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

Optimal. Leaf size=452 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (-2 b e g+3 c d g+c e f) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 c^2 g \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 c^2 g \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{3 c} \]

[Out]

(2*e*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(3*c) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*
e*f + 3*c*d*g - 2*b*e*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)])/(3*c^2*g*S
qrt[(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]*e*(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))]*Elli
pticF[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^2*g*Sqrt[f +
g*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.10603, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (-2 b e g+3 c d g+c e f) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 c^2 g \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{3 c^2 g \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(3*c) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*
e*f + 3*c*d*g - 2*b*e*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)])/(3*c^2*g*S
qrt[(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]*e*(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))]*Elli
pticF[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^2*g*Sqrt[f +
g*x]*Sqrt[a + b*x + c*x^2])

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

[Out]

Timed out

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Mathematica [C]  time = 10.5844, size = 766, normalized size = 1.69 \[ \frac{2 e \sqrt{f+g x} \left (a+b x+c x^2\right )}{3 c \sqrt{a+x (b+c x)}}+\frac{2 (f+g x)^{3/2} \sqrt{a+b x+c x^2} \left (\left (\frac{g \left (\frac{a g}{f+g x}-\frac{b f}{f+g x}+b\right )}{f+g x}+c \left (\frac{f}{f+g x}-1\right )^2\right ) (-2 b e g+3 c d g+c e f)+\frac{i \sqrt{1-\frac{2 \left (g (a g-b f)+c f^2\right )}{(f+g x) \left (\sqrt{g^2 \left (b^2-4 a c\right )}-b g+2 c f\right )}} \sqrt{\frac{2 \left (g (a g-b f)+c f^2\right )}{(f+g x) \left (\sqrt{g^2 \left (b^2-4 a c\right )}+b g-2 c f\right )}+1} \left (\left (c \left (\sqrt{g^2 \left (b^2-4 a c\right )} (3 d g+e f)-2 a e g^2-3 b g (d g+e f)\right )+2 b e g \left (b g-\sqrt{g^2 \left (b^2-4 a c\right )}\right )+6 c^2 d f g\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right )|-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )+\left (\sqrt{g^2 \left (b^2-4 a c\right )}-b g+2 c f\right ) (2 b e g-c (3 d g+e f)) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c f^2-b g f+a g^2}{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}}}{\sqrt{f+g x}}\right )|-\frac{-2 c f+b g+\sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )\right )}{2 \sqrt{2} \sqrt{f+g x} \sqrt{\frac{g (a g-b f)+c f^2}{\sqrt{g^2 \left (b^2-4 a c\right )}+b g-2 c f}}}\right )}{3 c^2 g^2 \sqrt{a+x (b+c x)} \sqrt{\frac{(f+g x)^2 \left (\frac{g \left (\frac{a g}{f+g x}-\frac{b f}{f+g x}+b\right )}{f+g x}+c \left (\frac{f}{f+g x}-1\right )^2\right )}{g^2}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[f + g*x]*(a + b*x + c*x^2))/(3*c*Sqrt[a + x*(b + c*x)]) + (2*(f + g*x)
^(3/2)*Sqrt[a + b*x + c*x^2]*((c*e*f + 3*c*d*g - 2*b*e*g)*(c*(-1 + f/(f + g*x))^
2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*x)) + ((I/2)*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 + Sqrt[(b^2 - 4*a*c)*g^2])*(2*b*e*g - c*(e*f + 3*d*
g))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sq
rt[(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]))] + (6*c^2*d*f*g + 2*b*e*g*(b*g - Sq
rt[(b^2 - 4*a*c)*g^2]) + c*(-2*a*e*g^2 - 3*b*g*(e*f + d*g) + Sqrt[(b^2 - 4*a*c)*
g^2]*(e*f + 3*d*g)))*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]*Sqrt[(
c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[f + g*x
])))/(3*c^2*g^2*Sqrt[a + x*(b + c*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 [B]  time = 0.051, size = 3805, normalized size = 8.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)*(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)*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*e*g
^3-6*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*d*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)*a*e*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))*b^2*e*f*g^2+6*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*c*d*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)
*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*c*e*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)*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*e*f*g^2-6*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))*c^2*d*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)*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*e*f^
2*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)*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*e*g^3+6*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*d*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*e*f*g^2+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))*b^2*e*f*g^2-6*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*d*f*g^2-6*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*e*f^2*g+6*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
ipticE(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*d*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*e*f^3-2*x^3*c^2*e*g^3-2*x^2*b*c*e*g^3-2*x^2*c^2*
e*f*g^2-2*x*a*c*e*g^3-2*x*b*c*e*f*g^2-2*a*c*e*f*g^2)/c^2/(c*g*x^3+b*g*x^2+c*f*x^
2+a*g*x+b*f*x+a*f)/g^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out