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

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

[Out]

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))-41*(6-x*(1-I*23^(1/2)))*(I+23^(1/2))/(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)))*(I+23^(1/2))/(1-4*x-I*23^(1/2))/(7*
I+23^(1/2)))^(1/2)

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Rubi [A]  time = 0.68, antiderivative size = 652, normalized size of antiderivative = 1.00, 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 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 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 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*}

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Mathematica [A]  time = 0.61, size = 390, normalized size = 0.60 \[ \frac {\left (-4 x+i \sqrt {23}+1\right ) \left (10 i x+\sqrt {31}+3 i\right ) \sqrt {\frac {20 i x-2 \sqrt {31}+6 i}{\left (11 i+5 \sqrt {23}-2 \sqrt {31}\right ) \left (4 i x+\sqrt {23}-i\right )}} \sqrt {\frac {\left (-22-10 i \sqrt {23}+4 i \sqrt {31}\right ) x-\sqrt {713}-i \sqrt {31}-3 i \sqrt {23}+63}{\left (11 i+5 \sqrt {23}+2 \sqrt {31}\right ) \left (4 i x+\sqrt {23}-i\right )}} F\left (\sin ^{-1}\left (\sqrt {2} \sqrt {-\frac {2 \left (11+5 i \sqrt {23}-2 i \sqrt {31}\right ) x+\sqrt {713}+i \sqrt {31}+3 i \sqrt {23}-63}{\left (11 i+5 \sqrt {23}+2 \sqrt {31}\right ) \left (4 i x+\sqrt {23}-i\right )}}\right )|\frac {1}{484} \left (1197+41 \sqrt {713}\right )\right )}{\left (-11 i+5 \sqrt {23}-2 \sqrt {31}\right ) \sqrt {\frac {10 i x+\sqrt {31}+3 i}{\left (11 i+5 \sqrt {23}+2 \sqrt {31}\right ) \left (4 i x+\sqrt {23}-i\right )}} \sqrt {2 x^2-x+3} \sqrt {5 x^2+3 x+2}} \]

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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fricas [F]  time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {2 \, x^{2} - x + 3}}{10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {2 \, x^{2} - x + 3}}\,{d x} \]

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)

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maple [A]  time = 0.47, size = 418, normalized size = 0.64 \[ -\frac {4 i \sqrt {5 x^{2}+3 x +2}\, \sqrt {2 x^{2}-x +3}\, \left (5 i \sqrt {23}+2 i \sqrt {31}-11\right ) \sqrt {\frac {\left (5 i \sqrt {23}-2 i \sqrt {31}+11\right ) \left (4 x -1+i \sqrt {23}\right )}{\left (5 i \sqrt {23}+2 i \sqrt {31}-11\right ) \left (-4 x +i \sqrt {23}+1\right )}}\, \left (-4 x +i \sqrt {23}+1\right )^{2} \sqrt {-\frac {i \sqrt {23}\, \left (10 x +i \sqrt {31}+3\right )}{\left (5 i \sqrt {23}-2 i \sqrt {31}-11\right ) \left (-4 x +i \sqrt {23}+1\right )}}\, \sqrt {\frac {i \sqrt {23}\, \left (-10 x +i \sqrt {31}-3\right )}{\left (5 i \sqrt {23}+2 i \sqrt {31}-11\right ) \left (-4 x +i \sqrt {23}+1\right )}}\, \sqrt {23}\, \sqrt {10}\, \EllipticF \left (\sqrt {\frac {\left (5 i \sqrt {23}-2 i \sqrt {31}+11\right ) \left (4 x -1+i \sqrt {23}\right )}{\left (5 i \sqrt {23}+2 i \sqrt {31}-11\right ) \left (-4 x +i \sqrt {23}+1\right )}}, \sqrt {\frac {\left (5 i \sqrt {23}+2 i \sqrt {31}+11\right ) \left (5 i \sqrt {23}+2 i \sqrt {31}-11\right )}{\left (5 i \sqrt {23}-2 i \sqrt {31}-11\right ) \left (5 i \sqrt {23}-2 i \sqrt {31}+11\right )}}\right )}{23 \sqrt {10 x^{4}+x^{3}+16 x^{2}+7 x +6}\, \left (5 i \sqrt {23}-2 i \sqrt {31}+11\right ) \sqrt {\left (4 x -1+i \sqrt {23}\right ) \left (-4 x +i \sqrt {23}+1\right ) \left (10 x +i \sqrt {31}+3\right ) \left (-10 x +i \sqrt {31}-3\right )}} \]

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)*(5*I*23^(1/2)+2*I*31^(1/2)-11)*((5*I*23^(1/2)-2*I*31^(1/2)+11)*(
-1+4*x+I*23^(1/2))/(5*I*23^(1/2)+2*I*31^(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)/(5*I*23^(1/2)-2*I*31^(1/2)-11)/(I*23^(1/2)-4*x+1))^(1/2)*(I*23^(1/2)*(I*31^(1/2)-10*x-3)/(
5*I*23^(1/2)+2*I*31^(1/2)-11)/(I*23^(1/2)-4*x+1))^(1/2)*23^(1/2)*10^(1/2)*EllipticF(((5*I*23^(1/2)-2*I*31^(1/2
)+11)*(-1+4*x+I*23^(1/2))/(5*I*23^(1/2)+2*I*31^(1/2)-11)/(I*23^(1/2)-4*x+1))^(1/2),((5*I*23^(1/2)+2*I*31^(1/2)
+11)*(5*I*23^(1/2)+2*I*31^(1/2)-11)/(5*I*23^(1/2)-2*I*31^(1/2)-11)/(5*I*23^(1/2)-2*I*31^(1/2)+11))^(1/2))/(10*
x^4+x^3+16*x^2+7*x+6)^(1/2)/(5*I*23^(1/2)-2*I*31^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {2 \, x^{2} - x + 3}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {2\,x^2-x+3}\,\sqrt {5\,x^2+3\,x+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 x^{2} - x + 3} \sqrt {5 x^{2} + 3 x + 2}}\, dx \]

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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