3.889 \(\int \sqrt{f+g x} \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=513 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c f-b g) \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c g (3 a g+b f)+b^2 g^2+c^2 f^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{5 g}-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} (2 c f-b g)}{15 c g} \]

[Out]

(-2*(2*c*f - b*g)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(15*c*g) + (2*(f + g*x)^(
3/2)*Sqrt[a + b*x + c*x^2])/(5*g) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c^2*f^2 + b^2*
g^2 - c*g*(b*f + 3*a*g))*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c
))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqr
t[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(15*c^2*g^
2*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2])
+ (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*f - b*g)*(c*f^2 - b*f*g + a*g^2)*Sqrt[(c*(f
+ g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2
- 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*
c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(15
*c^2*g^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 1.33651, antiderivative size = 513, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c f-b g) \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c g (3 a g+b f)+b^2 g^2+c^2 f^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{5 g}-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} (2 c f-b g)}{15 c g} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2],x]

[Out]

(-2*(2*c*f - b*g)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(15*c*g) + (2*(f + g*x)^(
3/2)*Sqrt[a + b*x + c*x^2])/(5*g) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c^2*f^2 + b^2*
g^2 - c*g*(b*f + 3*a*g))*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c
))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqr
t[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(15*c^2*g^
2*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2])
+ (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*f - b*g)*(c*f^2 - b*f*g + a*g^2)*Sqrt[(c*(f
+ g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2
- 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*
c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(15
*c^2*g^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**(1/2)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 13.3026, size = 3384, normalized size = 6.6 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2],x]

[Out]

((2*(c*f + b*g))/(15*c*g) + (2*x)/5)*Sqrt[f + g*x]*Sqrt[a + x*(b + c*x)] + (Sqrt
[a + x*(b + c*x)]*((-4*(c^2*f^2 - b*c*f*g + b^2*g^2 - 3*a*c*g^2)*(f + g*x)^(3/2)
*(c + (c*f^2)/(f + g*x)^2 - (b*f*g)/(f + g*x)^2 + (a*g^2)/(f + g*x)^2 - (2*c*f)/
(f + g*x) + (b*g)/(f + g*x)))/(c*Sqrt[((f + g*x)^2*(c*(-1 + f/(f + g*x))^2 + (g*
(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)))/g^2]) + (2*(c*f^2 - b*f*g +
a*g^2)*(f + g*x)*Sqrt[c + (c*f^2)/(f + g*x)^2 - (b*f*g)/(f + g*x)^2 + (a*g^2)/(
f + g*x)^2 - (2*c*f)/(f + g*x) + (b*g)/(f + g*x)]*((I*c^2*f^2*(2*c*f - b*g + Sqr
t[b^2*g^2 - 4*a*c*g^2])*Sqrt[1 - (2*(c*f^2 - b*f*g + a*g^2))/((2*c*f - b*g - Sqr
t[b^2*g^2 - 4*a*c*g^2])*(f + g*x))]*Sqrt[1 - (2*(c*f^2 - b*f*g + a*g^2))/((2*c*f
- b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*(f + g*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sq
rt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f
+ g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2
- 4*a*c*g^2])] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*
c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^
2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])]))/(Sqrt[2]*(c*f^2
- b*f*g + a*g^2)*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4
*a*c*g^2]))]*Sqrt[c + (c*f^2 - b*f*g + a*g^2)/(f + g*x)^2 + (-2*c*f + b*g)/(f +
g*x)]) - (I*b*c*f*g*(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*Sqrt[1 - (2*(c*f^2
- b*f*g + a*g^2))/((2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])*(f + g*x))]*Sqrt[1
- (2*(c*f^2 - b*f*g + a*g^2))/((2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*(f + g
*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g
- Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*
a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])] - EllipticF[I*ArcSinh[(Sqrt
[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))])/
Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^
2*g^2 - 4*a*c*g^2])]))/(Sqrt[2]*(c*f^2 - b*f*g + a*g^2)*Sqrt[-((c*f^2 - b*f*g +
a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))]*Sqrt[c + (c*f^2 - b*f*g + a*g
^2)/(f + g*x)^2 + (-2*c*f + b*g)/(f + g*x)]) + (I*b^2*g^2*(2*c*f - b*g + Sqrt[b^
2*g^2 - 4*a*c*g^2])*Sqrt[1 - (2*(c*f^2 - b*f*g + a*g^2))/((2*c*f - b*g - Sqrt[b^
2*g^2 - 4*a*c*g^2])*(f + g*x))]*Sqrt[1 - (2*(c*f^2 - b*f*g + a*g^2))/((2*c*f - b
*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*(f + g*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-
((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f + g
*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*
a*c*g^2])] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f
- b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^
2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])]))/(Sqrt[2]*(c*f^2 - b
*f*g + a*g^2)*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c
*g^2]))]*Sqrt[c + (c*f^2 - b*f*g + a*g^2)/(f + g*x)^2 + (-2*c*f + b*g)/(f + g*x)
]) - ((3*I)*a*c*g^2*(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*Sqrt[1 - (2*(c*f^2
- b*f*g + a*g^2))/((2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])*(f + g*x))]*Sqrt[1
- (2*(c*f^2 - b*f*g + a*g^2))/((2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*(f + g
*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g
- Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*
a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])] - EllipticF[I*ArcSinh[(Sqrt
[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))])/
Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^
2*g^2 - 4*a*c*g^2])]))/(Sqrt[2]*(c*f^2 - b*f*g + a*g^2)*Sqrt[-((c*f^2 - b*f*g +
a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))]*Sqrt[c + (c*f^2 - b*f*g + a*g
^2)/(f + g*x)^2 + (-2*c*f + b*g)/(f + g*x)]) + (I*Sqrt[2]*c^2*f*Sqrt[1 - (2*(c*f
^2 - b*f*g + a*g^2))/((2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])*(f + g*x))]*Sqrt
[1 - (2*(c*f^2 - b*f*g + a*g^2))/((2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*(f +
g*x))]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g
- Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4
*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])])/(Sqrt[-((c*f^2 - b*f*g +
a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))]*Sqrt[c + (c*f^2 - b*f*g + a*g
^2)/(f + g*x)^2 + (-2*c*f + b*g)/(f + g*x)]) - (I*b*c*g*Sqrt[1 - (2*(c*f^2 - b*f
*g + a*g^2))/((2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])*(f + g*x))]*Sqrt[1 - (2*
(c*f^2 - b*f*g + a*g^2))/((2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*(f + g*x))]*
EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[
b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2
])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])])/(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g +
a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))]*Sqrt[c + (c*f^2 - b*f*g + a*g
^2)/(f + g*x)^2 + (-2*c*f + b*g)/(f + g*x)])))/(c*Sqrt[((f + g*x)^2*(c*(-1 + f/(
f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)))/g^2])))/(1
5*c*g^3*Sqrt[a + b*x + c*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.035, size = 4356, normalized size = 8.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2),x)

[Out]

1/15*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2*(2*x*b^2*c*f*g^3+12*2^(1/2)*(-(g*x+f)
*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c
*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b
^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+
b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(
1/2)))^(1/2))*a*c^2*f^2*g^2-3*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*
f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/
2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*Ellip
ticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^
2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*c*f^2*g^2-8*2^(
1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)
^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2)
)/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a
*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*
(-4*a*c+b^2)^(1/2)))^(1/2))*a*c^2*f^2*g^2+8*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^
(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^
2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f
))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(
-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*c
*f^2*g^2-8*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*
x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*
a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*x
+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)
/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*b*c^2*f^3*g+2^(1/2)*(-(g*x+f)*c/(g*(-4
*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*
(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)
+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f
))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1
/2))*(-4*a*c+b^2)^(1/2)*a*b*g^4-2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*
c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(
1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*Ell
ipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+
b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2
)*b^2*f*g^3+6*x^2*a*c^2*g^4+2*x^2*b^2*c*g^4+2*x^2*c^3*f^2*g^2+8*x^3*c^3*f*g^3+8*
x^3*b*c^2*g^4+2*a*c^2*f^2*g^2+12*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2
*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^
(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*El
lipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c
+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a^2*c*g^4+4*2^(1
/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^
(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))
/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*
c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(
-4*a*c+b^2)^(1/2)))^(1/2))*c^3*f^4+2*x*b*c^2*f^2*g^2+10*x^2*b*c^2*f*g^3+2*x*a*b*
c*g^4+8*x*a*c^2*f*g^3+6*x^4*c^3*g^4-12*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)
+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1
/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1
/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(
-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*c*f*g^
3+8*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*a
*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2
)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(
g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f
-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*c*f*g^3+3*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+
b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a
*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-
2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1
/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*
(-4*a*c+b^2)^(1/2)*b*c*f^2*g^2-2*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2
*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^
(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*El
lipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c
+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/
2)*a*c*f*g^3-3*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-
2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+
(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-
(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*
c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b^2*g^4+3*2^(1/2)*(-(g*x+f)*c/(g
*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*
g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(
1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2
*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2))
)^(1/2))*b^3*f*g^3-12*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2
)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b
+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(
1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)
+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a^2*c*g^4+4*2^(1/2)*(-(g*x+
f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2
*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c
+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2
)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)
^(1/2)))^(1/2))*a*b^2*g^4-4*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)
)^(1/2)*(g*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)
*(g*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*Ellipti
cE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)
^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*b^3*f*g^3+2*a*b*c*f*g
^3-2*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(g*(-2*c*x+(-4*
a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*a*c+b^
2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/
(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*
f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*c^2*f^3*g)/(c*g*x^3+b*g*x
^2+c*f*x^2+a*g*x+b*f*x+a*f)/g^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{2} + b x + a} \sqrt{g x + f}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{2} + b x + a} \sqrt{g x + f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{f + g x} \sqrt{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**(1/2)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(sqrt(f + g*x)*sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f),x, algorithm="giac")

[Out]

Timed out