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

Optimal. Leaf size=63 \[ \frac{\left (2 a d^2+c\right ) \sin ^{-1}(d x)}{2 d^3}-\frac{b \sqrt{1-d^2 x^2}}{d^2}-\frac{c x \sqrt{1-d^2 x^2}}{2 d^2} \]

[Out]

-((b*Sqrt[1 - d^2*x^2])/d^2) - (c*x*Sqrt[1 - d^2*x^2])/(2*d^2) + ((c + 2*a*d^2)*
ArcSin[d*x])/(2*d^3)

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Rubi [A]  time = 0.13165, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (2 a d^2+c\right ) \sin ^{-1}(d x)}{2 d^3}-\frac{b \sqrt{1-d^2 x^2}}{d^2}-\frac{c x \sqrt{1-d^2 x^2}}{2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*Sqrt[1 - d^2*x^2])/d^2) - (c*x*Sqrt[1 - d^2*x^2])/(2*d^2) + ((c + 2*a*d^2)*
ArcSin[d*x])/(2*d^3)

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Rubi in Sympy [A]  time = 16.3064, size = 41, normalized size = 0.65 \[ - \frac{\left (2 b + c x\right ) \sqrt{- d^{2} x^{2} + 1}}{2 d^{2}} + \frac{\left (2 a d^{2} + c\right ) \operatorname{asin}{\left (d x \right )}}{2 d^{3}} \]

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]

-(2*b + c*x)*sqrt(-d**2*x**2 + 1)/(2*d**2) + (2*a*d**2 + c)*asin(d*x)/(2*d**3)

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Mathematica [A]  time = 0.0606096, size = 45, normalized size = 0.71 \[ \frac{\left (2 a d^2+c\right ) \sin ^{-1}(d x)-d \sqrt{1-d^2 x^2} (2 b+c x)}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(-(d*(2*b + c*x)*Sqrt[1 - d^2*x^2]) + (c + 2*a*d^2)*ArcSin[d*x])/(2*d^3)

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Maple [C]  time = 0.023, size = 117, normalized size = 1.9 \[ -{\frac{{\it csgn} \left ( d \right ) }{2\,{d}^{3}}\sqrt{-dx+1}\sqrt{dx+1} \left ( cx\sqrt{-{d}^{2}{x}^{2}+1}{\it csgn} \left ( d \right ) d+2\,\sqrt{-{d}^{2}{x}^{2}+1}b{\it csgn} \left ( d \right ) d-2\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) a{d}^{2}-c\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]

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]

-1/2*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(c*x*(-d^2*x^2+1)^(1/2)*csgn(d)*d+2*(-d^2*x^2+
1)^(1/2)*b*csgn(d)*d-2*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a*d^2-c*arctan(csg
n(d)*d*x/(-d^2*x^2+1)^(1/2)))*csgn(d)/(-d^2*x^2+1)^(1/2)/d^3

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Maxima [A]  time = 0.767701, size = 105, normalized size = 1.67 \[ \frac{a \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}}} - \frac{\sqrt{-d^{2} x^{2} + 1} c x}{2 \, d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1} b}{d^{2}} + \frac{c \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{2}} \]

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]

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

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Fricas [A]  time = 0.272735, size = 244, normalized size = 3.87 \[ \frac{2 \, c d^{3} x^{3} + 2 \, b d^{3} x^{2} - 2 \, c d x -{\left (c d^{3} x^{3} + 2 \, b d^{3} x^{2} - 2 \, c d x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 2 \,{\left (4 \, a d^{2} -{\left (2 \, a d^{4} + c d^{2}\right )} x^{2} - 2 \,{\left (2 \, a d^{2} + c\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 2 \, c\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{2 \,{\left (d^{5} x^{2} + 2 \, \sqrt{d x + 1} \sqrt{-d x + 1} d^{3} - 2 \, d^{3}\right )}} \]

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]

1/2*(2*c*d^3*x^3 + 2*b*d^3*x^2 - 2*c*d*x - (c*d^3*x^3 + 2*b*d^3*x^2 - 2*c*d*x)*s
qrt(d*x + 1)*sqrt(-d*x + 1) + 2*(4*a*d^2 - (2*a*d^4 + c*d^2)*x^2 - 2*(2*a*d^2 +
c)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*c)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)
/(d*x)))/(d^5*x^2 + 2*sqrt(d*x + 1)*sqrt(-d*x + 1)*d^3 - 2*d^3)

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Sympy [A]  time = 55.1506, size = 282, normalized size = 4.48 \[ - \frac{i a{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} + \frac{a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{b{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{i c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} + \frac{c{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} \]

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]

-I*a*meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), 1/(
d**2*x**2))/(4*pi**(3/2)*d) + a*meijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ()), ((-1
/4, 1/4), (-1/2, 0, 0, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*d) - I*
b*meijerg(((-1/4, 1/4), (0, 0, 1/2, 1)), ((-1/2, -1/4, 0, 1/4, 1/2, 0), ()), 1/(
d**2*x**2))/(4*pi**(3/2)*d**2) - b*meijerg(((-1, -3/4, -1/2, -1/4, 0, 1), ()), (
(-3/4, -1/4), (-1, -1/2, -1/2, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/2)
*d**2) - I*c*meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4,
0, 0), ()), 1/(d**2*x**2))/(4*pi**(3/2)*d**3) + c*meijerg(((-3/2, -5/4, -1, -3/
4, -1/2, 1), ()), ((-5/4, -3/4), (-3/2, -1, -1, 0)), exp_polar(-2*I*pi)/(d**2*x*
*2))/(4*pi**(3/2)*d**3)

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GIAC/XCAS [A]  time = 0.285598, size = 97, normalized size = 1.54 \[ -\frac{{\left ({\left (d x + 1\right )} c d^{4} + 2 \, b d^{5} - c d^{4}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 2 \,{\left (2 \, a d^{6} + c d^{4}\right )} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{192 \, d} \]

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]

-1/192*(((d*x + 1)*c*d^4 + 2*b*d^5 - c*d^4)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 2*(2*
a*d^6 + c*d^4)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))/d