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

Optimal. Leaf size=519 \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} (b e g-10 c d g+8 c e f) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c g (-6 a e g-5 b d g+3 b e f)+2 b^2 e g^2-2 c^2 f (4 e f-5 d g)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} (-b e g-5 c d g+4 c e f-3 c e g x)}{15 c g^2} \]

[Out]

(-2*Sqrt[f + g*x]*(4*c*e*f - 5*c*d*g - b*e*g - 3*c*e*g*x)*Sqrt[a + b*x + c*x^2])
/(15*c*g^2) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*b^2*e*g^2 - 2*c^2*f*(4*e*f - 5*d*g)
+ c*g*(3*b*e*f - 5*b*d*g - 6*a*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]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]
)/(15*c^2*g^3*Sqrt[(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]*(8*c*e*f - 10*c*d*g + b*e*g)*(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)])/(15*c^2*g^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.34783, antiderivative size = 519, 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 \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} (b e g-10 c d g+8 c e f) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c g (-6 a e g-5 b d g+3 b e f)+2 b^2 e g^2-2 c^2 f (4 e f-5 d g)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{15 c^2 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} (-b e g-5 c d g+4 c e f-3 c e g x)}{15 c g^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[f + g*x]*(4*c*e*f - 5*c*d*g - b*e*g - 3*c*e*g*x)*Sqrt[a + b*x + c*x^2])
/(15*c*g^2) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*b^2*e*g^2 - 2*c^2*f*(4*e*f - 5*d*g)
+ c*g*(3*b*e*f - 5*b*d*g - 6*a*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]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]
)/(15*c^2*g^3*Sqrt[(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]*(8*c*e*f - 10*c*d*g + b*e*g)*(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)])/(15*c^2*g^3*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)*(c*x**2+b*x+a)**(1/2)/(g*x+f)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 13.9726, size = 4921, normalized size = 9.48 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

((2*(-4*c*e*f + 5*c*d*g + b*e*g))/(15*c*g^2) + (2*e*x)/(5*g))*Sqrt[f + g*x]*Sqrt
[a + x*(b + c*x)] + (2*Sqrt[a + x*(b + c*x)]*(((8*c^2*e*f^2 - 10*c^2*d*f*g - 3*b
*c*e*f*g + 5*b*c*d*g^2 - 2*b^2*e*g^2 + 6*a*c*e*g^2)*(f + g*x)^(3/2)*(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)))/(c*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]) - ((c*f^2 - b*f*g + a*g^2)*(f + g*
x)*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)]*(((2*I)*Sqrt[2]*c^2*e*f^2*(2*c*f - b*g + Sqrt
[b^2*g^2 - 4*a*c*g^2])*Sqrt[1 - (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])*(f + g*x))]*Sqrt[1 - (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])*(f + g*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqr
t[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f
+ g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^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*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2
*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])]))/((c*f^2 - b*f*g
+ a*g^2)*Sqrt[-((c*f^2 - b*f*g + a*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)]) -
((5*I)*c^2*d*f*g*(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*Sqrt[1 - (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])*(f + g*x))]*Sqrt[1 -
(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])*(f + g*x)
)]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((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]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c
*g^2])/(2*c*f - b*g + Sqrt[b^2*g^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*g^2 - 4*a*c*g^2]))])/Sqr
t[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g
^2 - 4*a*c*g^2])]))/(Sqrt[2]*(c*f^2 - b*f*g + a*g^2)*Sqrt[-((c*f^2 - b*f*g + a*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)]) - (((3*I)/2)*b*c*e*f*g*(2*c*f - b*g +
Sqrt[b^2*g^2 - 4*a*c*g^2])*Sqrt[1 - (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])*(f + g*x))]*Sqrt[1 - (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])*(f + g*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]
*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqr
t[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g
^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*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt
[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])]))/(Sqrt[2]*(c*
f^2 - b*f*g + a*g^2)*Sqrt[-((c*f^2 - b*f*g + a*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)]) + (((5*I)/2)*b*c*d*g^2*(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*Sqrt[
1 - (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])*(f +
g*x))]*Sqrt[1 - (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])*(f + g*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/
(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt
[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^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*g^2 - 4*
a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f -
b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])]))/(Sqrt[2]*(c*f^2 - b*f*g + a*g^2)*Sqrt[-((c*f
^2 - b*f*g + a*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)]) - (I*b^2*e*g^2*(2*c*f
- b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*Sqrt[1 - (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])*(f + g*x))]*Sqrt[1 - (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])*(f + g*x))]*(EllipticE[I*ArcSinh[
(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]
))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sq
rt[b^2*g^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*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*
g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])]))/(Sqr
t[2]*(c*f^2 - b*f*g + a*g^2)*Sqrt[-((c*f^2 - b*f*g + a*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)]) + ((3*I)*a*c*e*g^2*(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])*S
qrt[1 - (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])*(
f + g*x))]*Sqrt[1 - (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])*(f + g*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g + a*g
^2)/(2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g -
Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])] - Elliptic
F[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*f^2 - b*f*g + a*g^2)/(2*c*f - b*g - Sqrt[b^2*g^2
- 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*
f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])]))/(Sqrt[2]*(c*f^2 - b*f*g + a*g^2)*Sqrt[-(
(c*f^2 - b*f*g + a*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)]) + ((4*I)*Sqrt[2]*c
^2*e*f*Sqrt[1 - (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])*(f + g*x))]*Sqrt[1 - (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])*(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*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f -
b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])])/(S
qrt[-((c*f^2 - b*f*g + a*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)]) - ((5*I)*Sqr
t[2]*c^2*d*g*Sqrt[1 - (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])*(f + g*x))]*Sqrt[1 - (2*(c*f^2 - b*f*g + a*g^2))/((2*c*f - b*g + Sq
rt[b^2*g^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*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2
*c*f - b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2]
)])/(Sqrt[-((c*f^2 - b*f*g + a*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)]) + (I*b
*c*e*g*Sqrt[1 - (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])*(f + g*x))]*Sqrt[1 - (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])*(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*g^2 - 4*a*c*g^2]))])/Sqrt[f + g*x]], (2*c*f -
b*g - Sqrt[b^2*g^2 - 4*a*c*g^2])/(2*c*f - b*g + Sqrt[b^2*g^2 - 4*a*c*g^2])])/(S
qrt[2]*Sqrt[-((c*f^2 - b*f*g + a*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)])))/(c
*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])))/(15*c*g^4*Sqrt[a + b*x + c*x^2])

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Maple [B]  time = 0.045, size = 6207, normalized size = 12. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)*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)*(e*x + d)/sqrt(g*x + f),x, algorithm="giac")

[Out]

Timed out