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

Optimal. Leaf size=168 \[ \frac{x \sqrt{1-d^2 x^2} \left (4 A d^2 e+B f+C e\right )}{8 d^2}-\frac{\left (1-d^2 x^2\right )^{3/2} \left (4 \left (5 d^2 f (A f+B e)-C \left (3 d^2 e^2-2 f^2\right )\right )-3 d^2 f x (3 C e-5 B f)\right )}{60 d^4 f}+\frac{\sin ^{-1}(d x) \left (4 A d^2 e+B f+C e\right )}{8 d^3}-\frac{C \left (1-d^2 x^2\right )^{3/2} (e+f x)^2}{5 d^2 f} \]

[Out]

((C*e + 4*A*d^2*e + B*f)*x*Sqrt[1 - d^2*x^2])/(8*d^2) - (C*(e + f*x)^2*(1 - d^2*
x^2)^(3/2))/(5*d^2*f) - ((4*(5*d^2*f*(B*e + A*f) - C*(3*d^2*e^2 - 2*f^2)) - 3*d^
2*f*(3*C*e - 5*B*f)*x)*(1 - d^2*x^2)^(3/2))/(60*d^4*f) + ((C*e + 4*A*d^2*e + B*f
)*ArcSin[d*x])/(8*d^3)

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Rubi [A]  time = 0.498443, antiderivative size = 170, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{x \sqrt{1-d^2 x^2} \left (4 A d^2 e+B f+C e\right )}{8 d^2}-\frac{\left (1-d^2 x^2\right )^{3/2} \left (4 \left (5 d^2 f (A f+B e)-C \left (3 d^2 e^2-2 f^2\right )\right )-3 d^2 f x (3 C e-5 B f)\right )}{60 d^4 f}+\frac{\sin ^{-1}(d x) \left (4 A d^2 e+B f+C e\right )}{8 d^3}-\frac{C \left (1-d^2 x^2\right )^{3/2} (e+f x)^2}{5 d^2 f} \]

Antiderivative was successfully verified.

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

[Out]

((C*e + 4*A*d^2*e + B*f)*x*Sqrt[1 - d^2*x^2])/(8*d^2) - (C*(e + f*x)^2*(1 - d^2*
x^2)^(3/2))/(5*d^2*f) - ((4*(5*d^2*f*(B*e + A*f) - C*(3*d^2*e^2 - 2*f^2)) - 3*d^
2*f*(3*C*e - 5*B*f)*x)*(1 - d^2*x^2)^(3/2))/(60*d^4*f) + ((C*e + 4*A*d^2*e + B*f
)*ArcSin[d*x])/(8*d^3)

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Rubi in Sympy [A]  time = 45.9748, size = 155, normalized size = 0.92 \[ - \frac{C \left (e + f x\right )^{2} \left (- d^{2} x^{2} + 1\right )^{\frac{3}{2}}}{5 d^{2} f} + \frac{x \sqrt{- d^{2} x^{2} + 1} \left (4 A d^{2} e + B f + C e\right )}{8 d^{2}} + \frac{\left (4 A d^{2} e + B f + C e\right ) \operatorname{asin}{\left (d x \right )}}{8 d^{3}} - \frac{\left (- d^{2} x^{2} + 1\right )^{\frac{3}{2}} \left (4 d^{2} e \left (5 B f - 3 C e\right ) + 3 d^{2} f x \left (5 B f - 3 C e\right ) + 4 f^{2} \left (5 A d^{2} + 2 C\right )\right )}{60 d^{4} f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-C*(e + f*x)**2*(-d**2*x**2 + 1)**(3/2)/(5*d**2*f) + x*sqrt(-d**2*x**2 + 1)*(4*A
*d**2*e + B*f + C*e)/(8*d**2) + (4*A*d**2*e + B*f + C*e)*asin(d*x)/(8*d**3) - (-
d**2*x**2 + 1)**(3/2)*(4*d**2*e*(5*B*f - 3*C*e) + 3*d**2*f*x*(5*B*f - 3*C*e) + 4
*f**2*(5*A*d**2 + 2*C))/(60*d**4*f)

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Mathematica [A]  time = 0.2406, size = 138, normalized size = 0.82 \[ \frac{15 d \sin ^{-1}(d x) \left (4 A d^2 e+B f+C e\right )+\sqrt{1-d^2 x^2} \left (8 d^2 x^2 \left (5 A d^2 f+5 B d^2 e-C f\right )-15 d^2 x \left (-4 A d^2 e+B f+C e\right )-8 \left (5 A d^2 f+5 B d^2 e+2 C f\right )+30 d^4 x^3 (B f+C e)+24 C d^4 f x^4\right )}{120 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - d^2*x^2]*(-8*(5*B*d^2*e + 2*C*f + 5*A*d^2*f) - 15*d^2*(C*e - 4*A*d^2*e
+ B*f)*x + 8*d^2*(5*B*d^2*e - C*f + 5*A*d^2*f)*x^2 + 30*d^4*(C*e + B*f)*x^3 + 2
4*C*d^4*f*x^4) + 15*d*(C*e + 4*A*d^2*e + B*f)*ArcSin[d*x])/(120*d^4)

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Maple [C]  time = 0.018, size = 377, normalized size = 2.2 \[{\frac{{\it csgn} \left ( d \right ) }{120\,{d}^{4}}\sqrt{-dx+1}\sqrt{dx+1} \left ( 24\,C{\it csgn} \left ( d \right ){x}^{4}{d}^{4}f\sqrt{-{d}^{2}{x}^{2}+1}+30\,B{\it csgn} \left ( d \right ){x}^{3}{d}^{4}f\sqrt{-{d}^{2}{x}^{2}+1}+30\,C{\it csgn} \left ( d \right ){x}^{3}{d}^{4}e\sqrt{-{d}^{2}{x}^{2}+1}+40\,A{\it csgn} \left ( d \right ){x}^{2}{d}^{4}f\sqrt{-{d}^{2}{x}^{2}+1}+40\,B{\it csgn} \left ( d \right ){x}^{2}{d}^{4}e\sqrt{-{d}^{2}{x}^{2}+1}+60\,A{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}x{d}^{4}e-8\,C{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}{x}^{2}{d}^{2}f-15\,B{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}x{d}^{2}f-15\,C{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}x{d}^{2}e-40\,A{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}{d}^{2}f+60\,A\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){d}^{3}e-40\,B{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}{d}^{2}e+15\,B\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) df-16\,C{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}f+15\,C\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) de \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/120*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(24*C*csgn(d)*x^4*d^4*f*(-d^2*x^2+1)^(1/2)+30
*B*csgn(d)*x^3*d^4*f*(-d^2*x^2+1)^(1/2)+30*C*csgn(d)*x^3*d^4*e*(-d^2*x^2+1)^(1/2
)+40*A*csgn(d)*x^2*d^4*f*(-d^2*x^2+1)^(1/2)+40*B*csgn(d)*x^2*d^4*e*(-d^2*x^2+1)^
(1/2)+60*A*csgn(d)*(-d^2*x^2+1)^(1/2)*x*d^4*e-8*C*csgn(d)*(-d^2*x^2+1)^(1/2)*x^2
*d^2*f-15*B*csgn(d)*(-d^2*x^2+1)^(1/2)*x*d^2*f-15*C*csgn(d)*(-d^2*x^2+1)^(1/2)*x
*d^2*e-40*A*csgn(d)*(-d^2*x^2+1)^(1/2)*d^2*f+60*A*arctan(csgn(d)*d*x/(-d^2*x^2+1
)^(1/2))*d^3*e-40*B*csgn(d)*(-d^2*x^2+1)^(1/2)*d^2*e+15*B*arctan(csgn(d)*d*x/(-d
^2*x^2+1)^(1/2))*d*f-16*C*csgn(d)*(-d^2*x^2+1)^(1/2)*f+15*C*arctan(csgn(d)*d*x/(
-d^2*x^2+1)^(1/2))*d*e)*csgn(d)/d^4/(-d^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.51607, size = 263, normalized size = 1.57 \[ \frac{1}{2} \, \sqrt{-d^{2} x^{2} + 1} A e x - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f x^{2}}{5 \, d^{2}} + \frac{A e \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} B e}{3 \, d^{2}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} A f}{3 \, d^{2}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (C e + B f\right )} x}{4 \, d^{2}} + \frac{\sqrt{-d^{2} x^{2} + 1}{\left (C e + B f\right )} x}{8 \, d^{2}} - \frac{2 \,{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f}{15 \, d^{4}} + \frac{{\left (C e + B f\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{2}} \]

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)*(f*x + e),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.234449, size = 921, normalized size = 5.48 \[ \text{result too large to display} \]

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),x, algorithm="fricas")

[Out]

1/120*(24*C*d^9*f*x^10 + 30*(C*d^9*e + B*d^9*f)*x^9 + 40*(B*d^9*e + (A*d^9 - 8*C
*d^7)*f)*x^8 - 15*(27*B*d^7*f - (4*A*d^9 - 27*C*d^7)*e)*x^7 - 40*(14*B*d^7*e + (
14*A*d^7 - 19*C*d^5)*f)*x^6 + 15*(69*B*d^5*f - (52*A*d^7 - 69*C*d^5)*e)*x^5 + 48
0*(3*B*d^5*e + (3*A*d^5 - C*d^3)*f)*x^4 - 60*(15*B*d^3*f - (28*A*d^5 - 15*C*d^3)
*e)*x^3 - 960*(B*d^3*e + A*d^3*f)*x^2 + 5*(24*C*d^7*f*x^8 + 30*(C*d^7*e + B*d^7*
f)*x^7 + 8*(5*B*d^7*e + (5*A*d^7 - 13*C*d^5)*f)*x^6 - 15*(9*B*d^5*f - (4*A*d^7 -
9*C*d^5)*e)*x^5 - 96*(2*B*d^5*e + (2*A*d^5 - C*d^3)*f)*x^4 + 12*(13*B*d^3*f - (
20*A*d^5 - 13*C*d^3)*e)*x^3 + 192*(B*d^3*e + A*d^3*f)*x^2 - 48*(B*d*f - (4*A*d^3
- C*d)*e)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 240*(B*d*f - (4*A*d^3 - C*d)*e)*x -
30*(5*(B*d^4*f + (4*A*d^6 + C*d^4)*e)*x^4 - 20*(B*d^2*f + (4*A*d^4 + C*d^2)*e)*
x^2 - ((B*d^4*f + (4*A*d^6 + C*d^4)*e)*x^4 - 12*(B*d^2*f + (4*A*d^4 + C*d^2)*e)*
x^2 + 16*(4*A*d^2 + C)*e + 16*B*f)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 16*(4*A*d^2 +
C)*e + 16*B*f)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(5*d^7*x^4 - 20
*d^5*x^2 + 16*d^3 - (d^7*x^4 - 12*d^5*x^2 + 16*d^3)*sqrt(d*x + 1)*sqrt(-d*x + 1)
)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.249742, size = 429, normalized size = 2.55 \[ \frac{8 \,{\left ({\left (d x + 1\right )}{\left (3 \,{\left (d x + 1\right )}{\left (\frac{d x + 1}{d^{3}} - \frac{4}{d^{3}}\right )} + \frac{17}{d^{3}}\right )} - \frac{10}{d^{3}}\right )}{\left (d x + 1\right )}^{\frac{3}{2}} \sqrt{-d x + 1} C f + \frac{40 \,{\left (d x + 1\right )}^{\frac{3}{2}}{\left (d x - 1\right )} \sqrt{-d x + 1} A f}{d} + \frac{40 \,{\left (d x + 1\right )}^{\frac{3}{2}}{\left (d x - 1\right )} \sqrt{-d x + 1} B e}{d} + 15 \,{\left ({\left ({\left (d x + 1\right )}{\left (2 \,{\left (d x + 1\right )}{\left (\frac{d x + 1}{d^{2}} - \frac{3}{d^{2}}\right )} + \frac{5}{d^{2}}\right )} - \frac{1}{d^{2}}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + \frac{2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{d^{2}}\right )} B f + 60 \,{\left (\sqrt{d x + 1} \sqrt{-d x + 1} d x + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )\right )} A e + 15 \,{\left ({\left ({\left (d x + 1\right )}{\left (2 \,{\left (d x + 1\right )}{\left (\frac{d x + 1}{d^{2}} - \frac{3}{d^{2}}\right )} + \frac{5}{d^{2}}\right )} - \frac{1}{d^{2}}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + \frac{2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{d^{2}}\right )} C e}{120 \, d} \]

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)*(f*x + e),x, algorithm="giac")

[Out]

1/120*(8*((d*x + 1)*(3*(d*x + 1)*((d*x + 1)/d^3 - 4/d^3) + 17/d^3) - 10/d^3)*(d*
x + 1)^(3/2)*sqrt(-d*x + 1)*C*f + 40*(d*x + 1)^(3/2)*(d*x - 1)*sqrt(-d*x + 1)*A*
f/d + 40*(d*x + 1)^(3/2)*(d*x - 1)*sqrt(-d*x + 1)*B*e/d + 15*(((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
*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^2)*B*f + 60*(sqrt(d*x + 1)*sqrt(-d*x + 1)*d
*x + 2*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))*A*e + 15*(((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*arcsin(1/
2*sqrt(2)*sqrt(d*x + 1))/d^2)*C*e)/d