\(\int \frac {1}{\sqrt {3-x+2 x^2} (2+3 x+5 x^2)^2} \, dx\) [118]

Optimal result

Integrand size = 27, antiderivative size = 188 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{682} \left (2343727+1678700 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (2343727+1678700 \sqrt {2}\right )}} \left (2119+1816 \sqrt {2}+\left (5751+3935 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1364}-\frac {\sqrt {\frac {1}{682} \left (-2343727+1678700 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-2343727+1678700 \sqrt {2}\right )}} \left (2119-1816 \sqrt {2}+\left (5751-3935 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1364} \] Output:

(4+65*x)*(2*x^2-x+3)^(1/2)/(3410*x^2+2046*x+1364)+1/930248*(1598421814+114 
4873400*2^(1/2))^(1/2)*arctan(11^(1/2)/(72655537+52039700*2^(1/2))^(1/2)*( 
2119+1816*2^(1/2)+(5751+3935*2^(1/2))*x)/(2*x^2-x+3)^(1/2))-1/930248*(-159 
8421814+1144873400*2^(1/2))^(1/2)*arctanh(11^(1/2)/(-72655537+52039700*2^( 
1/2))^(1/2)*(2119-1816*2^(1/2)+(5751-3935*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 = 0.57 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{682 \left (2+3 x+5 x^2\right )}+\frac {\text {RootSum}\left [-10580-2024 \sqrt {2} \text {$\#$1}+68 \text {$\#$1}^2+44 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-9430 \sqrt {2} \log \left (\sqrt {2} (-1+4 x)-4 \sqrt {3-x+2 x^2}+\text {$\#$1}\right )+4492 \log \left (\sqrt {2} (-1+4 x)-4 \sqrt {3-x+2 x^2}+\text {$\#$1}\right ) \text {$\#$1}+205 \sqrt {2} \log \left (\sqrt {2} (-1+4 x)-4 \sqrt {3-x+2 x^2}+\text {$\#$1}\right ) \text {$\#$1}^2}{-506 \sqrt {2}+34 \text {$\#$1}+33 \sqrt {2} \text {$\#$1}^2-5 \text {$\#$1}^3}\&\right ]}{682 \sqrt {2}} \] Input:

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

Output:

((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + RootSum[-10580 
- 2024*Sqrt[2]*#1 + 68*#1^2 + 44*Sqrt[2]*#1^3 - 5*#1^4 & , (-9430*Sqrt[2]* 
Log[Sqrt[2]*(-1 + 4*x) - 4*Sqrt[3 - x + 2*x^2] + #1] + 4492*Log[Sqrt[2]*(- 
1 + 4*x) - 4*Sqrt[3 - x + 2*x^2] + #1]*#1 + 205*Sqrt[2]*Log[Sqrt[2]*(-1 + 
4*x) - 4*Sqrt[3 - x + 2*x^2] + #1]*#1^2)/(-506*Sqrt[2] + 34*#1 + 33*Sqrt[2 
]*#1^2 - 5*#1^3) & ]/(682*Sqrt[2])
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1305, 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.

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

\(\Big \downarrow \) 1305

\(\displaystyle \frac {(65 x+4) \sqrt {2 x^2-x+3}}{682 \left (5 x^2+3 x+2\right )}-\frac {\int -\frac {11 (332-205 x)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{7502}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {332-205 x}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1368

\(\displaystyle \frac {\frac {\int -\frac {11 \left (-\left (\left (127-205 \sqrt {2}\right ) x\right )-332 \sqrt {2}+537\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int -\frac {11 \left (-\left (\left (127+205 \sqrt {2}\right ) x\right )+332 \sqrt {2}+537\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-\left (\left (127+205 \sqrt {2}\right ) x\right )+332 \sqrt {2}+537}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {-\left (\left (127-205 \sqrt {2}\right ) x\right )-332 \sqrt {2}+537}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 1362

\(\displaystyle \frac {\frac {\left (2343727-1678700 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119\right )^2}{2 x^2-x+3}-31 \left (2343727-1678700 \sqrt {2}\right )}d\frac {\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (2343727+1678700 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (5751+3935 \sqrt {2}\right ) x+1816 \sqrt {2}+2119\right )^2}{2 x^2-x+3}-31 \left (2343727+1678700 \sqrt {2}\right )}d\frac {\left (5751+3935 \sqrt {2}\right ) x+1816 \sqrt {2}+2119}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\left (2343727-1678700 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119\right )^2}{2 x^2-x+3}-31 \left (2343727-1678700 \sqrt {2}\right )}d\frac {\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (2343727+1678700 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (2343727+1678700 \sqrt {2}\right )}} \left (\left (5751+3935 \sqrt {2}\right ) x+1816 \sqrt {2}+2119\right )}{\sqrt {2 x^2-x+3}}\right )}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {\frac {1}{682} \left (2343727+1678700 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (2343727+1678700 \sqrt {2}\right )}} \left (\left (5751+3935 \sqrt {2}\right ) x+1816 \sqrt {2}+2119\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (2343727-1678700 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (1678700 \sqrt {2}-2343727\right )}} \left (\left (5751-3935 \sqrt {2}\right ) x-1816 \sqrt {2}+2119\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (1678700 \sqrt {2}-2343727\right )}}}{1364}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{682 \left (5 x^2+3 x+2\right )}\)

Input:

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

Output:

((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(682*(2 + 3*x + 5*x^2)) + (Sqrt[(2343727 
+ 1678700*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(2343727 + 1678700*Sqrt[2]))]* 
(2119 + 1816*Sqrt[2] + (5751 + 3935*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + (( 
2343727 - 1678700*Sqrt[2])*ArcTanh[(Sqrt[11/(31*(-2343727 + 1678700*Sqrt[2 
]))]*(2119 - 1816*Sqrt[2] + (5751 - 3935*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] 
)/Sqrt[682*(-2343727 + 1678700*Sqrt[2])])/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]
 

Maple [C] (warning: unable to verify)

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

Time = 4.72 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.52

method	result	size
trager	\(\frac {\left (4+65 x \right ) \sqrt {2 x^{2}-x +3}}{3410 x^{2}+2046 x +1364}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right ) \ln \left (-\frac {2055399700464 x \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right )[phone] \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right ) x +1369841493940634484 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {2 x^{2}-x +3}-12361058255700 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right )-40592603336832 \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right ) x +16211894834030320271 \sqrt {2 x^{2}-x +3}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}[phone]\right )}{49104 x \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+534991 x -67921}\right )}{930248}-\frac {3 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right ) \ln \left (\frac {-394636742489088 x \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{5}-13779789867949248 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{3} x +32137043846114592 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {2 x^{2}-x +3}-2373323185094400 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{3}-96300551793960525 x \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )+386610578208666875 \sqrt {2 x^{2}-x +3}-59939356016807400 \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )}{98208 x \operatorname {RootOf}\left (1205601408 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}[phone]\right )^{2}+1273745 x +135842}\right )}{341}\)	\(473\)
risch	\(\frac {\left (4+65 x \right ) \sqrt {2 x^{2}-x +3}}{3410 x^{2}+2046 x +1364}+\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 (153463 \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}}+217330 \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}-300787234 \,\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 )}{28837688 \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}}}\)	\(716\)
default	\(\text {Expression too large to display}\)	\(5225\)

Input:

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

Output:

1/682*(4+65*x)/(5*x^2+3*x+2)*(2*x^2-x+3)^(1/2)-1/930248*RootOf(_Z^2+669778 
56*RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)^2+1598421814)*ln( 
-(2055399700464*x*RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)^4* 
RootOf(_Z^2+66977856*RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625) 
^2+1598421814)[phone]*RootOf(1205601408*_Z^4+28771592652*_Z^2+1761 
27105625)^2*RootOf(_Z^2+66977856*RootOf(1205601408*_Z^4+28771592652*_Z^2+1 
76127105625)^2+1598421814)*x+1369841493940634484*RootOf(1205601408*_Z^4+28 
771592652*_Z^2+176127105625)^2*(2*x^2-x+3)^(1/2)-12361058255700*RootOf(120 
5601408*_Z^4+28771592652*_Z^2+176127105625)^2*RootOf(_Z^2+66977856*RootOf( 
1205601408*_Z^4+28771592652*_Z^2+176127105625)^2+1598421814)-4059260333683 
2*RootOf(_Z^2+66977856*RootOf(1205601408*_Z^4+28771592652*_Z^2+17612710562 
5)^2+1598421814)*x+16211894834030320271*(2*x^2-x+3)^(1/2)[phone]*R 
ootOf(_Z^2+66977856*RootOf(1205601408*_Z^4+28771592652*_Z^2+176127105625)^ 
2+1598421814))/(49104*x*RootOf(1205601408*_Z^4+28771592652*_Z^2+1761271056 
25)^2+534991*x-67921))-3/341*RootOf(1205601408*_Z^4+28771592652*_Z^2+17612 
7105625)*ln((-394636742489088*x*RootOf(1205601408*_Z^4+28771592652*_Z^2+17 
6127105625)^5-13779789867949248*RootOf(1205601408*_Z^4+28771592652*_Z^2+17 
6127105625)^3*x+32137043846114592*RootOf(1205601408*_Z^4+28771592652*_Z^2+ 
176127105625)^2*(2*x^2-x+3)^(1/2)-2373323185094400*RootOf(1205601408*_Z^4+ 
28771592652*_Z^2+176127105625)^3-96300551793960525*x*RootOf(1205601408*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (139) = 278\).

Time = 0.09 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.80 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/5456*(2*(5*x^2 + 3*x + 2)*sqrt(839350/341*sqrt(2) + 2343727/682)*arctan 
(-22/67921*(4*(15838*x^3 - 36948*x^2 - sqrt(2)*(12195*x^3 - 27017*x^2 - 73 
44*x + 12888) - 15152*x + 15936)*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(83 
9350/341*sqrt(2) - 2343727/682))*sqrt(839350/341*sqrt(2) + 2343727/682)/(3 
43*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) - 2*(5*x^2 + 3*x + 2)*sqrt(839 
350/341*sqrt(2) + 2343727/682)*arctan(22/67921*(4*(15838*x^3 - 36948*x^2 - 
 sqrt(2)*(12195*x^3 - 27017*x^2 - 7344*x + 12888) - 15152*x + 15936)*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(839350/341*sqrt(2) - 2343727/682))*sqrt( 
839350/341*sqrt(2) + 2343727/682)/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 
576)) + (5*x^2 + 3*x + 2)*sqrt(839350/341*sqrt(2) - 2343727/682)*log((3328 
129*x^2 + 22*sqrt(2*x^2 - x + 3)*(sqrt(2)*(14227*x - 36625) + 22398*x - 50 
852)*sqrt(839350/341*sqrt(2) - 2343727/682) + 2988524*sqrt(2)*(2*x^2 - x + 
 3) - 10256071*x + 13584200)/x^2) - (5*x^2 + 3*x + 2)*sqrt(839350/341*sqrt 
(2) - 2343727/682)*log((3328129*x^2 - 22*sqrt(2*x^2 - x + 3)*(sqrt(2)*(142 
27*x - 36625) + 22398*x - 50852)*sqrt(839350/341*sqrt(2) - 2343727/682) + 
2988524*sqrt(2)*(2*x^2 - x + 3) - 10256071*x + 13584200)/x^2) - 8*sqrt(2*x 
^2 - x + 3)*(65*x + 4))/(5*x^2 + 3*x + 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\sqrt {2 x^{2}-x +3}}{50 x^{6}+35 x^{5}+103 x^{4}+85 x^{3}+83 x^{2}+32 x +12}d x \] Input:

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

Output:

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)