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

Optimal. Leaf size=444 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 c g^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c f-b g) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 c g^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{3 g} \]

[Out]

(2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(3*g) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*
f - b*g)*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[Ar
cSin[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)])/(3*c*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]) + (4*Sqrt[2]*Sqrt[
b^2 - 4*a*c]*(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)])/(3*c*g^2*Sqrt[f + g*x]*Sqrt[a + b*x
+ c*x^2])

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Rubi [A]  time = 0.962545, antiderivative size = 444, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 c g^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c f-b g) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 c g^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{3 g} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(3*g) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*
f - b*g)*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[Ar
cSin[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)])/(3*c*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]) + (4*Sqrt[2]*Sqrt[
b^2 - 4*a*c]*(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)])/(3*c*g^2*Sqrt[f + g*x]*Sqrt[a + b*x
+ c*x^2])

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Rubi in Sympy [A]  time = 157.082, size = 418, normalized size = 0.94 \[ \frac{2 \sqrt{f + g x} \sqrt{a + b x + c x^{2}}}{3 g} + \frac{4 \sqrt{2} \sqrt{\frac{c \left (- f - g x\right )}{b g - 2 c f + g \sqrt{- 4 a c + b^{2}}}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{- 4 a c + b^{2}} \left (a g^{2} - b f g + c f^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 g \sqrt{- 4 a c + b^{2}}}{b g - 2 c f + g \sqrt{- 4 a c + b^{2}}}\right )}{3 c g^{2} \sqrt{f + g x} \sqrt{a + b x + c x^{2}}} + \frac{\sqrt{2} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{f + g x} \sqrt{- 4 a c + b^{2}} \left (b g - 2 c f\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 g \sqrt{- 4 a c + b^{2}}}{b g - 2 c f + g \sqrt{- 4 a c + b^{2}}}\right )}{3 c g^{2} \sqrt{\frac{c \left (- f - g x\right )}{b g - 2 c f + g \sqrt{- 4 a c + b^{2}}}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(f + g*x)*sqrt(a + b*x + c*x**2)/(3*g) + 4*sqrt(2)*sqrt(c*(-f - g*x)/(b*g
- 2*c*f + g*sqrt(-4*a*c + b**2)))*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*sqrt
(-4*a*c + b**2)*(a*g**2 - b*f*g + c*f**2)*elliptic_f(asin(sqrt(2)*sqrt((b + 2*c*
x + sqrt(-4*a*c + b**2))/sqrt(-4*a*c + b**2))/2), 2*g*sqrt(-4*a*c + b**2)/(b*g -
2*c*f + g*sqrt(-4*a*c + b**2)))/(3*c*g**2*sqrt(f + g*x)*sqrt(a + b*x + c*x**2))
+ sqrt(2)*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*sqrt(f + g*x)*sqrt(-4*a*c +
b**2)*(b*g - 2*c*f)*elliptic_e(asin(sqrt(2)*sqrt((b + 2*c*x + sqrt(-4*a*c + b**
2))/sqrt(-4*a*c + b**2))/2), 2*g*sqrt(-4*a*c + b**2)/(b*g - 2*c*f + g*sqrt(-4*a*
c + b**2)))/(3*c*g**2*sqrt(c*(-f - g*x)/(b*g - 2*c*f + g*sqrt(-4*a*c + b**2)))*s
qrt(a + b*x + c*x**2))

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Mathematica [C]  time = 11.8593, size = 591, normalized size = 1.33 \[ \frac{2 \left (\frac{i (f+g x) \sqrt{1-\frac{2 \left (g (a g-b f)+c f^2\right )}{(f+g x) \left (\sqrt{g^2 \left (b^2-4 a c\right )}-b g+2 c f\right )}} \sqrt{\frac{2 \left (g (a g-b f)+c f^2\right )}{(f+g x) \left (\sqrt{g^2 \left (b^2-4 a c\right )}+b g-2 c f\right )}+1} \left (\left (-2 c f \sqrt{g^2 \left (b^2-4 a c\right )}+b g \sqrt{g^2 \left (b^2-4 a c\right )}+4 a c g^2-b^2 g^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right )|-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )-(b g-2 c f) \left (\sqrt{g^2 \left (b^2-4 a c\right )}-b g+2 c f\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right )|-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )\right )}{2 \sqrt{2} c \sqrt{\frac{g (a g-b f)+c f^2}{\sqrt{g^2 \left (b^2-4 a c\right )}+b g-2 c f}}}+g^2 \sqrt{f+g x} (a+x (b+c x))+\frac{g^2 (a+x (b+c x)) (b g-2 c f)}{c \sqrt{f+g x}}\right )}{3 g^3 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*((g^2*(-2*c*f + b*g)*(a + x*(b + c*x)))/(c*Sqrt[f + g*x]) + g^2*Sqrt[f + g*x]
*(a + x*(b + c*x)) + ((I/2)*(f + g*x)*Sqrt[1 - (2*(c*f^2 + g*(-(b*f) + a*g)))/((
2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[1 + (2*(c*f^2 + g*(-(b*f
) + a*g)))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(f + g*x))]*(-((-2*c*f + b*
g)*(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*
f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]],
-((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2
]))]) + (-(b^2*g^2) + 4*a*c*g^2 - 2*c*f*Sqrt[(b^2 - 4*a*c)*g^2] + b*g*Sqrt[(b^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 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2
- 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))]))/(Sqrt[2]*c*Sqrt[(c*f^
2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])))/(3*g^3*Sqrt[a
+ x*(b + c*x)])

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Maple [B]  time = 0.032, size = 1854, normalized size = 4.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c*x^2+b*x+a)^(1/2)*(g*x+f)^(1/2)/c*(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*g^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*f*g^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*f^2*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)*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*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)*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*f*g^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)*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*f*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)*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*f^2*g-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)*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^2*f^3-x^3*c^2*g^3-x^2*b*c*g^3-x^2*c^2*f*g^2-x*a*c*g^3-x*b*c*f*g^2
-a*c*f*g^2)/(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. \[ \int \frac{\sqrt{c x^{2} + b x + a}}{\sqrt{g x + f}}\,{d x} \]

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. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}}{\sqrt{g x + f}}, x\right ) \]

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. \[ \int \frac{\sqrt{a + b x + c x^{2}}}{\sqrt{f + g x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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