3.36 \(\int \sqrt{1-d x} \sqrt{1+d x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx\)

Optimal. Leaf size=286 \[ \frac{\sin ^{-1}(d x) \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )}{16 d^5}+\frac{x \sqrt{1-d^2 x^2} \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )}{16 d^4}+\frac{\left (1-d^2 x^2\right )^{3/2} \left (8 \left (C \left (d^2 e^3-4 e f^2\right )-2 f \left (5 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-3 f x \left (5 f^2 \left (2 A d^2+C\right )-2 d^2 e (C e-2 B f)\right )\right )}{120 d^4 f}+\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^2 (C e-2 B f)}{10 d^2 f}-\frac{C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f} \]

[Out]

((C*(2*d^2*e^2 + f^2) + 2*d^2*(2*B*e*f + A*(4*d^2*e^2 + f^2)))*x*Sqrt[1 - d^2*x^
2])/(16*d^4) + ((C*e - 2*B*f)*(e + f*x)^2*(1 - d^2*x^2)^(3/2))/(10*d^2*f) - (C*(
e + f*x)^3*(1 - d^2*x^2)^(3/2))/(6*d^2*f) + ((8*(C*(d^2*e^3 - 4*e*f^2) - 2*f*(5*
A*d^2*e*f + B*(d^2*e^2 + f^2))) - 3*f*(5*(C + 2*A*d^2)*f^2 - 2*d^2*e*(C*e - 2*B*
f))*x)*(1 - d^2*x^2)^(3/2))/(120*d^4*f) + ((C*(2*d^2*e^2 + f^2) + 2*d^2*(2*B*e*f
+ A*(4*d^2*e^2 + f^2)))*ArcSin[d*x])/(16*d^5)

_______________________________________________________________________________________

Rubi [A]  time = 1.18619, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ \frac{\sin ^{-1}(d x) \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )}{16 d^5}+\frac{x \sqrt{1-d^2 x^2} \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )}{16 d^4}+\frac{\left (1-d^2 x^2\right )^{3/2} \left (8 \left (C \left (d^2 e^3-4 e f^2\right )-2 f \left (5 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-3 f x \left (5 f^2 \left (2 A d^2+C\right )-2 d^2 e (C e-2 B f)\right )\right )}{120 d^4 f}+\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^2 (C e-2 B f)}{10 d^2 f}-\frac{C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2*(A + B*x + C*x^2),x]

[Out]

((C*(2*d^2*e^2 + f^2) + 2*d^2*(2*B*e*f + A*(4*d^2*e^2 + f^2)))*x*Sqrt[1 - d^2*x^
2])/(16*d^4) + ((C*e - 2*B*f)*(e + f*x)^2*(1 - d^2*x^2)^(3/2))/(10*d^2*f) - (C*(
e + f*x)^3*(1 - d^2*x^2)^(3/2))/(6*d^2*f) + ((8*(C*(d^2*e^3 - 4*e*f^2) - 2*f*(5*
A*d^2*e*f + B*(d^2*e^2 + f^2))) - 3*f*(5*(C + 2*A*d^2)*f^2 - 2*d^2*e*(C*e - 2*B*
f))*x)*(1 - d^2*x^2)^(3/2))/(120*d^4*f) + ((C*(2*d^2*e^2 + f^2) + 2*d^2*(2*B*e*f
+ A*(4*d^2*e^2 + f^2)))*ArcSin[d*x])/(16*d^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 94.0848, size = 282, normalized size = 0.99 \[ - \frac{C \left (e + f x\right )^{3} \left (- d^{2} x^{2} + 1\right )^{\frac{3}{2}}}{6 d^{2} f} - \frac{\left (e + f x\right )^{2} \left (2 B f - C e\right ) \left (- d^{2} x^{2} + 1\right )^{\frac{3}{2}}}{10 d^{2} f} + \frac{x \sqrt{- d^{2} x^{2} + 1} \left (8 A d^{4} e^{2} + 2 A d^{2} f^{2} + 4 B d^{2} e f + 2 C d^{2} e^{2} + C f^{2}\right )}{16 d^{4}} - \frac{\left (- d^{2} x^{2} + 1\right )^{\frac{3}{2}} \left (240 A d^{2} e f^{2} + 48 B d^{2} e^{2} f + 48 B f^{3} - 24 C d^{2} e^{3} + 96 C e f^{2} + 9 f x \left (2 d^{2} e \left (2 B f - C e\right ) + 5 f^{2} \left (2 A d^{2} + C\right )\right )\right )}{360 d^{4} f} + \frac{\left (8 A d^{4} e^{2} + 2 A d^{2} f^{2} + 4 B d^{2} e f + 2 C d^{2} e^{2} + C f^{2}\right ) \operatorname{asin}{\left (d x \right )}}{16 d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x+e)**2*(C*x**2+B*x+A)*(-d*x+1)**(1/2)*(d*x+1)**(1/2),x)

[Out]

-C*(e + f*x)**3*(-d**2*x**2 + 1)**(3/2)/(6*d**2*f) - (e + f*x)**2*(2*B*f - C*e)*
(-d**2*x**2 + 1)**(3/2)/(10*d**2*f) + x*sqrt(-d**2*x**2 + 1)*(8*A*d**4*e**2 + 2*
A*d**2*f**2 + 4*B*d**2*e*f + 2*C*d**2*e**2 + C*f**2)/(16*d**4) - (-d**2*x**2 + 1
)**(3/2)*(240*A*d**2*e*f**2 + 48*B*d**2*e**2*f + 48*B*f**3 - 24*C*d**2*e**3 + 96
*C*e*f**2 + 9*f*x*(2*d**2*e*(2*B*f - C*e) + 5*f**2*(2*A*d**2 + C)))/(360*d**4*f)
+ (8*A*d**4*e**2 + 2*A*d**2*f**2 + 4*B*d**2*e*f + 2*C*d**2*e**2 + C*f**2)*asin(
d*x)/(16*d**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.456678, size = 243, normalized size = 0.85 \[ \frac{15 \sin ^{-1}(d x) \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )-d \sqrt{1-d^2 x^2} \left (-10 A d^2 \left (12 d^2 e^2 x+16 e f \left (d^2 x^2-1\right )+3 f^2 x \left (2 d^2 x^2-1\right )\right )+4 B \left (-2 d^4 x^2 \left (10 e^2+15 e f x+6 f^2 x^2\right )+d^2 \left (20 e^2+15 e f x+4 f^2 x^2\right )+8 f^2\right )+C \left (e^2 \left (30 d^2 x-60 d^4 x^3\right )+32 e f \left (-3 d^4 x^4+d^2 x^2+2\right )+5 f^2 x \left (-8 d^4 x^4+2 d^2 x^2+3\right )\right )\right )}{240 d^5} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^2*(A + B*x + C*x^2),x]

[Out]

(-(d*Sqrt[1 - d^2*x^2]*(-10*A*d^2*(12*d^2*e^2*x + 16*e*f*(-1 + d^2*x^2) + 3*f^2*
x*(-1 + 2*d^2*x^2)) + 4*B*(8*f^2 + d^2*(20*e^2 + 15*e*f*x + 4*f^2*x^2) - 2*d^4*x
^2*(10*e^2 + 15*e*f*x + 6*f^2*x^2)) + C*(e^2*(30*d^2*x - 60*d^4*x^3) + 5*f^2*x*(
3 + 2*d^2*x^2 - 8*d^4*x^4) + 32*e*f*(2 + d^2*x^2 - 3*d^4*x^4)))) + 15*(C*(2*d^2*
e^2 + f^2) + 2*d^2*(2*B*e*f + A*(4*d^2*e^2 + f^2)))*ArcSin[d*x])/(240*d^5)

_______________________________________________________________________________________

Maple [C]  time = 0.022, size = 652, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)^2*(C*x^2+B*x+A)*(-d*x+1)^(1/2)*(d*x+1)^(1/2),x)

[Out]

1/240*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(120*A*e^2*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1
/2))*d^4+30*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*A*f^2*d^2+30*arctan(csgn(d)*d
*x/(-d^2*x^2+1)^(1/2))*C*e^2*d^2+15*C*f^2*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))
-32*B*csgn(d)*d*(-d^2*x^2+1)^(1/2)*f^2+60*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))
*B*e*f*d^2-80*B*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*e^2-60*x*(-d^2*x^2+1)^(1/2)*B*e*f
*d^3*csgn(d)+96*C*csgn(d)*x^4*d^5*e*f*(-d^2*x^2+1)^(1/2)+120*B*csgn(d)*x^3*d^5*e
*f*(-d^2*x^2+1)^(1/2)+160*A*csgn(d)*x^2*d^5*e*f*(-d^2*x^2+1)^(1/2)-32*C*csgn(d)*
d^3*(-d^2*x^2+1)^(1/2)*x^2*e*f+40*C*csgn(d)*x^5*d^5*f^2*(-d^2*x^2+1)^(1/2)+48*B*
csgn(d)*x^4*d^5*f^2*(-d^2*x^2+1)^(1/2)+60*A*csgn(d)*x^3*d^5*f^2*(-d^2*x^2+1)^(1/
2)+60*C*csgn(d)*x^3*d^5*e^2*(-d^2*x^2+1)^(1/2)+80*B*csgn(d)*x^2*d^5*e^2*(-d^2*x^
2+1)^(1/2)-10*C*f^2*x^3*(-d^2*x^2+1)^(1/2)*d^3*csgn(d)-16*B*csgn(d)*d^3*(-d^2*x^
2+1)^(1/2)*x^2*f^2-160*A*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*e*f-64*C*csgn(d)*d*(-d^2
*x^2+1)^(1/2)*e*f+120*A*e^2*x*(-d^2*x^2+1)^(1/2)*d^5*csgn(d)-30*x*(-d^2*x^2+1)^(
1/2)*A*f^2*d^3*csgn(d)-30*x*(-d^2*x^2+1)^(1/2)*C*e^2*d^3*csgn(d)-15*C*f^2*x*(-d^
2*x^2+1)^(1/2)*csgn(d)*d)*csgn(d)/(-d^2*x^2+1)^(1/2)/d^5

_______________________________________________________________________________________

Maxima [A]  time = 1.50157, size = 459, normalized size = 1.6 \[ -\frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f^{2} x^{3}}{6 \, d^{2}} + \frac{1}{2} \, \sqrt{-d^{2} x^{2} + 1} A e^{2} x + \frac{A e^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} B e^{2}}{3 \, d^{2}} - \frac{2 \,{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} A e f}{3 \, d^{2}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (2 \, C e f + B f^{2}\right )} x^{2}}{5 \, d^{2}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} x}{4 \, d^{2}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f^{2} x}{8 \, d^{4}} + \frac{\sqrt{-d^{2} x^{2} + 1}{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} x}{8 \, d^{2}} + \frac{\sqrt{-d^{2} x^{2} + 1} C f^{2} x}{16 \, d^{4}} + \frac{{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{2}} + \frac{C f^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{16 \, \sqrt{d^{2}} d^{4}} - \frac{2 \,{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (2 \, C e f + B f^{2}\right )}}{15 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + 1)*sqrt(-d*x + 1)*(f*x + e)^2,x, algorithm="maxima")

[Out]

-1/6*(-d^2*x^2 + 1)^(3/2)*C*f^2*x^3/d^2 + 1/2*sqrt(-d^2*x^2 + 1)*A*e^2*x + 1/2*A
*e^2*arcsin(d^2*x/sqrt(d^2))/sqrt(d^2) - 1/3*(-d^2*x^2 + 1)^(3/2)*B*e^2/d^2 - 2/
3*(-d^2*x^2 + 1)^(3/2)*A*e*f/d^2 - 1/5*(-d^2*x^2 + 1)^(3/2)*(2*C*e*f + B*f^2)*x^
2/d^2 - 1/4*(-d^2*x^2 + 1)^(3/2)*(C*e^2 + 2*B*e*f + A*f^2)*x/d^2 - 1/8*(-d^2*x^2
+ 1)^(3/2)*C*f^2*x/d^4 + 1/8*sqrt(-d^2*x^2 + 1)*(C*e^2 + 2*B*e*f + A*f^2)*x/d^2
+ 1/16*sqrt(-d^2*x^2 + 1)*C*f^2*x/d^4 + 1/8*(C*e^2 + 2*B*e*f + A*f^2)*arcsin(d^
2*x/sqrt(d^2))/(sqrt(d^2)*d^2) + 1/16*C*f^2*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d
^4) - 2/15*(-d^2*x^2 + 1)^(3/2)*(2*C*e*f + B*f^2)/d^4

_______________________________________________________________________________________

Fricas [A]  time = 0.24753, size = 1705, normalized size = 5.96 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + 1)*sqrt(-d*x + 1)*(f*x + e)^2,x, algorithm="fricas")

[Out]

-1/240*(240*C*d^11*f^2*x^11 + 288*(2*C*d^11*e*f + B*d^11*f^2)*x^10 + 20*(18*C*d^
11*e^2 + 36*B*d^11*e*f + (18*A*d^11 - 79*C*d^9)*f^2)*x^9 + 480*(B*d^11*e^2 - 4*B
*d^9*f^2 + 2*(A*d^11 - 4*C*d^9)*e*f)*x^8 - 30*(164*B*d^9*e*f - 2*(12*A*d^11 - 41
*C*d^9)*e^2 + (82*A*d^9 - 95*C*d^7)*f^2)*x^7 - 80*(43*B*d^9*e^2 - 44*B*d^7*f^2 +
2*(43*A*d^9 - 44*C*d^7)*e*f)*x^6 + 30*(332*B*d^7*e*f - 2*(76*A*d^9 - 83*C*d^7)*
e^2 + (166*A*d^7 - 45*C*d^5)*f^2)*x^5 + 960*(7*B*d^7*e^2 - 2*B*d^5*f^2 + 2*(7*A*
d^7 - 2*C*d^5)*e*f)*x^4 - 640*(12*B*d^5*e*f - 6*(2*A*d^7 - C*d^5)*e^2 + (6*A*d^5
+ C*d^3)*f^2)*x^3 - 3840*(B*d^5*e^2 + 2*A*d^5*e*f)*x^2 - (40*C*d^11*f^2*x^11 +
48*(2*C*d^11*e*f + B*d^11*f^2)*x^10 + 10*(6*C*d^11*e^2 + 12*B*d^11*e*f + (6*A*d^
11 - 73*C*d^9)*f^2)*x^9 + 80*(B*d^11*e^2 - 11*B*d^9*f^2 + 2*(A*d^11 - 11*C*d^9)*
e*f)*x^8 - 15*(148*B*d^9*e*f - 2*(4*A*d^11 - 37*C*d^9)*e^2 + (74*A*d^9 - 139*C*d
^7)*f^2)*x^7 - 80*(19*B*d^9*e^2 - 32*B*d^7*f^2 + 2*(19*A*d^9 - 32*C*d^7)*e*f)*x^
6 + 10*(684*B*d^7*e*f - 18*(12*A*d^9 - 19*C*d^7)*e^2 + (342*A*d^7 - 149*C*d^5)*f
^2)*x^5 + 960*(5*B*d^7*e^2 - 2*B*d^5*f^2 + 2*(5*A*d^7 - 2*C*d^5)*e*f)*x^4 - 80*(
84*B*d^5*e*f - 6*(12*A*d^7 - 7*C*d^5)*e^2 + (42*A*d^5 + 5*C*d^3)*f^2)*x^3 - 3840
*(B*d^5*e^2 + 2*A*d^5*e*f)*x^2 + 480*(4*B*d^3*e*f - 2*(4*A*d^5 - C*d^3)*e^2 + (2
*A*d^3 + C*d)*f^2)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 480*(4*B*d^3*e*f - 2*(4*A*d
^5 - C*d^3)*e^2 + (2*A*d^3 + C*d)*f^2)*x + 30*((4*B*d^8*e*f + 2*(4*A*d^10 + C*d^
8)*e^2 + (2*A*d^8 + C*d^6)*f^2)*x^6 - 128*B*d^2*e*f - 18*(4*B*d^6*e*f + 2*(4*A*d
^8 + C*d^6)*e^2 + (2*A*d^6 + C*d^4)*f^2)*x^4 - 64*(4*A*d^4 + C*d^2)*e^2 - 32*(2*
A*d^2 + C)*f^2 + 48*(4*B*d^4*e*f + 2*(4*A*d^6 + C*d^4)*e^2 + (2*A*d^4 + C*d^2)*f
^2)*x^2 + 2*(64*B*d^2*e*f + 3*(4*B*d^6*e*f + 2*(4*A*d^8 + C*d^6)*e^2 + (2*A*d^6
+ C*d^4)*f^2)*x^4 + 32*(4*A*d^4 + C*d^2)*e^2 + 16*(2*A*d^2 + C)*f^2 - 16*(4*B*d^
4*e*f + 2*(4*A*d^6 + C*d^4)*e^2 + (2*A*d^4 + C*d^2)*f^2)*x^2)*sqrt(d*x + 1)*sqrt
(-d*x + 1))*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^11*x^6 - 18*d^9
*x^4 + 48*d^7*x^2 - 32*d^5 + 2*(3*d^9*x^4 - 16*d^7*x^2 + 16*d^5)*sqrt(d*x + 1)*s
qrt(-d*x + 1))

_______________________________________________________________________________________

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((f*x+e)**2*(C*x**2+B*x+A)*(-d*x+1)**(1/2)*(d*x+1)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.2736, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + 1)*sqrt(-d*x + 1)*(f*x + e)^2,x, algorithm="giac")

[Out]

Done