3.793 \(\int \frac{\left (a+b x+c x^2\right )^2}{\sqrt{1-d x} \sqrt{1+d x}} \, dx\)

Optimal. Leaf size=166 \[ \frac{\sin ^{-1}(d x) \left (8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right )}{8 d^5}-\frac{x \sqrt{1-d^2 x^2} \left (c \left (8 a+\frac{3 c}{d^2}\right )+4 b^2\right )}{8 d^2}-\frac{2 b \sqrt{1-d^2 x^2} \left (3 a d^2+2 c\right )}{3 d^4}-\frac{2 b c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{c^2 x^3 \sqrt{1-d^2 x^2}}{4 d^2} \]

[Out]

(-2*b*(2*c + 3*a*d^2)*Sqrt[1 - d^2*x^2])/(3*d^4) - ((4*b^2 + c*(8*a + (3*c)/d^2)
)*x*Sqrt[1 - d^2*x^2])/(8*d^2) - (2*b*c*x^2*Sqrt[1 - d^2*x^2])/(3*d^2) - (c^2*x^
3*Sqrt[1 - d^2*x^2])/(4*d^2) + ((3*c^2 + 4*b^2*d^2 + 8*a*c*d^2 + 8*a^2*d^4)*ArcS
in[d*x])/(8*d^5)

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Rubi [A]  time = 0.535479, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sin ^{-1}(d x) \left (8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right )}{8 d^5}-\frac{x \sqrt{1-d^2 x^2} \left (c \left (8 a+\frac{3 c}{d^2}\right )+4 b^2\right )}{8 d^2}-\frac{2 b \sqrt{1-d^2 x^2} \left (3 a d^2+2 c\right )}{3 d^4}-\frac{2 b c x^2 \sqrt{1-d^2 x^2}}{3 d^2}-\frac{c^2 x^3 \sqrt{1-d^2 x^2}}{4 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^2/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]

[Out]

(-2*b*(2*c + 3*a*d^2)*Sqrt[1 - d^2*x^2])/(3*d^4) - ((4*b^2 + c*(8*a + (3*c)/d^2)
)*x*Sqrt[1 - d^2*x^2])/(8*d^2) - (2*b*c*x^2*Sqrt[1 - d^2*x^2])/(3*d^2) - (c^2*x^
3*Sqrt[1 - d^2*x^2])/(4*d^2) + ((3*c^2 + 4*b^2*d^2 + 8*a*c*d^2 + 8*a^2*d^4)*ArcS
in[d*x])/(8*d^5)

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Rubi in Sympy [A]  time = 44.7715, size = 128, normalized size = 0.77 \[ - \frac{\left (5 b + 3 c x\right ) \sqrt{- d^{2} x^{2} + 1} \left (a + b x + c x^{2}\right )}{12 d^{2}} - \frac{\left (2 b \left (19 a d^{2} + 16 c\right ) + x \left (2 b^{2} d^{2} + 9 c \left (2 a d^{2} + c\right )\right )\right ) \sqrt{- d^{2} x^{2} + 1}}{24 d^{4}} + \frac{\left (8 a^{2} d^{4} + 8 a c d^{2} + 4 b^{2} d^{2} + 3 c^{2}\right ) \operatorname{asin}{\left (d x \right )}}{8 d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**2/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

-(5*b + 3*c*x)*sqrt(-d**2*x**2 + 1)*(a + b*x + c*x**2)/(12*d**2) - (2*b*(19*a*d*
*2 + 16*c) + x*(2*b**2*d**2 + 9*c*(2*a*d**2 + c)))*sqrt(-d**2*x**2 + 1)/(24*d**4
) + (8*a**2*d**4 + 8*a*c*d**2 + 4*b**2*d**2 + 3*c**2)*asin(d*x)/(8*d**5)

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Mathematica [A]  time = 0.171887, size = 114, normalized size = 0.69 \[ \frac{3 \sin ^{-1}(d x) \left (8 a^2 d^4+8 a c d^2+4 b^2 d^2+3 c^2\right )-d \sqrt{1-d^2 x^2} \left (16 b \left (3 a d^2+c d^2 x^2+2 c\right )+3 c x \left (8 a d^2+2 c d^2 x^2+3 c\right )+12 b^2 d^2 x\right )}{24 d^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^2/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]

[Out]

(-(d*Sqrt[1 - d^2*x^2]*(12*b^2*d^2*x + 16*b*(2*c + 3*a*d^2 + c*d^2*x^2) + 3*c*x*
(3*c + 8*a*d^2 + 2*c*d^2*x^2))) + 3*(3*c^2 + 4*b^2*d^2 + 8*a*c*d^2 + 8*a^2*d^4)*
ArcSin[d*x])/(24*d^5)

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Maple [C]  time = 0.033, size = 291, normalized size = 1.8 \[ -{\frac{{\it csgn} \left ( d \right ) }{24\,{d}^{5}}\sqrt{-dx+1}\sqrt{dx+1} \left ( 6\,{\it csgn} \left ( d \right ){x}^{3}{c}^{2}{d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}+16\,{\it csgn} \left ( d \right ){x}^{2}bc{d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}+24\,acx\sqrt{-{d}^{2}{x}^{2}+1}{d}^{3}{\it csgn} \left ( d \right ) +12\,{b}^{2}x\sqrt{-{d}^{2}{x}^{2}+1}{d}^{3}{\it csgn} \left ( d \right ) +48\,\sqrt{-{d}^{2}{x}^{2}+1}ab{d}^{3}{\it csgn} \left ( d \right ) -24\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){a}^{2}{d}^{4}+9\,{c}^{2}x\sqrt{-{d}^{2}{x}^{2}+1}{\it csgn} \left ( d \right ) d+32\,\sqrt{-{d}^{2}{x}^{2}+1}bc{\it csgn} \left ( d \right ) d-24\,ac\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){d}^{2}-12\,{b}^{2}\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){d}^{2}-9\,{c}^{2}\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^2/(-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*csgn(d)*x^3*c^2*d^3*(-d^2*x^2+1)^(1/2)+16*
csgn(d)*x^2*b*c*d^3*(-d^2*x^2+1)^(1/2)+24*a*c*x*(-d^2*x^2+1)^(1/2)*d^3*csgn(d)+1
2*b^2*x*(-d^2*x^2+1)^(1/2)*d^3*csgn(d)+48*(-d^2*x^2+1)^(1/2)*a*b*d^3*csgn(d)-24*
arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a^2*d^4+9*c^2*x*(-d^2*x^2+1)^(1/2)*csgn(d
)*d+32*(-d^2*x^2+1)^(1/2)*b*c*csgn(d)*d-24*a*c*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(
1/2))*d^2-12*b^2*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d^2-9*c^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 = 0.779709, size = 275, normalized size = 1.66 \[ -\frac{\sqrt{-d^{2} x^{2} + 1} c^{2} x^{3}}{4 \, d^{2}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1} b c x^{2}}{3 \, d^{2}} + \frac{a^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1} a b}{d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1}{\left (b^{2} + 2 \, a c\right )} x}{2 \, d^{2}} - \frac{3 \, \sqrt{-d^{2} x^{2} + 1} c^{2} x}{8 \, d^{4}} + \frac{{\left (b^{2} + 2 \, a c\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{2}} - \frac{4 \, \sqrt{-d^{2} x^{2} + 1} b c}{3 \, d^{4}} + \frac{3 \, c^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(sqrt(d*x + 1)*sqrt(-d*x + 1)),x, algorithm="maxima")

[Out]

-1/4*sqrt(-d^2*x^2 + 1)*c^2*x^3/d^2 - 2/3*sqrt(-d^2*x^2 + 1)*b*c*x^2/d^2 + a^2*a
rcsin(d^2*x/sqrt(d^2))/sqrt(d^2) - 2*sqrt(-d^2*x^2 + 1)*a*b/d^2 - 1/2*sqrt(-d^2*
x^2 + 1)*(b^2 + 2*a*c)*x/d^2 - 3/8*sqrt(-d^2*x^2 + 1)*c^2*x/d^4 + 1/2*(b^2 + 2*a
*c)*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^2) - 4/3*sqrt(-d^2*x^2 + 1)*b*c/d^4 + 3
/8*c^2*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^4)

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Fricas [A]  time = 0.280622, size = 717, normalized size = 4.32 \[ \frac{24 \, c^{2} d^{7} x^{7} + 64 \, b c d^{7} x^{6} - 192 \, a b d^{5} x^{2} + 12 \,{\left (4 \,{\left (b^{2} + 2 \, a c\right )} d^{7} - 3 \, c^{2} d^{5}\right )} x^{5} + 48 \,{\left (3 \, a b d^{7} - 2 \, b c d^{5}\right )} x^{4} - 12 \,{\left (12 \,{\left (b^{2} + 2 \, a c\right )} d^{5} + 5 \, c^{2} d^{3}\right )} x^{3} -{\left (6 \, c^{2} d^{7} x^{7} + 16 \, b c d^{7} x^{6} - 192 \, a b d^{5} x^{2} + 3 \,{\left (4 \,{\left (b^{2} + 2 \, a c\right )} d^{7} - 13 \, c^{2} d^{5}\right )} x^{5} + 48 \,{\left (a b d^{7} - 2 \, b c d^{5}\right )} x^{4} - 24 \,{\left (4 \,{\left (b^{2} + 2 \, a c\right )} d^{5} + c^{2} d^{3}\right )} x^{3} + 24 \,{\left (4 \,{\left (b^{2} + 2 \, a c\right )} d^{3} + 3 \, c^{2} d\right )} x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 24 \,{\left (4 \,{\left (b^{2} + 2 \, a c\right )} d^{3} + 3 \, c^{2} d\right )} x - 6 \,{\left (64 \, a^{2} d^{4} +{\left (8 \, a^{2} d^{8} + 4 \,{\left (b^{2} + 2 \, a c\right )} d^{6} + 3 \, c^{2} d^{4}\right )} x^{4} + 32 \,{\left (b^{2} + 2 \, a c\right )} d^{2} - 8 \,{\left (8 \, a^{2} d^{6} + 4 \,{\left (b^{2} + 2 \, a c\right )} d^{4} + 3 \, c^{2} d^{2}\right )} x^{2} - 4 \,{\left (16 \, a^{2} d^{4} + 8 \,{\left (b^{2} + 2 \, a c\right )} d^{2} -{\left (8 \, a^{2} d^{6} + 4 \,{\left (b^{2} + 2 \, a c\right )} d^{4} + 3 \, c^{2} d^{2}\right )} x^{2} + 6 \, c^{2}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 24 \, c^{2}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{24 \,{\left (d^{9} x^{4} - 8 \, d^{7} x^{2} + 8 \, d^{5} + 4 \,{\left (d^{7} x^{2} - 2 \, d^{5}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(sqrt(d*x + 1)*sqrt(-d*x + 1)),x, algorithm="fricas")

[Out]

1/24*(24*c^2*d^7*x^7 + 64*b*c*d^7*x^6 - 192*a*b*d^5*x^2 + 12*(4*(b^2 + 2*a*c)*d^
7 - 3*c^2*d^5)*x^5 + 48*(3*a*b*d^7 - 2*b*c*d^5)*x^4 - 12*(12*(b^2 + 2*a*c)*d^5 +
5*c^2*d^3)*x^3 - (6*c^2*d^7*x^7 + 16*b*c*d^7*x^6 - 192*a*b*d^5*x^2 + 3*(4*(b^2
+ 2*a*c)*d^7 - 13*c^2*d^5)*x^5 + 48*(a*b*d^7 - 2*b*c*d^5)*x^4 - 24*(4*(b^2 + 2*a
*c)*d^5 + c^2*d^3)*x^3 + 24*(4*(b^2 + 2*a*c)*d^3 + 3*c^2*d)*x)*sqrt(d*x + 1)*sqr
t(-d*x + 1) + 24*(4*(b^2 + 2*a*c)*d^3 + 3*c^2*d)*x - 6*(64*a^2*d^4 + (8*a^2*d^8
+ 4*(b^2 + 2*a*c)*d^6 + 3*c^2*d^4)*x^4 + 32*(b^2 + 2*a*c)*d^2 - 8*(8*a^2*d^6 + 4
*(b^2 + 2*a*c)*d^4 + 3*c^2*d^2)*x^2 - 4*(16*a^2*d^4 + 8*(b^2 + 2*a*c)*d^2 - (8*a
^2*d^6 + 4*(b^2 + 2*a*c)*d^4 + 3*c^2*d^2)*x^2 + 6*c^2)*sqrt(d*x + 1)*sqrt(-d*x +
1) + 24*c^2)*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((c*x**2+b*x+a)**2/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.292303, size = 243, normalized size = 1.46 \[ -\frac{{\left (48 \, a b d^{19} - 12 \, b^{2} d^{18} - 24 \, a c d^{18} + 48 \, b c d^{17} - 15 \, c^{2} d^{16} +{\left (12 \, b^{2} d^{18} + 24 \, a c d^{18} - 32 \, b c d^{17} + 27 \, c^{2} d^{16} + 2 \,{\left (3 \,{\left (d x + 1\right )} c^{2} d^{16} + 8 \, b c d^{17} - 9 \, c^{2} d^{16}\right )}{\left (d x + 1\right )}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 6 \,{\left (8 \, a^{2} d^{20} + 4 \, b^{2} d^{18} + 8 \, a c d^{18} + 3 \, c^{2} d^{16}\right )} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{344064 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(sqrt(d*x + 1)*sqrt(-d*x + 1)),x, algorithm="giac")

[Out]

-1/344064*((48*a*b*d^19 - 12*b^2*d^18 - 24*a*c*d^18 + 48*b*c*d^17 - 15*c^2*d^16
+ (12*b^2*d^18 + 24*a*c*d^18 - 32*b*c*d^17 + 27*c^2*d^16 + 2*(3*(d*x + 1)*c^2*d^
16 + 8*b*c*d^17 - 9*c^2*d^16)*(d*x + 1))*(d*x + 1))*sqrt(d*x + 1)*sqrt(-d*x + 1)
- 6*(8*a^2*d^20 + 4*b^2*d^18 + 8*a*c*d^18 + 3*c^2*d^16)*arcsin(1/2*sqrt(2)*sqrt
(d*x + 1)))/d