3.892 \(\int \frac{\sqrt{f+g x} \sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=1034 \[ -\frac{\sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right ) (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))}{4 \sqrt{2} e^2 \left (c d^2-b e d+a e^2\right ) (e f-d g) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{c x^2+b x+a}}+\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a} (c d (2 e f-3 d g)-e (b e f-2 b d g+a e g))}{4 e \left (c d^2-b e d+a e^2\right ) (e f-d g) (d+e x)}-\frac{\sqrt{b^2-4 a c} (e (b e f+4 b d g-5 a e g)-c d (2 e f+3 d g)) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{b^2-4 a c} g}{\left (b+\sqrt{b^2-4 a c}\right ) g-2 c f}\right )}{2 \sqrt{2} e^3 \left (c d^2+e (a e-b d)\right ) \sqrt{f+g x} \sqrt{a+x (b+c x)}}+\frac{\sqrt{2 c f-b g+\sqrt{b^2-4 a c} g} \left (b^2 f^2 e^4+a^2 g^2 e^4-2 a c \left (2 e^2 f^2-6 d e g f+3 d^2 g^2\right ) e^2-2 b g \left (a f e^3+c d^2 (3 e f-2 d g)\right ) e+c^2 d^3 g (4 e f-3 d g)\right ) \sqrt{\frac{g \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}{2 c f+\left (\sqrt{b^2-4 a c}-b\right ) g}} \sqrt{\frac{g \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (b+\sqrt{b^2-4 a c}\right ) g-2 c f}} \Pi \left (\frac{2 c e f-b e g+\sqrt{b^2-4 a c} e g}{2 c e f-2 c d g};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{f+g x}}{\sqrt{2 c f-b g+\sqrt{b^2-4 a c} g}}\right )|\frac{2 c f+\left (\sqrt{b^2-4 a c}-b\right ) g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{4 \sqrt{2} \sqrt{c} e^3 \left (c d^2+e (a e-b d)\right ) (e f-d g)^2 \sqrt{a+x (b+c x)}}-\frac{\sqrt{f+g x} \sqrt{c x^2+b x+a}}{2 e (d+e x)^2} \]

[Out]

-(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(2*e*(d + e*x)^2) + ((c*d*(2*e*f - 3*d*g)
- e*(b*e*f - 2*b*d*g + a*e*g))*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(4*e*(c*d^2
- b*d*e + a*e^2)*(e*f - d*g)*(d + e*x)) - (Sqrt[b^2 - 4*a*c]*(c*d*(2*e*f - 3*d*
g) - e*(b*e*f - 2*b*d*g + 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)])/(
4*Sqrt[2]*e^2*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*Sqrt[(c*(f + g*x))/(2*c*f - (b
+ Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) - (Sqrt[b^2 - 4*a*c]*(-(c*d*(2*
e*f + 3*d*g)) + e*(b*e*f + 4*b*d*g - 5*a*e*g))*Sqrt[(c*(f + g*x))/(2*c*f - (b +
Sqrt[b^2 - 4*a*c])*g)]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*EllipticF[ArcS
in[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)])/(2*Sqrt[2]*e^3*(c*d^2 + e*(-
(b*d) + a*e))*Sqrt[f + g*x]*Sqrt[a + x*(b + c*x)]) + (Sqrt[2*c*f - b*g + Sqrt[b^
2 - 4*a*c]*g]*(b^2*e^4*f^2 + a^2*e^4*g^2 + c^2*d^3*g*(4*e*f - 3*d*g) - 2*a*c*e^2
*(2*e^2*f^2 - 6*d*e*f*g + 3*d^2*g^2) - 2*b*e*g*(a*e^3*f + c*d^2*(3*e*f - 2*d*g))
)*Sqrt[(g*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)
]*Sqrt[(g*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]
*EllipticPi[(2*c*e*f - b*e*g + Sqrt[b^2 - 4*a*c]*e*g)/(2*c*e*f - 2*c*d*g), ArcSi
n[(Sqrt[2]*Sqrt[c]*Sqrt[f + g*x])/Sqrt[2*c*f - b*g + Sqrt[b^2 - 4*a*c]*g]], (2*c
*f + (-b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(4*Sqrt[2
]*Sqrt[c]*e^3*(c*d^2 + e*(-(b*d) + a*e))*(e*f - d*g)^2*Sqrt[a + x*(b + c*x)])

_______________________________________________________________________________________

Rubi [A]  time = 18.6669, antiderivative size = 1705, normalized size of antiderivative = 1.65, number of steps used = 25, number of rules used = 11, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.355 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]  Int[(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(d + e*x)^3,x]

[Out]

-(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(2*e*(d + e*x)^2) + ((c*d*(2*e*f - 3*d*g)
- e*(b*e*f - 2*b*d*g + a*e*g))*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(4*e*(c*d^2
- b*d*e + a*e^2)*(e*f - d*g)*(d + e*x)) - (Sqrt[b^2 - 4*a*c]*(c*d*(2*e*f - 3*d*
g) - e*(b*e*f - 2*b*d*g + 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)])/(
4*Sqrt[2]*e^2*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*Sqrt[(c*(f + g*x))/(2*c*f - (b
+ Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) + (3*Sqrt[b^2 - 4*a*c]*g*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)])
/(Sqrt[2]*e^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]) + (Sqrt[b^2 - 4*a*c]*f*(c*d*(
2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*Sqrt[(c*(f + g*x))/(2*c*f - (b + S
qrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[Arc
Sin[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)])/(2*Sqrt[2]*e^2*(c*d^2 - b*d
*e + a*e^2)*(e*f - d*g)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]) - (Sqrt[b^2 - 4*a*c
]*d*g*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*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))]*E
llipticF[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)])/(2*Sqrt[2]*e^3*
(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]) - (Sqrt
[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]*(c*e*f - 3*c*d*g + b*e*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[2]*Sqrt[c]*e^3*(e*f - d*g)*Sqrt[a + b*x + c*x^2]) + (Sqrt[2*c*f
- (b - Sqrt[b^2 - 4*a*c])*g]*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g)
)^2*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 + Sqr
t[b^2 - 4*a*c]*g))/(2*c*(e*f - d*g)), ArcSin[(Sqrt[2]*Sqrt[c]*Sqrt[f + g*x])/Sqr
t[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)])/(4*Sqrt[2]*Sqrt[c]*e^3*(c*d^2 - b*d*e + a*e^2)*
(e*f - d*g)^2*Sqrt[a + b*x + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{f + g x} \sqrt{a + b x + c x^{2}}}{\left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**(1/2)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**3,x)

[Out]

Integral(sqrt(f + g*x)*sqrt(a + b*x + c*x**2)/(d + e*x)**3, x)

_______________________________________________________________________________________

Mathematica [C]  time = 19.8898, size = 33765, normalized size = 32.65 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(d + e*x)^3,x]

[Out]

Result too large to show

_______________________________________________________________________________________

Maple [B]  time = 0.146, size = 55360, normalized size = 53.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^3,x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a} \sqrt{g x + f}}{{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f)/(e*x + d)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f)/(e*x + d)^3, x)

_______________________________________________________________________________________

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(c*x^2 + b*x + a)*sqrt(g*x + f)/(e*x + d)^3,x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**(1/2)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

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)*sqrt(g*x + f)/(e*x + d)^3,x, algorithm="giac")

[Out]

Timed out