\(\int \frac {1}{\sqrt {3-x+2 x^2} (2+3 x+5 x^2)} \, dx\) [117]

Optimal result

Integrand size = 27, antiderivative size = 148 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx=\sqrt {\frac {1}{682} \left (13+10 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (13+10 \sqrt {2}\right )}} \left (7+3 \sqrt {2}+\left (13+10 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\sqrt {\frac {1}{682} \left (-13+10 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (-13+10 \sqrt {2}\right )}} \left (7-3 \sqrt {2}+\left (13-10 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \] Output:

1/682*(8866+6820*2^(1/2))^(1/2)*arctan(11^(1/2)/(403+310*2^(1/2))^(1/2)*(7 
+3*2^(1/2)+(13+10*2^(1/2))*x)/(2*x^2-x+3)^(1/2))-1/682*(-8866+6820*2^(1/2) 
)^(1/2)*arctanh(11^(1/2)/(-403+310*2^(1/2))^(1/2)*(7-3*2^(1/2)+(13-10*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 = 0.40 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx=\text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+2 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ] \] Input:

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

Output:

RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (Log[- 
(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 2*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqr 
t[3 - x + 2*x^2] - #1]*#1)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3 
) & ]
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1317, 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.

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

\(\Big \downarrow \) 1317

\(\displaystyle \frac {\int \frac {11 \left (-x+\sqrt {2}+1\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int \frac {11 \left (-x-\sqrt {2}+1\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1362

\(\displaystyle \frac {\left (13-10 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (13-10 \sqrt {2}\right ) x-3 \sqrt {2}+7\right )^2}{2 x^2-x+3}-31 \left (13-10 \sqrt {2}\right )}d\frac {\left (13-10 \sqrt {2}\right ) x-3 \sqrt {2}+7}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}-\frac {\left (13+10 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (13+10 \sqrt {2}\right ) x+3 \sqrt {2}+7\right )^2}{2 x^2-x+3}-31 \left (13+10 \sqrt {2}\right )}d\frac {\left (13+10 \sqrt {2}\right ) x+3 \sqrt {2}+7}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (13-10 \sqrt {2}\right ) \int \frac {1}{-\frac {11 \left (\left (13-10 \sqrt {2}\right ) x-3 \sqrt {2}+7\right )^2}{2 x^2-x+3}-31 \left (13-10 \sqrt {2}\right )}d\frac {\left (13-10 \sqrt {2}\right ) x-3 \sqrt {2}+7}{\sqrt {2 x^2-x+3}}}{\sqrt {2}}+\sqrt {\frac {1}{682} \left (13+10 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (13+10 \sqrt {2}\right )}} \left (\left (13+10 \sqrt {2}\right ) x+3 \sqrt {2}+7\right )}{\sqrt {2 x^2-x+3}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \sqrt {\frac {1}{682} \left (13+10 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (13+10 \sqrt {2}\right )}} \left (\left (13+10 \sqrt {2}\right ) x+3 \sqrt {2}+7\right )}{\sqrt {2 x^2-x+3}}\right )+\frac {\left (13-10 \sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (10 \sqrt {2}-13\right )}} \left (\left (13-10 \sqrt {2}\right ) x-3 \sqrt {2}+7\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (10 \sqrt {2}-13\right )}}\)

Input:

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

Output:

Sqrt[(13 + 10*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(13 + 10*Sqrt[2]))]*(7 + 3 
*Sqrt[2] + (13 + 10*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]] + ((13 - 10*Sqrt[2]) 
*ArcTanh[(Sqrt[11/(31*(-13 + 10*Sqrt[2]))]*(7 - 3*Sqrt[2] + (13 - 10*Sqrt[ 
2])*x))/Sqrt[3 - x + 2*x^2]])/Sqrt[682*(-13 + 10*Sqrt[2])]
 

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 1317

Int[1/(((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[(c*d - a*f + q + (c*e - b*f)*x)/((a + b*x + c*x^2 
)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q)   Int[(c*d - a*f - q + (c*e 
 - b*f)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, 
b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[c*e 
 - b*f, 0] && NegQ[b^2 - 4*a*c]
 

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]
 

Maple [C] (warning: unable to verify)

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

Time = 4.32 (sec) , antiderivative size = 443, normalized size of antiderivative = 2.99

method	result	size
trager	\(-\operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right ) \ln \left (-\frac {-74884964 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{5} x -3976060 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{3} x +391468 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2} \sqrt {2 x^{2}-x +3}+954800 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{3}-41625 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right ) x +3650 \sqrt {2 x^{2}-x +3}+37000 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )}{682 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2} x +5 x -2}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+465124 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right ) \ln \left (-\frac {18721241 \operatorname {RootOf}\left (\textit {\_Z}^{2}+465124 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right ) \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{4} x -280302 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+465124 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right ) x +238700 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+465124 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right )-66745294 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2} \sqrt {2 x^{2}-x +3}-1739 \operatorname {RootOf}\left (\textit {\_Z}^{2}+465124 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right ) x -4700 \operatorname {RootOf}\left (\textit {\_Z}^{2}+465124 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2}+8866\right )-649946 \sqrt {2 x^{2}-x +3}}{341 \operatorname {RootOf}\left (232562 \textit {\_Z}^{4}+4433 \textit {\_Z}^{2}+25\right )^{2} x +4 x +1}\right )}{682}\)	\(443\)
default	\(\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 (369 \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}}+520 \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}}+465124 \,\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}-866822 \,\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 )}{21142 \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}}}\)	\(684\)

Input:

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

Output:

-RootOf(232562*_Z^4+4433*_Z^2+25)*ln(-(-74884964*RootOf(232562*_Z^4+4433*_ 
Z^2+25)^5*x-3976060*RootOf(232562*_Z^4+4433*_Z^2+25)^3*x+391468*RootOf(232 
562*_Z^4+4433*_Z^2+25)^2*(2*x^2-x+3)^(1/2)+954800*RootOf(232562*_Z^4+4433* 
_Z^2+25)^3-41625*RootOf(232562*_Z^4+4433*_Z^2+25)*x+3650*(2*x^2-x+3)^(1/2) 
+37000*RootOf(232562*_Z^4+4433*_Z^2+25))/(682*RootOf(232562*_Z^4+4433*_Z^2 
+25)^2*x+5*x-2))+1/682*RootOf(_Z^2+465124*RootOf(232562*_Z^4+4433*_Z^2+25) 
^2+8866)*ln(-(18721241*RootOf(_Z^2+465124*RootOf(232562*_Z^4+4433*_Z^2+25) 
^2+8866)*RootOf(232562*_Z^4+4433*_Z^2+25)^4*x-280302*RootOf(232562*_Z^4+44 
33*_Z^2+25)^2*RootOf(_Z^2+465124*RootOf(232562*_Z^4+4433*_Z^2+25)^2+8866)* 
x+238700*RootOf(232562*_Z^4+4433*_Z^2+25)^2*RootOf(_Z^2+465124*RootOf(2325 
62*_Z^4+4433*_Z^2+25)^2+8866)-66745294*RootOf(232562*_Z^4+4433*_Z^2+25)^2* 
(2*x^2-x+3)^(1/2)-1739*RootOf(_Z^2+465124*RootOf(232562*_Z^4+4433*_Z^2+25) 
^2+8866)*x-4700*RootOf(_Z^2+465124*RootOf(232562*_Z^4+4433*_Z^2+25)^2+8866 
)-649946*(2*x^2-x+3)^(1/2))/(341*RootOf(232562*_Z^4+4433*_Z^2+25)^2*x+4*x+ 
1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (108) = 216\).

Time = 0.08 (sec) , antiderivative size = 454, normalized size of antiderivative = 3.07 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx=\frac {1}{2} \, \sqrt {\frac {5}{341} \, \sqrt {2} + \frac {13}{682}} \arctan \left (-\frac {22 \, {\left (4 \, {\left (44 \, x^{3} - 94 \, x^{2} - \sqrt {2} {\left (25 \, x^{3} - 61 \, x^{2} - 32 \, x + 24\right )} - 16 \, x + 48\right )} \sqrt {2 \, x^{2} - x + 3} + {\left (171 \, x^{4} + 1212 \, x^{3} - 1640 \, x^{2} - 176 \, \sqrt {2} {\left (6 \, x^{4} + 5 \, x^{3} + 5 \, x^{2} + 12 \, x\right )} - 3936 \, x\right )} \sqrt {\frac {5}{341} \, \sqrt {2} - \frac {13}{682}}\right )} \sqrt {\frac {5}{341} \, \sqrt {2} + \frac {13}{682}}}{343 \, x^{4} - 400 \, x^{3} + 1136 \, x^{2} + 384 \, x - 576}\right ) - \frac {1}{2} \, \sqrt {\frac {5}{341} \, \sqrt {2} + \frac {13}{682}} \arctan \left (\frac {22 \, {\left (4 \, {\left (44 \, x^{3} - 94 \, x^{2} - \sqrt {2} {\left (25 \, x^{3} - 61 \, x^{2} - 32 \, x + 24\right )} - 16 \, x + 48\right )} \sqrt {2 \, x^{2} - x + 3} - {\left (171 \, x^{4} + 1212 \, x^{3} - 1640 \, x^{2} - 176 \, \sqrt {2} {\left (6 \, x^{4} + 5 \, x^{3} + 5 \, x^{2} + 12 \, x\right )} - 3936 \, x\right )} \sqrt {\frac {5}{341} \, \sqrt {2} - \frac {13}{682}}\right )} \sqrt {\frac {5}{341} \, \sqrt {2} + \frac {13}{682}}}{343 \, x^{4} - 400 \, x^{3} + 1136 \, x^{2} + 384 \, x - 576}\right ) + \frac {1}{4} \, \sqrt {\frac {5}{341} \, \sqrt {2} - \frac {13}{682}} \log \left (\frac {49 \, x^{2} + 22 \, \sqrt {2 \, x^{2} - x + 3} {\left (\sqrt {2} {\left (41 \, x - 85\right )} + 44 \, x - 126\right )} \sqrt {\frac {5}{341} \, \sqrt {2} - \frac {13}{682}} + 44 \, \sqrt {2} {\left (2 \, x^{2} - x + 3\right )} - 151 \, x + 200}{x^{2}}\right ) - \frac {1}{4} \, \sqrt {\frac {5}{341} \, \sqrt {2} - \frac {13}{682}} \log \left (\frac {49 \, x^{2} - 22 \, \sqrt {2 \, x^{2} - x + 3} {\left (\sqrt {2} {\left (41 \, x - 85\right )} + 44 \, x - 126\right )} \sqrt {\frac {5}{341} \, \sqrt {2} - \frac {13}{682}} + 44 \, \sqrt {2} {\left (2 \, x^{2} - x + 3\right )} - 151 \, x + 200}{x^{2}}\right ) \] Input:

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

Output:

1/2*sqrt(5/341*sqrt(2) + 13/682)*arctan(-22*(4*(44*x^3 - 94*x^2 - sqrt(2)* 
(25*x^3 - 61*x^2 - 32*x + 24) - 16*x + 48)*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(5/341*sqrt(2) - 13/682))*sqrt(5/341*sqrt(2) + 13/682)/(343*x^4 - 4 
00*x^3 + 1136*x^2 + 384*x - 576)) - 1/2*sqrt(5/341*sqrt(2) + 13/682)*arcta 
n(22*(4*(44*x^3 - 94*x^2 - sqrt(2)*(25*x^3 - 61*x^2 - 32*x + 24) - 16*x + 
48)*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(5/341*sqrt(2) - 13/682))*sqrt(5 
/341*sqrt(2) + 13/682)/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) + 1/4 
*sqrt(5/341*sqrt(2) - 13/682)*log((49*x^2 + 22*sqrt(2*x^2 - x + 3)*(sqrt(2 
)*(41*x - 85) + 44*x - 126)*sqrt(5/341*sqrt(2) - 13/682) + 44*sqrt(2)*(2*x 
^2 - x + 3) - 151*x + 200)/x^2) - 1/4*sqrt(5/341*sqrt(2) - 13/682)*log((49 
*x^2 - 22*sqrt(2*x^2 - x + 3)*(sqrt(2)*(41*x - 85) + 44*x - 126)*sqrt(5/34 
1*sqrt(2) - 13/682) + 44*sqrt(2)*(2*x^2 - x + 3) - 151*x + 200)/x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx=\int \frac {\sqrt {2 x^{2}-x +3}}{10 x^{4}+x^{3}+16 x^{2}+7 x +6}d x \] Input:

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

Output:

int(sqrt(2*x**2 - x + 3)/(10*x**4 + x**3 + 16*x**2 + 7*x + 6),x)