \left (5 x^2+3 x+2\right )}}{1364}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {1}{341} \left (\frac {961}{22} \left (\frac {45 \arctan \left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {23}}-\frac {1}{2} \log \left (2 x^2-x+3\right )\right )+\frac {1}{22} \left (\frac {847793 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{\sqrt {31}}+\frac {961}{2} \log \left (5 x^2+3 x+2\right )\right )\right )+\frac {21605 x+7923}{341 \left (5 x^2+3 x+2\right )}}{1364}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}\)

Input:

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

Output:

(4 + 65*x)/(1364*(2 + 3*x + 5*x^2)^2) + ((7923 + 21605*x)/(341*(2 + 3*x + 
5*x^2)) + ((961*((45*ArcTan[(-1 + 4*x)/Sqrt[23]])/Sqrt[23] - Log[3 - x + 2 
*x^2]/2))/22 + ((847793*ArcTan[(3 + 10*x)/Sqrt[31]])/Sqrt[31] + (961*Log[2 
 + 3*x + 5*x^2])/2)/22)/341)/1364
 

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 = 2.92 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77

method	result	size
default	\(-\frac {\ln \left (2 x^{2}-x +3\right )}{21296}+\frac {45 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{244904}+\frac {\frac {108025}{465124} x^{3}+\frac {52215}{232562} x^{2}+\frac {2026}{10571} x +\frac {8605}{232562}}{\left (5 x^{2}+3 x +2\right )^{2}}+\frac {\ln \left (5 x^{2}+3 x +2\right )}{21296}+\frac {847793 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{317214568}\)	\(89\)
risch	\(\frac {\frac {108025}{465124} x^{3}+\frac {52215}{232562} x^{2}+\frac {2026}{10571} x +\frac {8605}{232562}}{\left (5 x^{2}+3 x +2\right )^{2}}+\frac {\ln \left (100 x^{2}+60 x +40\right )}{21296}+\frac {847793 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{317214568}-\frac {\ln \left (16 x^{2}-8 x +24\right )}{21296}+\frac {45 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{244904}\)	\(89\)

Input:

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

Output:

-1/21296*ln(2*x^2-x+3)+45/244904*23^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))+25 
/10648*(95062/961*x^3+459492/4805*x^2+1961168/24025*x+75724/4805)/(5*x^2+3 
*x+2)^2+1/21296*ln(5*x^2+3*x+2)+847793/317214568*arctan(1/31*(10*x+3)*31^( 
1/2))*31^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {3388960300 \, x^{3} + 38998478 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 2681190 \, \sqrt {23} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 3276177960 \, x^{2} + 685193 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 685193 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (2 \, x^{2} - x + 3\right ) + 2796625568 \, x + 539912120}{14591870128 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:

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

Output:

1/14591870128*(3388960300*x^3 + 38998478*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^ 
2 + 12*x + 4)*arctan(1/31*sqrt(31)*(10*x + 3)) + 2681190*sqrt(23)*(25*x^4 
+ 30*x^3 + 29*x^2 + 12*x + 4)*arctan(1/23*sqrt(23)*(4*x - 1)) + 3276177960 
*x^2 + 685193*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*log(5*x^2 + 3*x + 2) - 
 685193*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*log(2*x^2 - x + 3) + 2796625 
568*x + 539912120)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
 

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {108025 x^{3} + 104430 x^{2} + 89144 x + 17210}{11628100 x^{4} + 13953720 x^{3} + 13488596 x^{2} + 5581488 x + 1860496} - \frac {\log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{21296} + \frac {\log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{21296} + \frac {45 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{244904} + \frac {847793 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{317214568} \] Input:

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

Output:

(108025*x**3 + 104430*x**2 + 89144*x + 17210)/(11628100*x**4 + 13953720*x* 
*3 + 13488596*x**2 + 5581488*x + 1860496) - log(x**2 - x/2 + 3/2)/21296 + 
log(x**2 + 3*x/5 + 2/5)/21296 + 45*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23 
)/23)/244904 + 847793*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/3172 
14568
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {847793}{317214568} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {45}{244904} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {108025 \, x^{3} + 104430 \, x^{2} + 89144 \, x + 17210}{465124 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} + \frac {1}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac {1}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:

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

Output:

847793/317214568*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 45/244904*sqr 
t(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/465124*(108025*x^3 + 104430*x^2 
+ 89144*x + 17210)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 1/21296*log(5*x 
^2 + 3*x + 2) - 1/21296*log(2*x^2 - x + 3)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {847793}{317214568} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {45}{244904} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {108025 \, x^{3} + 104430 \, x^{2} + 89144 \, x + 17210}{465124 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} + \frac {1}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac {1}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:

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

Output:

847793/317214568*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 45/244904*sqr 
t(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/465124*(108025*x^3 + 104430*x^2 
+ 89144*x + 17210)/(5*x^2 + 3*x + 2)^2 + 1/21296*log(5*x^2 + 3*x + 2) - 1/ 
21296*log(2*x^2 - x + 3)
 

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {4321\,x^3}{465124}+\frac {10443\,x^2}{1162810}+\frac {2026\,x}{264275}+\frac {1721}{1162810}}{x^4+\frac {6\,x^3}{5}+\frac {29\,x^2}{25}+\frac {12\,x}{25}+\frac {4}{25}}+\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{21296}+\frac {\sqrt {23}\,45{}\mathrm {i}}{489808}\right )-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {1}{21296}+\frac {\sqrt {31}\,847793{}\mathrm {i}}{634429136}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {1}{21296}+\frac {\sqrt {31}\,847793{}\mathrm {i}}{634429136}\right )-\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {1}{21296}+\frac {\sqrt {23}\,45{}\mathrm {i}}{489808}\right ) \] Input:

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

Output:

log(x + (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*45i)/489808 - 1/21296) - log(x - 
 (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*45i)/489808 + 1/21296) - log(x - (31^(1 
/2)*1i)/10 + 3/10)*((31^(1/2)*847793i)/634429136 - 1/21296) + log(x + (31^ 
(1/2)*1i)/10 + 3/10)*((31^(1/2)*847793i)/634429136 + 1/21296) + ((2026*x)/ 
264275 + (10443*x^2)/1162810 + (4321*x^3)/465124 + 1721/1162810)/((12*x)/2 
5 + (29*x^2)/25 + (6*x^3)/5 + x^4 + 4/25)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 358, normalized size of antiderivative = 3.11 \[ \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {2924885850 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{4}[phone] \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{3}[phone] \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{2}[phone] \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x [phone] \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right )[phone] \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{4}[phone] \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{3}[phone] \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{2}[phone] \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x [phone] \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right )[phone] \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{4}[phone] \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{3}[phone] \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{2}[phone] \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x +8222316 \,\mathrm {log}\left (5 x^{2}+3 x +2\right )-51389475 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{4}-61667370 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{3}-59611791 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{2}-24666948 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x -8222316 \,\mathrm {log}\left (2 x^{2}-x +3\right )-8472400750 x^{4}+549010 x^{2}[phone] x [phone]}{1094390259600 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

(2924885850*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**4 + 3509863020*sqrt(31)* 
atan((10*x + 3)/sqrt(31))*x**3 + 3392867586*sqrt(31)*atan((10*x + 3)/sqrt( 
31))*x**2 + 1403945208*sqrt(31)*atan((10*x + 3)/sqrt(31))*x + 467981736*sq 
rt(31)*atan((10*x + 3)/sqrt(31)) + 201089250*sqrt(23)*atan((4*x - 1)/sqrt( 
23))*x**4 + 241307100*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**3 + 233263530*s 
qrt(23)*atan((4*x - 1)/sqrt(23))*x**2 + 96522840*sqrt(23)*atan((4*x - 1)/s 
qrt(23))*x + 32174280*sqrt(23)*atan((4*x - 1)/sqrt(23)) + 51389475*log(5*x 
**2 + 3*x + 2)*x**4 + 61667370*log(5*x**2 + 3*x + 2)*x**3 + 59611791*log(5 
*x**2 + 3*x + 2)*x**2 + 24666948*log(5*x**2 + 3*x + 2)*x + 8222316*log(5*x 
**2 + 3*x + 2) - 51389475*log(2*x**2 - x + 3)*x**4 - 61667370*log(2*x**2 - 
 x + 3)*x**3 - 59611791*log(2*x**2 - x + 3)*x**2 - 24666948*log(2*x**2 - x 
 + 3)*x - 8222316*log(2*x**2 - x + 3) - 8472400750*x**4 + 549010*x**2 + 43 
23124344*x + 264152240)/(43775610384*(25*x**4 + 30*x**3 + 29*x**2 + 12*x + 
 4))