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

Optimal. Leaf size=475 \[ \frac{\sqrt{2} (d+e x) \sqrt{-\sqrt{b^2-4 a c}+b+2 c x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )} \sqrt{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) (e f-d g)}{(d+e x) \left (2 c f-g \left (\sqrt{b^2-4 a c}+b\right )\right )}} \sqrt{\frac{\left (x \left (\sqrt{b^2-4 a c}+b\right )+2 a\right ) (e f-d g)}{(d+e x) \left (f \sqrt{b^2-4 a c}-2 a g+b f\right )}} \Pi \left (\frac{e \left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) g};\sin ^{-1}\left (\frac{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \sqrt{f+g x}}{\sqrt{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g} \sqrt{d+e x}}\right )|\frac{\left (b d+\sqrt{b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b f+\sqrt{b^2-4 a c} f-2 a g\right )}\right )}{g \sqrt{\frac{2 a c}{\sqrt{b^2-4 a c}+b}+c x} \sqrt{a+b x+c x^2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \]

[Out]

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

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Rubi [A]  time = 0.88333, antiderivative size = 475, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03 \[ \frac{\sqrt{2} (d+e x) \sqrt{-\sqrt{b^2-4 a c}+b+2 c x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )} \sqrt{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) (e f-d g)}{(d+e x) \left (2 c f-g \left (\sqrt{b^2-4 a c}+b\right )\right )}} \sqrt{\frac{\left (x \left (\sqrt{b^2-4 a c}+b\right )+2 a\right ) (e f-d g)}{(d+e x) \left (f \sqrt{b^2-4 a c}-2 a g+b f\right )}} \Pi \left (\frac{e \left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) g};\sin ^{-1}\left (\frac{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \sqrt{f+g x}}{\sqrt{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g} \sqrt{d+e x}}\right )|\frac{\left (b d+\sqrt{b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b f+\sqrt{b^2-4 a c} f-2 a g\right )}\right )}{g \sqrt{\frac{2 a c}{\sqrt{b^2-4 a c}+b}+c x} \sqrt{a+b x+c x^2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \]

Antiderivative was successfully verified.

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 13.5702, size = 2493, normalized size = 5.25 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]*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))*(e - (e
*f)/(f + g*x) + (d*g)/(f + g*x))]*Sqrt[d + ((f + g*x)*(e - (e*f)/(f + g*x)))/g]*
(-((e*f*Sqrt[(-(e/(e*f - d*g)) + (f + g*x)^(-1))/(-(e/(e*f - d*g)) + (2*c*f - b*
g + Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*(c*f^2 - b*f*g + a*g^2)))]*Sqrt[(-(2*c*f - b*g
- Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*(c*f^2 - b*f*g + a*g^2)) + (f + g*x)^(-1))/(-(2
*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(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])/(2*(c*f^2 - b*f*g + a*g^2)))]*(-(2*c*f - b*g + S
qrt[b^2*g^2 - 4*a*c*g^2])/(2*(c*f^2 - b*f*g + a*g^2)) + (f + g*x)^(-1))*Elliptic
F[ArcSin[Sqrt[(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2] - (2*c*f^2)/(f + g*x) + (2*
b*f*g)/(f + g*x) - (2*a*g^2)/(f + g*x))/Sqrt[(b^2 - 4*a*c)*g^2]]/Sqrt[2]], (2*Sq
rt[(b^2 - 4*a*c)*g^2]*(e*f - d*g))/(-2*c*d*f*g + b*e*f*g + b*d*g^2 - 2*a*e*g^2 +
e*f*Sqrt[(b^2 - 4*a*c)*g^2] - d*g*Sqrt[(b^2 - 4*a*c)*g^2])])/(Sqrt[(-(2*c*f - b
*g + Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*(c*f^2 - b*f*g + a*g^2)) + (f + g*x)^(-1))/((
2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(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])/(2*(c*f^2 - b*f*g + a*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))*(e + (-(e*f) + d*g)/(f
+ g*x))])) + (d*g*Sqrt[(-(e/(e*f - d*g)) + (f + g*x)^(-1))/(-(e/(e*f - d*g)) + (
2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*(c*f^2 - b*f*g + a*g^2)))]*Sqrt[(-(2
*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*(c*f^2 - b*f*g + a*g^2)) + (f + g*x)^
(-1))/(-(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(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])/(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])/(2*(c*f^2 - b*f*g + a*g^2)) + (f + g*x)^(-1)
)*EllipticF[ArcSin[Sqrt[(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2] - (2*c*f^2)/(f +
g*x) + (2*b*f*g)/(f + g*x) - (2*a*g^2)/(f + g*x))/Sqrt[(b^2 - 4*a*c)*g^2]]/Sqrt[
2]], (2*Sqrt[(b^2 - 4*a*c)*g^2]*(e*f - d*g))/(-2*c*d*f*g + b*e*f*g + b*d*g^2 - 2
*a*e*g^2 + e*f*Sqrt[(b^2 - 4*a*c)*g^2] - d*g*Sqrt[(b^2 - 4*a*c)*g^2])])/(Sqrt[(-
(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*(c*f^2 - b*f*g + a*g^2)) + (f + g*x
)^(-1))/((2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(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])/(2*(c*f^2 - b*f*g + a*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))*(e + (-(e*f)
+ d*g)/(f + g*x))]) - (2*e*(c*f^2 - b*f*g + a*g^2)*(-(2*c*f - b*g - Sqrt[b^2*g^2
- 4*a*c*g^2])/(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])/(2*(c*f^2 - b*f*g + a*g^2)))*Sqrt[(-(e/(e*f - d*g)) + (f + g*x)^(-1))/(-(
e/(e*f - d*g)) + (2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*(c*f^2 - b*f*g + a
*g^2)))]*Sqrt[-(((-(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*(c*f^2 - b*f*g +
a*g^2)) + (f + g*x)^(-1))*(-(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*(c*f^2
- b*f*g + a*g^2)) + (f + g*x)^(-1)))/(-(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]
)/(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])/(2*(c*
f^2 - b*f*g + a*g^2)))^2)]*EllipticPi[(2*Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g +
Sqrt[(b^2 - 4*a*c)*g^2]), ArcSin[Sqrt[(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2] -
(2*c*f^2)/(f + g*x) + (2*b*f*g)/(f + g*x) - (2*a*g^2)/(f + g*x))/Sqrt[(b^2 - 4*a
*c)*g^2]]/Sqrt[2]], (2*Sqrt[(b^2 - 4*a*c)*g^2]*(e*f - d*g))/(-2*c*d*f*g + b*e*f*
g + b*d*g^2 - 2*a*e*g^2 + e*f*Sqrt[(b^2 - 4*a*c)*g^2] - d*g*Sqrt[(b^2 - 4*a*c)*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))*(e + (-(e*f) + d*g)/(f + g*x))])))/(
g*Sqrt[a + x*(b + c*x)]*(e - (e*f)/(f + g*x) + (d*g)/(f + g*x))*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])

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Maple [A]  time = 0.192, size = 645, normalized size = 1.4 \[ 4\,{\frac{\sqrt{ex+d}\sqrt{gx+f}\sqrt{c{x}^{2}+bx+a} \left ( \sqrt{-4\,ac+{b}^{2}}{x}^{2}{e}^{2}g+b{e}^{2}g{x}^{2}-2\,c{e}^{2}f{x}^{2}+2\,\sqrt{-4\,ac+{b}^{2}}xdeg+2\,xbdeg-4\,xcdef+\sqrt{-4\,ac+{b}^{2}}{d}^{2}g+b{d}^{2}g-2\,c{d}^{2}f \right ) }{g \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \sqrt{ceg{x}^{4}+beg{x}^{3}+cdg{x}^{3}+cef{x}^{3}+aeg{x}^{2}+bdg{x}^{2}+bef{x}^{2}+cdf{x}^{2}+adgx+aefx+bdfx+adf}}\sqrt{{\frac{ \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( gx+f \right ) }{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( ex+d \right ) }}}\sqrt{{\frac{ \left ( dg-ef \right ) \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) }{ \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) \left ( ex+d \right ) }}}\sqrt{{\frac{ \left ( dg-ef \right ) \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) }{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( ex+d \right ) }}}{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( gx+f \right ) }{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( ex+d \right ) }}},{\frac{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) e}{g \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) }},\sqrt{{\frac{ \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) }{ \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) }}} \right ){\frac{1}{\sqrt{-{\frac{ \left ( ex+d \right ) \left ( gx+f \right ) \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) }{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/g*((e*(-4*a*c+b^2)^(1/2)+b*e-2
*c*d)*(g*x+f)/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(e*x+d))^(1/2)*((d*g-e*f)*(-2*c*x
+(-4*a*c+b^2)^(1/2)-b)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2))/(e*x+d))^(1/2)*((d*g-e*f
)*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(e*x+d))^(1/2)*E
llipticPi(((e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)*(g*x+f)/(g*(-4*a*c+b^2)^(1/2)+b*g-2*
c*f)/(e*x+d))^(1/2),(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)*e/g/(e*(-4*a*c+b^2)^(1/2)+b
*e-2*c*d),((e*(-4*a*c+b^2)^(1/2)-b*e+2*c*d)*(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))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2))*((-4*a*c+
b^2)^(1/2)*x^2*e^2*g+b*e^2*g*x^2-2*c*e^2*f*x^2+2*(-4*a*c+b^2)^(1/2)*x*d*e*g+2*x*
b*d*e*g-4*x*c*d*e*f+(-4*a*c+b^2)^(1/2)*d^2*g+b*d^2*g-2*c*d^2*f)/(-1/c*(g*x+f)*(e
*x+d)*(-2*c*x+(-4*a*c+b^2)^(1/2)-b)*(b+2*c*x+(-4*a*c+b^2)^(1/2)))^(1/2)/(e*(-4*a
*c+b^2)^(1/2)+b*e-2*c*d)/(c*e*g*x^4+b*e*g*x^3+c*d*g*x^3+c*e*f*x^3+a*e*g*x^2+b*d*
g*x^2+b*e*f*x^2+c*d*f*x^2+a*d*g*x+a*e*f*x+b*d*f*x+a*d*f)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(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. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x + a} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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