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

Optimal. Leaf size=95 \[ \frac{x \sqrt{1-d^2 x^2} \left (4 A d^2+C\right )}{8 d^2}+\frac{\left (4 A d^2+C\right ) \sin ^{-1}(d x)}{8 d^3}-\frac{B \left (1-d^2 x^2\right )^{3/2}}{3 d^2}-\frac{C x \left (1-d^2 x^2\right )^{3/2}}{4 d^2} \]

[Out]

((C + 4*A*d^2)*x*Sqrt[1 - d^2*x^2])/(8*d^2) - (B*(1 - d^2*x^2)^(3/2))/(3*d^2) -
(C*x*(1 - d^2*x^2)^(3/2))/(4*d^2) + ((C + 4*A*d^2)*ArcSin[d*x])/(8*d^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.154758, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x \sqrt{1-d^2 x^2} \left (4 A d^2+C\right )}{8 d^2}+\frac{\left (4 A d^2+C\right ) \sin ^{-1}(d x)}{8 d^3}-\frac{B \left (1-d^2 x^2\right )^{3/2}}{3 d^2}-\frac{C x \left (1-d^2 x^2\right )^{3/2}}{4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

((C + 4*A*d^2)*x*Sqrt[1 - d^2*x^2])/(8*d^2) - (B*(1 - d^2*x^2)^(3/2))/(3*d^2) -
(C*x*(1 - d^2*x^2)^(3/2))/(4*d^2) + ((C + 4*A*d^2)*ArcSin[d*x])/(8*d^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.6266, size = 70, normalized size = 0.74 \[ \frac{x \left (4 A d^{2} + C\right ) \sqrt{- d^{2} x^{2} + 1}}{8 d^{2}} - \frac{\left (4 B + 3 C x\right ) \left (- d^{2} x^{2} + 1\right )^{\frac{3}{2}}}{12 d^{2}} + \frac{\left (4 A d^{2} + C\right ) \operatorname{asin}{\left (d x \right )}}{8 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(4*A*d**2 + C)*sqrt(-d**2*x**2 + 1)/(8*d**2) - (4*B + 3*C*x)*(-d**2*x**2 + 1)*
*(3/2)/(12*d**2) + (4*A*d**2 + C)*asin(d*x)/(8*d**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0874555, size = 71, normalized size = 0.75 \[ \frac{d \sqrt{1-d^2 x^2} \left (12 A d^2 x+8 B d^2 x^2-8 B+6 C d^2 x^3-3 C x\right )+3 \left (4 A d^2+C\right ) \sin ^{-1}(d x)}{24 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(d*Sqrt[1 - d^2*x^2]*(-8*B - 3*C*x + 12*A*d^2*x + 8*B*d^2*x^2 + 6*C*d^2*x^3) + 3
*(C + 4*A*d^2)*ArcSin[d*x])/(24*d^3)

_______________________________________________________________________________________

Maple [C]  time = 0.015, size = 185, normalized size = 2. \[{\frac{{\it csgn} \left ( d \right ) }{24\,{d}^{3}}\sqrt{-dx+1}\sqrt{dx+1} \left ( 6\,C{\it csgn} \left ( d \right ){x}^{3}{d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}+8\,B{\it csgn} \left ( d \right ){x}^{2}{d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}+12\,Ax\sqrt{-{d}^{2}{x}^{2}+1}{d}^{3}{\it csgn} \left ( d \right ) -3\,Cx\sqrt{-{d}^{2}{x}^{2}+1}{\it csgn} \left ( d \right ) d+12\,A\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){d}^{2}-8\,B\sqrt{-{d}^{2}{x}^{2}+1}{\it csgn} \left ( d \right ) d+3\,C\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(6*C*csgn(d)*x^3*d^3*(-d^2*x^2+1)^(1/2)+8*B*cs
gn(d)*x^2*d^3*(-d^2*x^2+1)^(1/2)+12*A*x*(-d^2*x^2+1)^(1/2)*d^3*csgn(d)-3*C*x*(-d
^2*x^2+1)^(1/2)*csgn(d)*d+12*A*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d^2-8*B*(-
d^2*x^2+1)^(1/2)*csgn(d)*d+3*C*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2)))*csgn(d)/(
-d^2*x^2+1)^(1/2)/d^3

_______________________________________________________________________________________

Maxima [A]  time = 1.51922, size = 154, normalized size = 1.62 \[ \frac{1}{2} \, \sqrt{-d^{2} x^{2} + 1} A x - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C x}{4 \, d^{2}} + \frac{A \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} B}{3 \, d^{2}} + \frac{\sqrt{-d^{2} x^{2} + 1} C x}{8 \, d^{2}} + \frac{C \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{2}} \]

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),x, algorithm="maxima")

[Out]

1/2*sqrt(-d^2*x^2 + 1)*A*x - 1/4*(-d^2*x^2 + 1)^(3/2)*C*x/d^2 + 1/2*A*arcsin(d^2
*x/sqrt(d^2))/sqrt(d^2) - 1/3*(-d^2*x^2 + 1)^(3/2)*B/d^2 + 1/8*sqrt(-d^2*x^2 + 1
)*C*x/d^2 + 1/8*C*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.228682, size = 487, normalized size = 5.13 \[ -\frac{24 \, C d^{7} x^{7} + 32 \, B d^{7} x^{6} - 120 \, B d^{5} x^{4} + 96 \, B d^{3} x^{2} + 12 \,{\left (4 \, A d^{7} - 7 \, C d^{5}\right )} x^{5} - 12 \,{\left (12 \, A d^{5} - 7 \, C d^{3}\right )} x^{3} -{\left (6 \, C d^{7} x^{7} + 8 \, B d^{7} x^{6} - 72 \, B d^{5} x^{4} + 96 \, B d^{3} x^{2} + 3 \,{\left (4 \, A d^{7} - 17 \, C d^{5}\right )} x^{5} - 24 \,{\left (4 \, A d^{5} - 3 \, C d^{3}\right )} x^{3} + 24 \,{\left (4 \, A d^{3} - C d\right )} x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 24 \,{\left (4 \, A d^{3} - C d\right )} x + 6 \,{\left ({\left (4 \, A d^{6} + C d^{4}\right )} x^{4} + 32 \, A d^{2} - 8 \,{\left (4 \, A d^{4} + C d^{2}\right )} x^{2} - 4 \,{\left (8 \, A d^{2} -{\left (4 \, A d^{4} + C d^{2}\right )} x^{2} + 2 \, C\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 8 \, C\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{24 \,{\left (d^{7} x^{4} - 8 \, d^{5} x^{2} + 8 \, d^{3} + 4 \,{\left (d^{5} x^{2} - 2 \, d^{3}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}\right )}} \]

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),x, algorithm="fricas")

[Out]

-1/24*(24*C*d^7*x^7 + 32*B*d^7*x^6 - 120*B*d^5*x^4 + 96*B*d^3*x^2 + 12*(4*A*d^7
- 7*C*d^5)*x^5 - 12*(12*A*d^5 - 7*C*d^3)*x^3 - (6*C*d^7*x^7 + 8*B*d^7*x^6 - 72*B
*d^5*x^4 + 96*B*d^3*x^2 + 3*(4*A*d^7 - 17*C*d^5)*x^5 - 24*(4*A*d^5 - 3*C*d^3)*x^
3 + 24*(4*A*d^3 - C*d)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 24*(4*A*d^3 - C*d)*x +
6*((4*A*d^6 + C*d^4)*x^4 + 32*A*d^2 - 8*(4*A*d^4 + C*d^2)*x^2 - 4*(8*A*d^2 - (4*
A*d^4 + C*d^2)*x^2 + 2*C)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 8*C)*arctan((sqrt(d*x +
1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^7*x^4 - 8*d^5*x^2 + 8*d^3 + 4*(d^5*x^2 - 2*d^
3)*sqrt(d*x + 1)*sqrt(-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((C*x**2+B*x+A)*(-d*x+1)**(1/2)*(d*x+1)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.226715, size = 198, normalized size = 2.08 \[ \frac{\frac{8 \,{\left (d x + 1\right )}^{\frac{3}{2}}{\left (d x - 1\right )} \sqrt{-d x + 1} B}{d} + 12 \,{\left (\sqrt{d x + 1} \sqrt{-d x + 1} d x + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )\right )} A + 3 \,{\left ({\left ({\left (d x + 1\right )}{\left (2 \,{\left (d x + 1\right )}{\left (\frac{d x + 1}{d^{2}} - \frac{3}{d^{2}}\right )} + \frac{5}{d^{2}}\right )} - \frac{1}{d^{2}}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + \frac{2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{d^{2}}\right )} C}{24 \, d} \]

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),x, algorithm="giac")

[Out]

1/24*(8*(d*x + 1)^(3/2)*(d*x - 1)*sqrt(-d*x + 1)*B/d + 12*(sqrt(d*x + 1)*sqrt(-d
*x + 1)*d*x + 2*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*A + 3*(((d*x + 1)*(2*(d*x + 1
)*((d*x + 1)/d^2 - 3/d^2) + 5/d^2) - 1/d^2)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*arc
sin(1/2*sqrt(2)*sqrt(d*x + 1))/d^2)*C)/d