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

Optimal. Leaf size=280 \[ -\frac{\sqrt{2} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \Pi \left (\frac{e \left (2 c f-b g+\sqrt{b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right )}{\sqrt{c} \sqrt{a+b x+c x^2} (e f-d g)} \]

[Out]

-((Sqrt[2]*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[1 - (2*c*(f + g*x))/(2*c
*f - (b - Sqrt[b^2 - 4*a*c])*g)]*Sqrt[1 - (2*c*(f + g*x))/(2*c*f - (b + Sqrt[b^2
- 4*a*c])*g)]*EllipticPi[(e*(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g))/(2*c*(e*f - d*
g)), ArcSin[(Sqrt[2]*Sqrt[c]*Sqrt[f + g*x])/Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])
*g]], (b - Sqrt[b^2 - 4*a*c] - (2*c*f)/g)/(b + Sqrt[b^2 - 4*a*c] - (2*c*f)/g)])/
(Sqrt[c]*(e*f - d*g)*Sqrt[a + b*x + c*x^2]))

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Rubi [A]  time = 4.03091, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{\sqrt{2} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \Pi \left (\frac{e \left (2 c f-b g+\sqrt{b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-\left (b-\sqrt{b^2-4 a c}\right ) g}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 c f}{g}}{b+\sqrt{b^2-4 a c}-\frac{2 c f}{g}}\right )}{\sqrt{c} \sqrt{a+b x+c x^2} (e f-d g)} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2]*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[1 - (2*c*(f + g*x))/(2*c
*f - (b - Sqrt[b^2 - 4*a*c])*g)]*Sqrt[1 - (2*c*(f + g*x))/(2*c*f - (b + Sqrt[b^2
- 4*a*c])*g)]*EllipticPi[(e*(2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g))/(2*c*(e*f - d*
g)), ArcSin[(Sqrt[2]*Sqrt[c]*Sqrt[f + g*x])/Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])
*g]], (b - Sqrt[b^2 - 4*a*c] - (2*c*f)/g)/(b + Sqrt[b^2 - 4*a*c] - (2*c*f)/g)])/
(Sqrt[c]*(e*f - d*g)*Sqrt[a + b*x + c*x^2]))

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Rubi in Sympy [A]  time = 32.8874, size = 262, normalized size = 0.94 \[ \frac{\sqrt{2} \sqrt{\frac{g \left (- b - 2 c x + \sqrt{- 4 a c + b^{2}}\right )}{- b g + 2 c f + g \sqrt{- 4 a c + b^{2}}}} \sqrt{\frac{g \left (b + 2 c x + \sqrt{- 4 a c + b^{2}}\right )}{b g - 2 c f + g \sqrt{- 4 a c + b^{2}}}} \Pi \left (\frac{e \left (b g - 2 c f - g \sqrt{- 4 a c + b^{2}}\right )}{2 c \left (d g - e f\right )}; \operatorname{asin}{\left (\sqrt{2} \sqrt{\frac{c}{- b g + 2 c f + g \sqrt{- 4 a c + b^{2}}}} \sqrt{f + g x} \right )}\middle | \frac{b g - 2 c f - g \sqrt{- 4 a c + b^{2}}}{b g - 2 c f + g \sqrt{- 4 a c + b^{2}}}\right )}{\sqrt{\frac{c}{- b g + 2 c f + g \sqrt{- 4 a c + b^{2}}}} \left (d g - e f\right ) \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*sqrt(g*(-b - 2*c*x + sqrt(-4*a*c + b**2))/(-b*g + 2*c*f + g*sqrt(-4*a*c
+ b**2)))*sqrt(g*(b + 2*c*x + sqrt(-4*a*c + b**2))/(b*g - 2*c*f + g*sqrt(-4*a*c
+ b**2)))*elliptic_pi(e*(b*g - 2*c*f - g*sqrt(-4*a*c + b**2))/(2*c*(d*g - e*f)),
asin(sqrt(2)*sqrt(c/(-b*g + 2*c*f + g*sqrt(-4*a*c + b**2)))*sqrt(f + g*x)), (b*
g - 2*c*f - g*sqrt(-4*a*c + b**2))/(b*g - 2*c*f + g*sqrt(-4*a*c + b**2)))/(sqrt(
c/(-b*g + 2*c*f + g*sqrt(-4*a*c + b**2)))*(d*g - e*f)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 2.28219, size = 499, normalized size = 1.78 \[ \frac{i (f+g x) \sqrt{2-\frac{4 \left (g (a g-b f)+c f^2\right )}{(f+g x) \left (\sqrt{g^2 \left (b^2-4 a c\right )}-b g+2 c f\right )}} \sqrt{\frac{2 \left (g (a g-b f)+c f^2\right )}{(f+g x) \left (\sqrt{g^2 \left (b^2-4 a c\right )}+b g-2 c f\right )}+1} \left (F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right )|-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )-\Pi \left (\frac{(e f-d g) \left (2 c f-b g-\sqrt{\left (b^2-4 a c\right ) g^2}\right )}{2 e \left (c f^2+g (a g-b f)\right )};i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right )|-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )\right )}{\sqrt{a+x (b+c x)} (d g-e f) \sqrt{\frac{g (a g-b f)+c f^2}{\sqrt{g^2 \left (b^2-4 a c\right )}+b g-2 c f}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*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])
)] - EllipticPi[((e*f - d*g)*(2*c*f - b*g - Sqrt[(b^2 - 4*a*c)*g^2]))/(2*e*(c*f^
2 + g*(-(b*f) + a*g))), 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]))]))/((-(e*f) + d*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 = 0.057, size = 330, normalized size = 1.2 \[{\frac{\sqrt{2}}{ \left ( dg-ef \right ) c \left ( cg{x}^{3}+bg{x}^{2}+cf{x}^{2}+agx+bfx+fa \right ) } \left ( -g\sqrt{-4\,ac+{b}^{2}}-bg+2\,cf \right ){\it EllipticPi} \left ( \sqrt{2}\sqrt{-{ \left ( gx+f \right ) c \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}},{\frac{e}{2\, \left ( dg-ef \right ) c} \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) },\sqrt{-{1 \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \right ) \sqrt{{g \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}}\sqrt{{g \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}}\sqrt{-{ \left ( gx+f \right ) c \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}}\sqrt{c{x}^{2}+bx+a}\sqrt{gx+f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(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)*EllipticPi(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)
^(1/2)+b*g-2*c*f))^(1/2),1/2*(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)*e/c/(d*g-e*f),(-(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)/c*(c*x^2+b*x+a)^(1/2)*(g*x+f)^(1/2)/(d*g-e*
f)/(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. \[ \int \frac{1}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right ) \sqrt{f + g x} \sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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