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

Optimal. Leaf size=79 \[ -\frac{\sqrt{1-d^2 x^2} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac{b \sin ^{-1}(d x)}{2 d^3}-\frac{c x^2 \sqrt{1-d^2 x^2}}{3 d^2} \]

[Out]

-(c*x^2*Sqrt[1 - d^2*x^2])/(3*d^2) - ((2*(2*c + 3*a*d^2) + 3*b*d^2*x)*Sqrt[1 - d
^2*x^2])/(6*d^4) + (b*ArcSin[d*x])/(2*d^3)

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Rubi [A]  time = 0.252987, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{\sqrt{1-d^2 x^2} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac{b \sin ^{-1}(d x)}{2 d^3}-\frac{c x^2 \sqrt{1-d^2 x^2}}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

-(c*x^2*Sqrt[1 - d^2*x^2])/(3*d^2) - ((2*(2*c + 3*a*d^2) + 3*b*d^2*x)*Sqrt[1 - d
^2*x^2])/(6*d^4) + (b*ArcSin[d*x])/(2*d^3)

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Rubi in Sympy [A]  time = 26.4531, size = 68, normalized size = 0.86 \[ \frac{b \operatorname{asin}{\left (d x \right )}}{2 d^{3}} - \frac{c x^{2} \sqrt{- d^{2} x^{2} + 1}}{3 d^{2}} - \frac{\sqrt{- d^{2} x^{2} + 1} \left (6 a d^{2} + 3 b d^{2} x + 4 c\right )}{6 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*asin(d*x)/(2*d**3) - c*x**2*sqrt(-d**2*x**2 + 1)/(3*d**2) - sqrt(-d**2*x**2 +
1)*(6*a*d**2 + 3*b*d**2*x + 4*c)/(6*d**4)

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Mathematica [A]  time = 0.0918098, size = 57, normalized size = 0.72 \[ \frac{3 b d \sin ^{-1}(d x)-\sqrt{1-d^2 x^2} \left (3 d^2 (2 a+b x)+2 c \left (d^2 x^2+2\right )\right )}{6 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[1 - d^2*x^2]*(3*d^2*(2*a + b*x) + 2*c*(2 + d^2*x^2))) + 3*b*d*ArcSin[d*x
])/(6*d^4)

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Maple [C]  time = 0., size = 139, normalized size = 1.8 \[ -{\frac{{\it csgn} \left ( d \right ) }{6\,{d}^{4}}\sqrt{-dx+1}\sqrt{dx+1} \left ( 2\,{\it csgn} \left ( d \right ){x}^{2}c{d}^{2}\sqrt{-{d}^{2}{x}^{2}+1}+3\,{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}xb{d}^{2}+6\,{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}a{d}^{2}+4\,{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}c-3\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) bd \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(2*csgn(d)*x^2*c*d^2*(-d^2*x^2+1)^(1/2)+3*csgn
(d)*(-d^2*x^2+1)^(1/2)*x*b*d^2+6*csgn(d)*(-d^2*x^2+1)^(1/2)*a*d^2+4*csgn(d)*(-d^
2*x^2+1)^(1/2)*c-3*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*b*d)*csgn(d)/d^4/(-d^2
*x^2+1)^(1/2)

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Maxima [A]  time = 1.5115, size = 134, normalized size = 1.7 \[ -\frac{\sqrt{-d^{2} x^{2} + 1} c x^{2}}{3 \, d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1} b x}{2 \, d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1} a}{d^{2}} + \frac{b \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{2}} - \frac{2 \, \sqrt{-d^{2} x^{2} + 1} c}{3 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233505, size = 306, normalized size = 3.87 \[ -\frac{2 \, c d^{5} x^{6} + 3 \, b d^{5} x^{5} - 15 \, b d^{3} x^{3} - 12 \, a d^{3} x^{2} + 6 \,{\left (a d^{5} - c d^{3}\right )} x^{4} + 12 \, b d x + 3 \,{\left (2 \, c d^{3} x^{4} + 3 \, b d^{3} x^{3} + 4 \, a d^{3} x^{2} - 4 \, b d x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 6 \,{\left (3 \, b d^{2} x^{2} -{\left (b d^{2} x^{2} - 4 \, b\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 4 \, b\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{6 \,{\left (3 \, d^{5} x^{2} - 4 \, d^{3} -{\left (d^{5} x^{2} - 4 \, d^{3}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*c*d^5*x^6 + 3*b*d^5*x^5 - 15*b*d^3*x^3 - 12*a*d^3*x^2 + 6*(a*d^5 - c*d^3
)*x^4 + 12*b*d*x + 3*(2*c*d^3*x^4 + 3*b*d^3*x^3 + 4*a*d^3*x^2 - 4*b*d*x)*sqrt(d*
x + 1)*sqrt(-d*x + 1) + 6*(3*b*d^2*x^2 - (b*d^2*x^2 - 4*b)*sqrt(d*x + 1)*sqrt(-d
*x + 1) - 4*b)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(3*d^5*x^2 - 4*
d^3 - (d^5*x^2 - 4*d^3)*sqrt(d*x + 1)*sqrt(-d*x + 1))

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Sympy [A]  time = 108.768, size = 313, normalized size = 3.96 \[ - \frac{i a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{a{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{i b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} + \frac{b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{i c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} - \frac{c{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*a*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) - a*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*b*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) + b*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) - I*c*meijerg(((-5/4, -3/4), (-1, -1, -1/2, 1)), ((-3
/2, -5/4, -1, -3/4, -1/2, 0), ()), 1/(d**2*x**2))/(4*pi**(3/2)*d**4) - c*meijerg
(((-2, -7/4, -3/2, -5/4, -1, 1), ()), ((-7/4, -5/4), (-2, -3/2, -3/2, 0)), exp_p
olar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*d**4)

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GIAC/XCAS [A]  time = 0.215714, size = 123, normalized size = 1.56 \[ \frac{6 \, b d^{10} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right ) -{\left (6 \, a d^{11} - 3 \, b d^{10} + 6 \, c d^{9} +{\left (2 \,{\left (d x + 1\right )} c d^{9} + 3 \, b d^{10} - 4 \, c d^{9}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}}{3840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3840*(6*b*d^10*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)) - (6*a*d^11 - 3*b*d^10 + 6*c*
d^9 + (2*(d*x + 1)*c*d^9 + 3*b*d^10 - 4*c*d^9)*(d*x + 1))*sqrt(d*x + 1)*sqrt(-d*
x + 1))/d