3.912 \(\int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=189 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{c \sqrt{f+g x} \sqrt{a+b x+c x^2}} \]

[Out]

(2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*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 + Sqr
t[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)])/(c*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.27869, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{c \sqrt{f+g x} \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*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 + Sqr
t[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)])/(c*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 50.6126, size = 178, normalized size = 0.94 \[ \frac{2 \sqrt{2} \sqrt{\frac{c \left (- f - g x\right )}{b g - 2 c f + g \sqrt{- 4 a c + b^{2}}}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{- 4 a c + b^{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 g \sqrt{- 4 a c + b^{2}}}{b g - 2 c f + g \sqrt{- 4 a c + b^{2}}}\right )}{c \sqrt{f + g x} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.39202, size = 308, normalized size = 1.63 \[ \frac{i (f+g x) \sqrt{2-\frac{4 \left (g (a g-b f)+c f^2\right )}{(f+g x) \left (\sqrt{g^2 \left (b^2-4 a c\right )}-b g+2 c f\right )}} \sqrt{\frac{2 \left (g (a g-b f)+c f^2\right )}{(f+g x) \left (\sqrt{g^2 \left (b^2-4 a c\right )}+b g-2 c f\right )}+1} F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right )|-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )}{g \sqrt{a+x (b+c x)} \sqrt{\frac{g (a g-b f)+c f^2}{\sqrt{g^2 \left (b^2-4 a c\right )}+b g-2 c f}}} \]

Antiderivative was successfully verified.

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

[Out]

(I*(f + g*x)*Sqrt[2 - (4*(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))]*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]))
])/(g*Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*
Sqrt[a + x*(b + c*x)])

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Maple [A]  time = 0.048, size = 287, normalized size = 1.5 \[{\frac{\sqrt{2}}{cg \left ( cg{x}^{3}+bg{x}^{2}+cf{x}^{2}+agx+bfx+fa \right ) } \left ( -g\sqrt{-4\,ac+{b}^{2}}-bg+2\,cf \right ){\it EllipticF} \left ( \sqrt{2}\sqrt{-{ \left ( gx+f \right ) c \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}},\sqrt{-{1 \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \right ) \sqrt{{g \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}}\sqrt{{g \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}}\sqrt{-{ \left ( gx+f \right ) c \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}}\sqrt{gx+f}\sqrt{c{x}^{2}+bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-g*(-4*a*c+b^2)^(1/2)-b*g+2*c*f)/c*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))*(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)*(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)*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)/g*(g*x+f)^(1/
2)*(c*x^2+b*x+a)^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x + a} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f)),x, algorithm="maxima")

[Out]

integrate(1/(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. \[{\rm integral}\left (\frac{1}{\sqrt{c x^{2} + b x + a} \sqrt{g x + f}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f)),x, algorithm="fricas")

[Out]

integral(1/(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. \[ \int \frac{1}{\sqrt{f + g x} \sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(f + g*x)*sqrt(a + b*x + c*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x + a} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f)), x)