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

Optimal. Leaf size=228 \[ \frac{\sin ^{-1}(d x) \left (4 d^2 \left (A \left (2 d^2 e^2+f^2\right )+2 B e f\right )+C \left (4 d^2 e^2+3 f^2\right )\right )}{8 d^5}+\frac{\sqrt{1-d^2 x^2} \left (4 \left (C \left (d^2 e^3-8 e f^2\right )-4 f \left (3 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-f x \left (3 f^2 \left (4 A d^2+3 C\right )-2 d^2 e (C e-4 B f)\right )\right )}{24 d^4 f}+\frac{\sqrt{1-d^2 x^2} (e+f x)^2 (C e-4 B f)}{12 d^2 f}-\frac{C \sqrt{1-d^2 x^2} (e+f x)^3}{4 d^2 f} \]

[Out]

((C*e - 4*B*f)*(e + f*x)^2*Sqrt[1 - d^2*x^2])/(12*d^2*f) - (C*(e + f*x)^3*Sqrt[1
- d^2*x^2])/(4*d^2*f) + ((4*(C*(d^2*e^3 - 8*e*f^2) - 4*f*(3*A*d^2*e*f + B*(d^2*
e^2 + f^2))) - f*(3*(3*C + 4*A*d^2)*f^2 - 2*d^2*e*(C*e - 4*B*f))*x)*Sqrt[1 - d^2
*x^2])/(24*d^4*f) + ((C*(4*d^2*e^2 + 3*f^2) + 4*d^2*(2*B*e*f + A*(2*d^2*e^2 + f^
2)))*ArcSin[d*x])/(8*d^5)

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Rubi [A]  time = 1.03938, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{\sin ^{-1}(d x) \left (4 d^2 \left (A \left (2 d^2 e^2+f^2\right )+2 B e f\right )+C \left (4 d^2 e^2+3 f^2\right )\right )}{8 d^5}+\frac{\sqrt{1-d^2 x^2} \left (4 \left (C \left (d^2 e^3-8 e f^2\right )-4 f \left (3 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-f x \left (3 f^2 \left (4 A d^2+3 C\right )-2 d^2 e (C e-4 B f)\right )\right )}{24 d^4 f}+\frac{\sqrt{1-d^2 x^2} (e+f x)^2 (C e-4 B f)}{12 d^2 f}-\frac{C \sqrt{1-d^2 x^2} (e+f x)^3}{4 d^2 f} \]

Antiderivative was successfully verified.

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

[Out]

((C*e - 4*B*f)*(e + f*x)^2*Sqrt[1 - d^2*x^2])/(12*d^2*f) - (C*(e + f*x)^3*Sqrt[1
- d^2*x^2])/(4*d^2*f) + ((4*(C*(d^2*e^3 - 8*e*f^2) - 4*f*(3*A*d^2*e*f + B*(d^2*
e^2 + f^2))) - f*(3*(3*C + 4*A*d^2)*f^2 - 2*d^2*e*(C*e - 4*B*f))*x)*Sqrt[1 - d^2
*x^2])/(24*d^4*f) + ((C*(4*d^2*e^2 + 3*f^2) + 4*d^2*(2*B*e*f + A*(2*d^2*e^2 + f^
2)))*ArcSin[d*x])/(8*d^5)

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Rubi in Sympy [A]  time = 86.4899, size = 218, normalized size = 0.96 \[ - \frac{C \left (e + f x\right )^{3} \sqrt{- d^{2} x^{2} + 1}}{4 d^{2} f} - \frac{\left (e + f x\right )^{2} \left (4 B f - C e\right ) \sqrt{- d^{2} x^{2} + 1}}{12 d^{2} f} - \frac{\sqrt{- d^{2} x^{2} + 1} \left (48 A d^{2} e f^{2} + 16 B d^{2} e^{2} f + 16 B f^{3} - 4 C d^{2} e^{3} + 32 C e f^{2} + f x \left (2 d^{2} e \left (4 B f - C e\right ) + f^{2} \left (12 A d^{2} + 9 C\right )\right )\right )}{24 d^{4} f} + \frac{\left (8 A d^{4} e^{2} + 4 A d^{2} f^{2} + 8 B d^{2} e f + 4 C d^{2} e^{2} + 3 C f^{2}\right ) \operatorname{asin}{\left (d x \right )}}{8 d^{5}} \]

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*sqrt(-d**2*x**2 + 1)/(4*d**2*f) - (e + f*x)**2*(4*B*f - C*e)*sqr
t(-d**2*x**2 + 1)/(12*d**2*f) - sqrt(-d**2*x**2 + 1)*(48*A*d**2*e*f**2 + 16*B*d*
*2*e**2*f + 16*B*f**3 - 4*C*d**2*e**3 + 32*C*e*f**2 + f*x*(2*d**2*e*(4*B*f - C*e
) + f**2*(12*A*d**2 + 9*C)))/(24*d**4*f) + (8*A*d**4*e**2 + 4*A*d**2*f**2 + 8*B*
d**2*e*f + 4*C*d**2*e**2 + 3*C*f**2)*asin(d*x)/(8*d**5)

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Mathematica [A]  time = 0.326114, size = 160, normalized size = 0.7 \[ \frac{3 \sin ^{-1}(d x) \left (4 d^2 \left (A \left (2 d^2 e^2+f^2\right )+2 B e f\right )+C \left (4 d^2 e^2+3 f^2\right )\right )-d \sqrt{1-d^2 x^2} \left (12 A d^2 f (4 e+f x)+8 B \left (d^2 \left (3 e^2+3 e f x+f^2 x^2\right )+2 f^2\right )+C \left (12 d^2 e^2 x+16 e f \left (d^2 x^2+2\right )+3 f^2 x \left (2 d^2 x^2+3\right )\right )\right )}{24 d^5} \]

Antiderivative was successfully verified.

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

[Out]

(-(d*Sqrt[1 - d^2*x^2]*(12*A*d^2*f*(4*e + f*x) + C*(12*d^2*e^2*x + 16*e*f*(2 + d
^2*x^2) + 3*f^2*x*(3 + 2*d^2*x^2)) + 8*B*(2*f^2 + d^2*(3*e^2 + 3*e*f*x + f^2*x^2
)))) + 3*(C*(4*d^2*e^2 + 3*f^2) + 4*d^2*(2*B*e*f + A*(2*d^2*e^2 + f^2)))*ArcSin[
d*x])/(24*d^5)

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Maple [C]  time = 0.034, size = 423, normalized size = 1.9 \[ -{\frac{{\it csgn} \left ( d \right ) }{24\,{d}^{5}}\sqrt{-dx+1}\sqrt{dx+1} \left ( 6\,C{f}^{2}{x}^{3}\sqrt{-{d}^{2}{x}^{2}+1}{d}^{3}{\it csgn} \left ( d \right ) +8\,B{\it csgn} \left ( d \right ){d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}{x}^{2}{f}^{2}+16\,C{\it csgn} \left ( d \right ){d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}{x}^{2}ef+12\,x\sqrt{-{d}^{2}{x}^{2}+1}A{f}^{2}{d}^{3}{\it csgn} \left ( d \right ) +24\,x\sqrt{-{d}^{2}{x}^{2}+1}Bef{d}^{3}{\it csgn} \left ( d \right ) +12\,x\sqrt{-{d}^{2}{x}^{2}+1}C{e}^{2}{d}^{3}{\it csgn} \left ( d \right ) +48\,A{\it csgn} \left ( d \right ){d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}ef-24\,A{e}^{2}\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){d}^{4}+24\,B{\it csgn} \left ( d \right ){d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}{e}^{2}+9\,C{f}^{2}x\sqrt{-{d}^{2}{x}^{2}+1}{\it csgn} \left ( d \right ) d-12\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) A{f}^{2}{d}^{2}+16\,B{\it csgn} \left ( d \right ) d\sqrt{-{d}^{2}{x}^{2}+1}{f}^{2}-24\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) Bef{d}^{2}+32\,C{\it csgn} \left ( d \right ) d\sqrt{-{d}^{2}{x}^{2}+1}ef-12\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) C{e}^{2}{d}^{2}-9\,C{f}^{2}\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)^2*(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*f^2*x^3*(-d^2*x^2+1)^(1/2)*d^3*csgn(d)+8
*B*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x^2*f^2+16*C*csgn(d)*d^3*(-d^2*x^2+1)^(1/2)*x^
2*e*f+12*x*(-d^2*x^2+1)^(1/2)*A*f^2*d^3*csgn(d)+24*x*(-d^2*x^2+1)^(1/2)*B*e*f*d^
3*csgn(d)+12*x*(-d^2*x^2+1)^(1/2)*C*e^2*d^3*csgn(d)+48*A*csgn(d)*d^3*(-d^2*x^2+1
)^(1/2)*e*f-24*A*e^2*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d^4+24*B*csgn(d)*d^3
*(-d^2*x^2+1)^(1/2)*e^2+9*C*f^2*x*(-d^2*x^2+1)^(1/2)*csgn(d)*d-12*arctan(csgn(d)
*d*x/(-d^2*x^2+1)^(1/2))*A*f^2*d^2+16*B*csgn(d)*d*(-d^2*x^2+1)^(1/2)*f^2-24*arct
an(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*B*e*f*d^2+32*C*csgn(d)*d*(-d^2*x^2+1)^(1/2)*e
*f-12*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*C*e^2*d^2-9*C*f^2*arctan(csgn(d)*d*
x/(-d^2*x^2+1)^(1/2)))*csgn(d)/(-d^2*x^2+1)^(1/2)/d^5

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Maxima [A]  time = 1.51715, size = 356, normalized size = 1.56 \[ -\frac{\sqrt{-d^{2} x^{2} + 1} C f^{2} x^{3}}{4 \, d^{2}} + \frac{A e^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}}} - \frac{\sqrt{-d^{2} x^{2} + 1} B e^{2}}{d^{2}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1} A e f}{d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1}{\left (2 \, C e f + B f^{2}\right )} x^{2}}{3 \, d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1}{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} x}{2 \, d^{2}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1} C f^{2} x}{8 \, d^{4}} + \frac{{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{2}} + \frac{3 \, C f^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{4}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1}{\left (2 \, C e f + B f^{2}\right )}}{3 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.245524, size = 1035, normalized size = 4.54 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(24*C*d^7*f^2*x^7 + 32*(2*C*d^7*e*f + B*d^7*f^2)*x^6 + 12*(4*C*d^7*e^2 + 8*
B*d^7*e*f + (4*A*d^7 - 3*C*d^5)*f^2)*x^5 + 24*(3*B*d^7*e^2 - 2*B*d^5*f^2 + 2*(3*
A*d^7 - 2*C*d^5)*e*f)*x^4 - 12*(12*C*d^5*e^2 + 24*B*d^5*e*f + (12*A*d^5 + 5*C*d^
3)*f^2)*x^3 - 96*(B*d^5*e^2 + 2*A*d^5*e*f)*x^2 - (6*C*d^7*f^2*x^7 + 8*(2*C*d^7*e
*f + B*d^7*f^2)*x^6 + 3*(4*C*d^7*e^2 + 8*B*d^7*e*f + (4*A*d^7 - 13*C*d^5)*f^2)*x
^5 + 24*(B*d^7*e^2 - 2*B*d^5*f^2 + 2*(A*d^7 - 2*C*d^5)*e*f)*x^4 - 24*(4*C*d^5*e^
2 + 8*B*d^5*e*f + (4*A*d^5 + C*d^3)*f^2)*x^3 - 96*(B*d^5*e^2 + 2*A*d^5*e*f)*x^2
+ 24*(4*C*d^3*e^2 + 8*B*d^3*e*f + (4*A*d^3 + 3*C*d)*f^2)*x)*sqrt(d*x + 1)*sqrt(-
d*x + 1) + 24*(4*C*d^3*e^2 + 8*B*d^3*e*f + (4*A*d^3 + 3*C*d)*f^2)*x - 6*(64*B*d^
2*e*f + (8*B*d^6*e*f + 4*(2*A*d^8 + C*d^6)*e^2 + (4*A*d^6 + 3*C*d^4)*f^2)*x^4 +
32*(2*A*d^4 + C*d^2)*e^2 + 8*(4*A*d^2 + 3*C)*f^2 - 8*(8*B*d^4*e*f + 4*(2*A*d^6 +
C*d^4)*e^2 + (4*A*d^4 + 3*C*d^2)*f^2)*x^2 - 4*(16*B*d^2*e*f + 8*(2*A*d^4 + C*d^
2)*e^2 + 2*(4*A*d^2 + 3*C)*f^2 - (8*B*d^4*e*f + 4*(2*A*d^6 + C*d^4)*e^2 + (4*A*d
^4 + 3*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^9*x^4 - 8*d^7*x^2 + 8*d^5 + 4*(d^7*x^2 - 2*d^5)*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)**2*(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 = 0.224605, size = 352, normalized size = 1.54 \[ -\frac{{\left (48 \, A d^{19} f e - 12 \, A d^{18} f^{2} + 24 \, B d^{19} e^{2} - 24 \, B d^{18} f e + 24 \, B d^{17} f^{2} - 12 \, C d^{18} e^{2} + 48 \, C d^{17} f e - 15 \, C d^{16} f^{2} +{\left (12 \, A d^{18} f^{2} + 24 \, B d^{18} f e - 16 \, B d^{17} f^{2} + 12 \, C d^{18} e^{2} - 32 \, C d^{17} f e + 27 \, C d^{16} f^{2} + 2 \,{\left (3 \,{\left (d x + 1\right )} C d^{16} f^{2} + 4 \, B d^{17} f^{2} + 8 \, C d^{17} f e - 9 \, C d^{16} f^{2}\right )}{\left (d x + 1\right )}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 6 \,{\left (8 \, A d^{20} e^{2} + 4 \, A d^{18} f^{2} + 8 \, B d^{18} f e + 4 \, C d^{18} e^{2} + 3 \, C d^{16} f^{2}\right )} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{86016 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/86016*((48*A*d^19*f*e - 12*A*d^18*f^2 + 24*B*d^19*e^2 - 24*B*d^18*f*e + 24*B*
d^17*f^2 - 12*C*d^18*e^2 + 48*C*d^17*f*e - 15*C*d^16*f^2 + (12*A*d^18*f^2 + 24*B
*d^18*f*e - 16*B*d^17*f^2 + 12*C*d^18*e^2 - 32*C*d^17*f*e + 27*C*d^16*f^2 + 2*(3
*(d*x + 1)*C*d^16*f^2 + 4*B*d^17*f^2 + 8*C*d^17*f*e - 9*C*d^16*f^2)*(d*x + 1))*(
d*x + 1))*sqrt(d*x + 1)*sqrt(-d*x + 1) - 6*(8*A*d^20*e^2 + 4*A*d^18*f^2 + 8*B*d^
18*f*e + 4*C*d^18*e^2 + 3*C*d^16*f^2)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))/d