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

Optimal. Leaf size=652 \[ \frac{\sqrt{\frac{23}{11}} \left (-4 x-i \sqrt{23}+1\right ) \sqrt{4 x+i \sqrt{23}-1} \sqrt{6-\left (1-i \sqrt{23}\right ) x} \sqrt{\frac{\left (-\sqrt{23}+11 i\right ) \left (5 x^2+3 x+2\right )}{\left (\sqrt{23}+7 i\right ) \left (-4 x-i \sqrt{23}+1\right )^2}} \left (1-\frac{\sqrt{-\frac{-\sqrt{23}+3 i}{\sqrt{23}+7 i}} \left (6-\left (1-i \sqrt{23}\right ) x\right )}{-4 x-i \sqrt{23}+1}\right ) \sqrt{\frac{-\frac{11 \left (-\sqrt{23}+3 i\right ) \left (6-\left (1-i \sqrt{23}\right ) x\right )^2}{\left (\sqrt{23}+7 i\right ) \left (-4 x-i \sqrt{23}+1\right )^2}-\frac{41 \left (\sqrt{23}+i\right ) \left (6-\left (1-i \sqrt{23}\right ) x\right )}{\left (\sqrt{23}+7 i\right ) \left (-4 x-i \sqrt{23}+1\right )}+11}{\left (1-\frac{\sqrt{-\frac{-\sqrt{23}+3 i}{\sqrt{23}+7 i}} \left (6-\left (1-i \sqrt{23}\right ) x\right )}{-4 x-i \sqrt{23}+1}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-\frac{-\sqrt{23}+3 i}{\sqrt{23}+7 i}} \sqrt{6-\left (1-i \sqrt{23}\right ) x}}{\sqrt{4 x+i \sqrt{23}-1}}\right ),\frac{1}{88} \left (44-\frac{41 \left (\sqrt{23}+i\right )}{\sqrt{11+i \sqrt{23}}}\right )\right )}{\left (23+i \sqrt{23}\right ) \sqrt [4]{-\frac{-\sqrt{23}+3 i}{\sqrt{23}+7 i}} \sqrt{2 x^2-x+3} \sqrt{5 x^2+3 x+2} \sqrt{-\frac{11 \left (-\sqrt{23}+3 i\right ) \left (6-\left (1-i \sqrt{23}\right ) x\right )^2}{\left (\sqrt{23}+7 i\right ) \left (-4 x-i \sqrt{23}+1\right )^2}-\frac{41 \left (\sqrt{23}+i\right ) \left (6-\left (1-i \sqrt{23}\right ) x\right )}{\left (\sqrt{23}+7 i\right ) \left (-4 x-i \sqrt{23}+1\right )}+11}} \]

[Out]

(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 - Sqr
t[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 + Sq
rt[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 + Sqr
t[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*Sqr
t[23])*x])/Sqrt[-1 + I*Sqrt[23] + 4*x]], (44 - (41*(I + Sqrt[23]))/Sqrt[11 + I*Sqrt[23]])/88])/((23 + I*Sqrt[2
3])*(-((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)])

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Rubi [A]  time = 0.676932, antiderivative size = 652, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {992, 935, 1103} \[ \frac{\sqrt{\frac{23}{11}} \left (-4 x-i \sqrt{23}+1\right ) \sqrt{4 x+i \sqrt{23}-1} \sqrt{6-\left (1-i \sqrt{23}\right ) x} \sqrt{\frac{\left (-\sqrt{23}+11 i\right ) \left (5 x^2+3 x+2\right )}{\left (\sqrt{23}+7 i\right ) \left (-4 x-i \sqrt{23}+1\right )^2}} \left (1-\frac{\sqrt{-\frac{-\sqrt{23}+3 i}{\sqrt{23}+7 i}} \left (6-\left (1-i \sqrt{23}\right ) x\right )}{-4 x-i \sqrt{23}+1}\right ) \sqrt{\frac{-\frac{11 \left (-\sqrt{23}+3 i\right ) \left (6-\left (1-i \sqrt{23}\right ) x\right )^2}{\left (\sqrt{23}+7 i\right ) \left (-4 x-i \sqrt{23}+1\right )^2}-\frac{41 \left (\sqrt{23}+i\right ) \left (6-\left (1-i \sqrt{23}\right ) x\right )}{\left (\sqrt{23}+7 i\right ) \left (-4 x-i \sqrt{23}+1\right )}+11}{\left (1-\frac{\sqrt{-\frac{-\sqrt{23}+3 i}{\sqrt{23}+7 i}} \left (6-\left (1-i \sqrt{23}\right ) x\right )}{-4 x-i \sqrt{23}+1}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-\frac{3 i-\sqrt{23}}{7 i+\sqrt{23}}} \sqrt{6-\left (1-i \sqrt{23}\right ) x}}{\sqrt{4 x+i \sqrt{23}-1}}\right )|\frac{1}{88} \left (44-\frac{41 \left (i+\sqrt{23}\right )}{\sqrt{11+i \sqrt{23}}}\right )\right )}{\left (23+i \sqrt{23}\right ) \sqrt [4]{-\frac{-\sqrt{23}+3 i}{\sqrt{23}+7 i}} \sqrt{2 x^2-x+3} \sqrt{5 x^2+3 x+2} \sqrt{-\frac{11 \left (-\sqrt{23}+3 i\right ) \left (6-\left (1-i \sqrt{23}\right ) x\right )^2}{\left (\sqrt{23}+7 i\right ) \left (-4 x-i \sqrt{23}+1\right )^2}-\frac{41 \left (\sqrt{23}+i\right ) \left (6-\left (1-i \sqrt{23}\right ) x\right )}{\left (\sqrt{23}+7 i\right ) \left (-4 x-i \sqrt{23}+1\right )}+11}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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 - Sqr
t[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 + Sq
rt[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 + Sqr
t[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*Sqr
t[23])*x])/Sqrt[-1 + I*Sqrt[23] + 4*x]], (44 - (41*(I + Sqrt[23]))/Sqrt[11 + I*Sqrt[23]])/88])/((23 + I*Sqrt[2
3])*(-((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)])

Rule 992

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]}, Dist[(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]

Rule 935

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :
> Dist[(-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)*Sqrt[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] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1103

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{3-x+2 x^2} \sqrt{2+3 x+5 x^2}} \, dx &=\frac{\left (\sqrt{-1+i \sqrt{23}+4 x} \sqrt{6+\left (-1+i \sqrt{23}\right ) x}\right ) \int \frac{1}{\sqrt{-1+i \sqrt{23}+4 x} \sqrt{6+\left (-1+i \sqrt{23}\right ) x} \sqrt{2+3 x+5 x^2}} \, dx}{\sqrt{3-x+2 x^2}}\\ &=-\frac{\left (2 \left (-1+i \sqrt{23}+4 x\right )^{3/2} \sqrt{6+\left (-1+i \sqrt{23}\right ) x} \sqrt{\frac{\left (24-\left (-1+i \sqrt{23}\right )^2\right )^2 \left (2+3 x+5 x^2\right )}{\left (180-18 \left (-1+i \sqrt{23}\right )+2 \left (-1+i \sqrt{23}\right )^2\right ) \left (-1+i \sqrt{23}+4 x\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{\left (-72+76 \left (-1+i \sqrt{23}\right )-3 \left (-1+i \sqrt{23}\right )^2\right ) x^2}{180-18 \left (-1+i \sqrt{23}\right )+2 \left (-1+i \sqrt{23}\right )^2}+\frac{\left (32-12 \left (-1+i \sqrt{23}\right )+5 \left (-1+i \sqrt{23}\right )^2\right ) x^4}{180-18 \left (-1+i \sqrt{23}\right )+2 \left (-1+i \sqrt{23}\right )^2}}} \, dx,x,\frac{\sqrt{6+\left (-1+i \sqrt{23}\right ) x}}{\sqrt{-1+i \sqrt{23}+4 x}}\right )}{\left (24-\left (-1+i \sqrt{23}\right )^2\right ) \sqrt{3-x+2 x^2} \sqrt{2+3 x+5 x^2}}\\ &=-\frac{\sqrt{\frac{23}{11}} \left (-1+i \sqrt{23}+4 x\right )^{3/2} \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}} F\left (2 \tan ^{-1}\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}}}\\ \end{align*}

Mathematica [A]  time = 0.61975, size = 390, normalized size = 0.6 \[ \frac{\left (-4 x+i \sqrt{23}+1\right ) \left (10 i x+\sqrt{31}+3 i\right ) \sqrt{\frac{20 i x-2 \sqrt{31}+6 i}{\left (11 i+5 \sqrt{23}-2 \sqrt{31}\right ) \left (4 i x+\sqrt{23}-i\right )}} \sqrt{\frac{\left (-22-10 i \sqrt{23}+4 i \sqrt{31}\right ) x-\sqrt{713}-i \sqrt{31}-3 i \sqrt{23}+63}{\left (11 i+5 \sqrt{23}+2 \sqrt{31}\right ) \left (4 i x+\sqrt{23}-i\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} \sqrt{-\frac{2 \left (11+5 i \sqrt{23}-2 i \sqrt{31}\right ) x+\sqrt{713}+i \sqrt{31}+3 i \sqrt{23}-63}{\left (11 i+5 \sqrt{23}+2 \sqrt{31}\right ) \left (4 i x+\sqrt{23}-i\right )}}\right ),\frac{1}{484} \left (1197+41 \sqrt{713}\right )\right )}{\left (-11 i+5 \sqrt{23}-2 \sqrt{31}\right ) \sqrt{\frac{10 i x+\sqrt{31}+3 i}{\left (11 i+5 \sqrt{23}+2 \sqrt{31}\right ) \left (4 i x+\sqrt{23}-i\right )}} \sqrt{2 x^2-x+3} \sqrt{5 x^2+3 x+2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((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
[23] + (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])

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Maple [A]  time = 3.535, size = 420, normalized size = 0.6 \begin{align*}{\frac{{\frac{4\,i}{23}} \left ( 2\,i\sqrt{31}+5\,i\sqrt{23}-11 \right ) \left ( i\sqrt{23}-4\,x+1 \right ) ^{2}\sqrt{23}\sqrt{10}}{2\,i\sqrt{31}-5\,i\sqrt{23}-11}\sqrt{5\,{x}^{2}+3\,x+2}\sqrt{2\,{x}^{2}-x+3}\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{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 ) }}}{\it EllipticF} \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 ){\frac{1}{\sqrt{10\,{x}^{4}+{x}^{3}+16\,{x}^{2}+7\,x+6}}}{\frac{1}{\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 ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/23*I*(5*x^2+3*x+2)^(1/2)*(2*x^2-x+3)^(1/2)*(2*I*31^(1/2)+5*I*23^(1/2)-11)*(-(2*I*31^(1/2)-5*I*23^(1/2)-11)*(
-1+4*x+I*23^(1/2))/(2*I*31^(1/2)+5*I*23^(1/2)-11)/(I*23^(1/2)-4*x+1))^(1/2)*(I*23^(1/2)-4*x+1)^2*(I*23^(1/2)*(
I*31^(1/2)+10*x+3)/(2*I*31^(1/2)-5*I*23^(1/2)+11)/(I*23^(1/2)-4*x+1))^(1/2)*(I*23^(1/2)*(I*31^(1/2)-10*x-3)/(2
*I*31^(1/2)+5*I*23^(1/2)-11)/(I*23^(1/2)-4*x+1))^(1/2)*23^(1/2)*10^(1/2)*EllipticF((-(2*I*31^(1/2)-5*I*23^(1/2
)-11)*(-1+4*x+I*23^(1/2))/(2*I*31^(1/2)+5*I*23^(1/2)-11)/(I*23^(1/2)-4*x+1))^(1/2),((2*I*31^(1/2)+5*I*23^(1/2)
+11)*(2*I*31^(1/2)+5*I*23^(1/2)-11)/(2*I*31^(1/2)-5*I*23^(1/2)+11)/(2*I*31^(1/2)-5*I*23^(1/2)-11))^(1/2))/(10*
x^4+x^3+16*x^2+7*x+6)^(1/2)/(2*I*31^(1/2)-5*I*23^(1/2)-11)/((-1+4*x+I*23^(1/2))*(I*23^(1/2)-4*x+1)*(I*31^(1/2)
+10*x+3)*(I*31^(1/2)-10*x-3))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{5 \, x^{2} + 3 \, x + 2} \sqrt{2 \, x^{2} - x + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{5 \, x^{2} + 3 \, x + 2} \sqrt{2 \, x^{2} - x + 3}}{10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(5*x^2 + 3*x + 2)*sqrt(2*x^2 - x + 3)/(10*x^4 + x^3 + 16*x^2 + 7*x + 6), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 x^{2} - x + 3} \sqrt{5 x^{2} + 3 x + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{5 \, x^{2} + 3 \, x + 2} \sqrt{2 \, x^{2} - x + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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