Integrand size = 29, antiderivative size = 652 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}} \, dx=\frac {\sqrt {\frac {23}{11}} \left (1-i \sqrt {23}-4 x\right ) \sqrt {-1+i \sqrt {23}+4 x} \sqrt {6-\left (1-i \sqrt {23}\right ) x} \sqrt {\frac {\left (11 i-\sqrt {23}\right ) \left (2+3 x+5 x^2\right )}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )^2}} \left (1-\frac {\sqrt {-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \left (6-\left (1-i \sqrt {23}\right ) x\right )}{1-i \sqrt {23}-4 x}\right ) \sqrt {\frac {11-\frac {41 \left (i+\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )}-\frac {11 \left (3 i-\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )^2}}{\left (1-\frac {\sqrt {-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \left (6-\left (1-i \sqrt {23}\right ) x\right )}{1-i \sqrt {23}-4 x}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \sqrt {6-\left (1-i \sqrt {23}\right ) x}}{\sqrt {-1+i \sqrt {23}+4 x}}\right ),\frac {1}{88} \left (44-\frac {41 \left (i+\sqrt {23}\right )}{\sqrt {11+i \sqrt {23}}}\right )\right )}{\left (23+i \sqrt {23}\right ) \sqrt [4]{-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2} \sqrt {11-\frac {41 \left (i+\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )}-\frac {11 \left (3 i-\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (7 i+\sqrt {23}\right ) \left (1-i \sqrt {23}-4 x\right )^2}}} \]
1/11*(cos(2*arctan(((-3*I+23^(1/2))/(7*I+23^(1/2)))^(1/4)*(6-x*(1-I*23^(1/ 2)))^(1/2)/(-1+4*x+I*23^(1/2))^(1/2)))^2)^(1/2)/cos(2*arctan(((-3*I+23^(1/ 2))/(7*I+23^(1/2)))^(1/4)*(6-x*(1-I*23^(1/2)))^(1/2)/(-1+4*x+I*23^(1/2))^( 1/2)))*EllipticF(sin(2*arctan(((-3*I+23^(1/2))/(7*I+23^(1/2)))^(1/4)*(6-x* (1-I*23^(1/2)))^(1/2)/(-1+4*x+I*23^(1/2))^(1/2))),1/22*11^(1/2)*((66*I-22* 23^(1/2)+41*(-23*(3*I-23^(1/2))/(7*I+23^(1/2)))^(1/2)+41*I*((-3*I+23^(1/2) )/(7*I+23^(1/2)))^(1/2))/(3*I-23^(1/2)))^(1/2))*253^(1/2)*(1-4*x-I*23^(1/2 ))*(6-x*(1-I*23^(1/2)))^(1/2)*(-1+4*x+I*23^(1/2))^(1/2)*(1-(6-x*(1-I*23^(1 /2)))*((-3*I+23^(1/2))/(7*I+23^(1/2)))^(1/2)/(1-4*x-I*23^(1/2)))*((5*x^2+3 *x+2)*(11*I-23^(1/2))/(1-4*x-I*23^(1/2))^2/(7*I+23^(1/2)))^(1/2)*((11-11*( 6-x*(1-I*23^(1/2)))^2*(3*I-23^(1/2))/(1-4*x-I*23^(1/2))^2/(7*I+23^(1/2))-4 1*(6-x*(1-I*23^(1/2)))*(23^(1/2)+I)/(1-4*x-I*23^(1/2))/(7*I+23^(1/2)))/(1- (6-x*(1-I*23^(1/2)))*((-3*I+23^(1/2))/(7*I+23^(1/2)))^(1/2)/(1-4*x-I*23^(1 /2)))^2)^(1/2)/(23+I*23^(1/2))/((-3*I+23^(1/2))/(7*I+23^(1/2)))^(1/4)/(2*x ^2-x+3)^(1/2)/(5*x^2+3*x+2)^(1/2)/(11-11*(6-x*(1-I*23^(1/2)))^2*(3*I-23^(1 /2))/(1-4*x-I*23^(1/2))^2/(7*I+23^(1/2))-41*(6-x*(1-I*23^(1/2)))*(23^(1/2) +I)/(1-4*x-I*23^(1/2))/(7*I+23^(1/2)))^(1/2)
Time = 2.25 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}} \, dx=\frac {\left (1+i \sqrt {23}-4 x\right ) \left (3 i+\sqrt {31}+10 i x\right ) \sqrt {\frac {6 i-2 \sqrt {31}+20 i x}{\left (11 i+5 \sqrt {23}-2 \sqrt {31}\right ) \left (-i+\sqrt {23}+4 i x\right )}} \sqrt {\frac {63-3 i \sqrt {23}-i \sqrt {31}-\sqrt {713}+\left (-22-10 i \sqrt {23}+4 i \sqrt {31}\right ) x}{\left (11 i+5 \sqrt {23}+2 \sqrt {31}\right ) \left (-i+\sqrt {23}+4 i x\right )}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} \sqrt {-\frac {-63+3 i \sqrt {23}+i \sqrt {31}+\sqrt {713}+2 \left (11+5 i \sqrt {23}-2 i \sqrt {31}\right ) x}{\left (11 i+5 \sqrt {23}+2 \sqrt {31}\right ) \left (-i+\sqrt {23}+4 i x\right )}}\right ),\frac {1}{484} \left (1197+41 \sqrt {713}\right )\right )}{\left (-11 i+5 \sqrt {23}-2 \sqrt {31}\right ) \sqrt {\frac {3 i+\sqrt {31}+10 i x}{\left (11 i+5 \sqrt {23}+2 \sqrt {31}\right ) \left (-i+\sqrt {23}+4 i x\right )}} \sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}} \]
((1 + I*Sqrt[23] - 4*x)*(3*I + Sqrt[31] + (10*I)*x)*Sqrt[(6*I - 2*Sqrt[31] + (20*I)*x)/((11*I + 5*Sqrt[23] - 2*Sqrt[31])*(-I + Sqrt[23] + (4*I)*x))] *Sqrt[(63 - (3*I)*Sqrt[23] - I*Sqrt[31] - Sqrt[713] + (-22 - (10*I)*Sqrt[2 3] + (4*I)*Sqrt[31])*x)/((11*I + 5*Sqrt[23] + 2*Sqrt[31])*(-I + Sqrt[23] + (4*I)*x))]*EllipticF[ArcSin[Sqrt[2]*Sqrt[-((-63 + (3*I)*Sqrt[23] + I*Sqrt [31] + Sqrt[713] + 2*(11 + (5*I)*Sqrt[23] - (2*I)*Sqrt[31])*x)/((11*I + 5* Sqrt[23] + 2*Sqrt[31])*(-I + Sqrt[23] + (4*I)*x)))]], (1197 + 41*Sqrt[713] )/484])/((-11*I + 5*Sqrt[23] - 2*Sqrt[31])*Sqrt[(3*I + Sqrt[31] + (10*I)*x )/((11*I + 5*Sqrt[23] + 2*Sqrt[31])*(-I + Sqrt[23] + (4*I)*x))]*Sqrt[3 - x + 2*x^2]*Sqrt[2 + 3*x + 5*x^2])
Time = 0.60 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1323, 1280, 1416}
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.
| \(\displaystyle \int \frac {1}{\sqrt {2 x^2-x+3} \sqrt {5 x^2+3 x+2}} \, dx\) |
| \(\Big \downarrow \) 1323 |
| \(\displaystyle \frac {\sqrt {4 x+i \sqrt {23}-1} \sqrt {6-\left (1-i \sqrt {23}\right ) x} \int \frac {1}{\sqrt {4 x+i \sqrt {23}-1} \sqrt {6-\left (1-i \sqrt {23}\right ) x} \sqrt {5 x^2+3 x+2}}dx}{\sqrt {2 x^2-x+3}}\) |
| \(\Big \downarrow \) 1280 |
| \(\displaystyle \frac {2 \sqrt {\frac {23}{11}} \left (-4 x-i \sqrt {23}+1\right ) \sqrt {4 x+i \sqrt {23}-1} \sqrt {6-\left (1-i \sqrt {23}\right ) x} \sqrt {\frac {\left (-\sqrt {23}+11 i\right ) \left (5 x^2+3 x+2\right )}{\left (\sqrt {23}+7 i\right ) \left (-4 x-i \sqrt {23}+1\right )^2}} \int \frac {1}{\sqrt {-\frac {\left (3 i-\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (7 i+\sqrt {23}\right ) \left (4 x+i \sqrt {23}-1\right )^2}+\frac {41 \left (i+\sqrt {23}\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{11 \left (7 i+\sqrt {23}\right ) \left (4 x+i \sqrt {23}-1\right )}+1}}d\frac {\sqrt {6-\left (1-i \sqrt {23}\right ) x}}{\sqrt {4 x+i \sqrt {23}-1}}}{\left (23+i \sqrt {23}\right ) \sqrt {2 x^2-x+3} \sqrt {5 x^2+3 x+2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\sqrt {\frac {23}{11}} \left (-4 x-i \sqrt {23}+1\right ) \sqrt {4 x+i \sqrt {23}-1} \sqrt {6-\left (1-i \sqrt {23}\right ) x} \sqrt {\frac {\left (-\sqrt {23}+11 i\right ) \left (5 x^2+3 x+2\right )}{\left (\sqrt {23}+7 i\right ) \left (-4 x-i \sqrt {23}+1\right )^2}} \left (1+\frac {\sqrt {-\frac {-\sqrt {23}+3 i}{\sqrt {23}+7 i}} \left (6-\left (1-i \sqrt {23}\right ) x\right )}{4 x+i \sqrt {23}-1}\right ) \sqrt {\frac {-\frac {11 \left (-\sqrt {23}+3 i\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (\sqrt {23}+7 i\right ) \left (4 x+i \sqrt {23}-1\right )^2}+\frac {41 \left (\sqrt {23}+i\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{\left (\sqrt {23}+7 i\right ) \left (4 x+i \sqrt {23}-1\right )}+11}{\left (1+\frac {\sqrt {-\frac {-\sqrt {23}+3 i}{\sqrt {23}+7 i}} \left (6-\left (1-i \sqrt {23}\right ) x\right )}{4 x+i \sqrt {23}-1}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{-\frac {3 i-\sqrt {23}}{7 i+\sqrt {23}}} \sqrt {6-\left (1-i \sqrt {23}\right ) x}}{\sqrt {4 x+i \sqrt {23}-1}}\right ),\frac {1}{88} \left (44-\frac {41 \left (i+\sqrt {23}\right )}{\sqrt {11+i \sqrt {23}}}\right )\right )}{\left (23+i \sqrt {23}\right ) \sqrt [4]{-\frac {-\sqrt {23}+3 i}{\sqrt {23}+7 i}} \sqrt {2 x^2-x+3} \sqrt {5 x^2+3 x+2} \sqrt {-\frac {11 \left (-\sqrt {23}+3 i\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )^2}{\left (\sqrt {23}+7 i\right ) \left (4 x+i \sqrt {23}-1\right )^2}+\frac {41 \left (\sqrt {23}+i\right ) \left (6-\left (1-i \sqrt {23}\right ) x\right )}{\left (\sqrt {23}+7 i\right ) \left (4 x+i \sqrt {23}-1\right )}+11}}\) |
(Sqrt[23/11]*(1 - I*Sqrt[23] - 4*x)*Sqrt[-1 + I*Sqrt[23] + 4*x]*Sqrt[6 - ( 1 - I*Sqrt[23])*x]*Sqrt[((11*I - Sqrt[23])*(2 + 3*x + 5*x^2))/((7*I + Sqrt [23])*(1 - I*Sqrt[23] - 4*x)^2)]*(1 + (Sqrt[-((3*I - Sqrt[23])/(7*I + Sqrt [23]))]*(6 - (1 - I*Sqrt[23])*x))/(-1 + I*Sqrt[23] + 4*x))*Sqrt[(11 + (41* (I + Sqrt[23])*(6 - (1 - I*Sqrt[23])*x))/((7*I + Sqrt[23])*(-1 + I*Sqrt[23 ] + 4*x)) - (11*(3*I - Sqrt[23])*(6 - (1 - I*Sqrt[23])*x)^2)/((7*I + Sqrt[ 23])*(-1 + I*Sqrt[23] + 4*x)^2))/(1 + (Sqrt[-((3*I - Sqrt[23])/(7*I + Sqrt [23]))]*(6 - (1 - I*Sqrt[23])*x))/(-1 + I*Sqrt[23] + 4*x))^2]*EllipticF[2* ArcTan[((-((3*I - Sqrt[23])/(7*I + Sqrt[23])))^(1/4)*Sqrt[6 - (1 - I*Sqrt[ 23])*x])/Sqrt[-1 + I*Sqrt[23] + 4*x]], (44 - (41*(I + Sqrt[23]))/Sqrt[11 + I*Sqrt[23]])/88])/((23 + I*Sqrt[23])*(-((3*I - Sqrt[23])/(7*I + Sqrt[23]) ))^(1/4)*Sqrt[3 - x + 2*x^2]*Sqrt[2 + 3*x + 5*x^2]*Sqrt[11 + (41*(I + Sqrt [23])*(6 - (1 - I*Sqrt[23])*x))/((7*I + Sqrt[23])*(-1 + I*Sqrt[23] + 4*x)) - (11*(3*I - Sqrt[23])*(6 - (1 - I*Sqrt[23])*x)^2)/((7*I + Sqrt[23])*(-1 + I*Sqrt[23] + 4*x)^2)])
3.2.23.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.) *(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[-2*(d + e*x)*(Sqrt[(e*f - d*g)^2* ((a + b*x + c*x^2)/((c*f^2 - b*f*g + a*g^2)*(d + e*x)^2))]/((e*f - d*g)*Sqr t[a + b*x + c*x^2])) Subst[Int[1/Sqrt[1 - (2*c*d*f - b*e*f - b*d*g + 2*a* e*g)*(x^2/(c*f^2 - b*f*g + a*g^2)) + (c*d^2 - b*d*e + a*e^2)*(x^4/(c*f^2 - b*f*g + a*g^2))], x], x, Sqrt[f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c , d, e, f, g}, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]*Sqrt[(d_) + (e_.)*(x_) + (f_. )*(x_)^2]), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b + r + 2 *c*x]*(Sqrt[2*a + (b + r)*x]/Sqrt[a + b*x + c*x^2]) Int[1/(Sqrt[b + r + 2 *c*x]*Sqrt[2*a + (b + r)*x]*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]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Time = 2.48 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.61
| method | result | size |
| elliptic | \(-\frac {i \sqrt {\left (2 x^{2}-x +3\right ) \left (5 x^{2}+3 x +2\right )}\, \left (\frac {11}{20}-\frac {i \sqrt {23}}{4}-\frac {i \sqrt {31}}{10}\right ) \sqrt {\frac {\left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}-\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}+\frac {i \sqrt {23}}{4}\right )}{\left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}+\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}}\, \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )^{2} \sqrt {\frac {i \sqrt {23}\, \left (x +\frac {3}{10}+\frac {i \sqrt {31}}{10}\right )}{\left (-\frac {11}{20}-\frac {i \sqrt {31}}{10}+\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}}\, \sqrt {\frac {i \sqrt {23}\, \left (x +\frac {3}{10}-\frac {i \sqrt {31}}{10}\right )}{\left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}+\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}}\, \sqrt {23}\, \sqrt {10}\, F\left (\sqrt {\frac {\left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}-\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}+\frac {i \sqrt {23}}{4}\right )}{\left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}+\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}}, \sqrt {\frac {\left (\frac {11}{20}+\frac {i \sqrt {23}}{4}+\frac {i \sqrt {31}}{10}\right ) \left (\frac {11}{20}-\frac {i \sqrt {23}}{4}-\frac {i \sqrt {31}}{10}\right )}{\left (\frac {11}{20}-\frac {i \sqrt {23}}{4}+\frac {i \sqrt {31}}{10}\right ) \left (\frac {11}{20}+\frac {i \sqrt {23}}{4}-\frac {i \sqrt {31}}{10}\right )}}\right )}{115 \sqrt {2 x^{2}-x +3}\, \sqrt {5 x^{2}+3 x +2}\, \left (-\frac {11}{20}+\frac {i \sqrt {31}}{10}-\frac {i \sqrt {23}}{4}\right ) \sqrt {\left (x -\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) \left (x -\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) \left (x +\frac {3}{10}+\frac {i \sqrt {31}}{10}\right ) \left (x +\frac {3}{10}-\frac {i \sqrt {31}}{10}\right )}}\) | \(395\) |
| default | \(\frac {4 i \sqrt {5 x^{2}+3 x +2}\, \sqrt {2 x^{2}-x +3}\, \left (2 i \sqrt {31}+5 i \sqrt {23}-11\right ) \sqrt {-\frac {\left (2 i \sqrt {31}-5 i \sqrt {23}-11\right ) \left (-1+4 x +i \sqrt {23}\right )}{\left (2 i \sqrt {31}+5 i \sqrt {23}-11\right ) \left (i \sqrt {23}-4 x +1\right )}}\, \left (i \sqrt {23}-4 x +1\right )^{2} \sqrt {\frac {i \sqrt {23}\, \left (i \sqrt {31}+10 x +3\right )}{\left (2 i \sqrt {31}-5 i \sqrt {23}+11\right ) \left (i \sqrt {23}-4 x +1\right )}}\, \sqrt {\frac {i \sqrt {23}\, \left (i \sqrt {31}-10 x -3\right )}{\left (2 i \sqrt {31}+5 i \sqrt {23}-11\right ) \left (i \sqrt {23}-4 x +1\right )}}\, \sqrt {23}\, \sqrt {10}\, F\left (\sqrt {-\frac {\left (2 i \sqrt {31}-5 i \sqrt {23}-11\right ) \left (-1+4 x +i \sqrt {23}\right )}{\left (2 i \sqrt {31}+5 i \sqrt {23}-11\right ) \left (i \sqrt {23}-4 x +1\right )}}, \sqrt {\frac {\left (2 i \sqrt {31}+5 i \sqrt {23}+11\right ) \left (2 i \sqrt {31}+5 i \sqrt {23}-11\right )}{\left (2 i \sqrt {31}-5 i \sqrt {23}+11\right ) \left (2 i \sqrt {31}-5 i \sqrt {23}-11\right )}}\right )}{23 \sqrt {10 x^{4}+x^{3}+16 x^{2}+7 x +6}\, \left (2 i \sqrt {31}-5 i \sqrt {23}-11\right ) \sqrt {\left (-1+4 x +i \sqrt {23}\right ) \left (i \sqrt {23}-4 x +1\right ) \left (i \sqrt {31}+10 x +3\right ) \left (i \sqrt {31}-10 x -3\right )}}\) | \(420\) |
-1/115*I*((2*x^2-x+3)*(5*x^2+3*x+2))^(1/2)/(2*x^2-x+3)^(1/2)/(5*x^2+3*x+2) ^(1/2)*(11/20-1/4*I*23^(1/2)-1/10*I*31^(1/2))*((-11/20+1/10*I*31^(1/2)-1/4 *I*23^(1/2))*(x-1/4+1/4*I*23^(1/2))/(-11/20+1/10*I*31^(1/2)+1/4*I*23^(1/2) )/(x-1/4-1/4*I*23^(1/2)))^(1/2)*(x-1/4-1/4*I*23^(1/2))^2*(I*23^(1/2)*(x+3/ 10+1/10*I*31^(1/2))/(-11/20-1/10*I*31^(1/2)+1/4*I*23^(1/2))/(x-1/4-1/4*I*2 3^(1/2)))^(1/2)*(I*23^(1/2)*(x+3/10-1/10*I*31^(1/2))/(-11/20+1/10*I*31^(1/ 2)+1/4*I*23^(1/2))/(x-1/4-1/4*I*23^(1/2)))^(1/2)/(-11/20+1/10*I*31^(1/2)-1 /4*I*23^(1/2))*23^(1/2)*10^(1/2)/((x-1/4+1/4*I*23^(1/2))*(x-1/4-1/4*I*23^( 1/2))*(x+3/10+1/10*I*31^(1/2))*(x+3/10-1/10*I*31^(1/2)))^(1/2)*EllipticF(( (-11/20+1/10*I*31^(1/2)-1/4*I*23^(1/2))*(x-1/4+1/4*I*23^(1/2))/(-11/20+1/1 0*I*31^(1/2)+1/4*I*23^(1/2))/(x-1/4-1/4*I*23^(1/2)))^(1/2),((11/20+1/4*I*2 3^(1/2)+1/10*I*31^(1/2))*(11/20-1/4*I*23^(1/2)-1/10*I*31^(1/2))/(11/20-1/4 *I*23^(1/2)+1/10*I*31^(1/2))/(11/20+1/4*I*23^(1/2)-1/10*I*31^(1/2)))^(1/2) )
\[ \int \frac {1}{\sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {2 \, x^{2} - x + 3}} \,d x } \]
\[ \int \frac {1}{\sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}} \, dx=\int \frac {1}{\sqrt {2 x^{2} - x + 3} \sqrt {5 x^{2} + 3 x + 2}}\, dx \]
\[ \int \frac {1}{\sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {2 \, x^{2} - x + 3}} \,d x } \]
\[ \int \frac {1}{\sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {2 \, x^{2} - x + 3}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {3-x+2 x^2} \sqrt {2+3 x+5 x^2}} \, dx=\int \frac {1}{\sqrt {2\,x^2-x+3}\,\sqrt {5\,x^2+3\,x+2}} \,d x \]