3.53 \(\int \frac{a+b x+c x^2}{x^3 \sqrt{1-d x} \sqrt{1+d x}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{1}{2} \left (a d^2+2 c\right ) \tanh ^{-1}\left (\sqrt{1-d^2 x^2}\right )-\frac{a \sqrt{1-d^2 x^2}}{2 x^2}-\frac{b \sqrt{1-d^2 x^2}}{x} \]

[Out]

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

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Rubi [A]  time = 0.345256, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{1}{2} \left (a d^2+2 c\right ) \tanh ^{-1}\left (\sqrt{1-d^2 x^2}\right )-\frac{a \sqrt{1-d^2 x^2}}{2 x^2}-\frac{b \sqrt{1-d^2 x^2}}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.871, size = 56, normalized size = 0.79 \[ - \frac{a \sqrt{- d^{2} x^{2} + 1}}{2 x^{2}} - \frac{b \sqrt{- d^{2} x^{2} + 1}}{x} - \left (\frac{a d^{2}}{2} + c\right ) \operatorname{atanh}{\left (\sqrt{- d^{2} x^{2} + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)/x**3/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

-a*sqrt(-d**2*x**2 + 1)/(2*x**2) - b*sqrt(-d**2*x**2 + 1)/x - (a*d**2/2 + c)*ata
nh(sqrt(-d**2*x**2 + 1))

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Mathematica [A]  time = 0.104087, size = 70, normalized size = 0.99 \[ \frac{1}{2} \left (-\frac{\sqrt{1-d^2 x^2} (a+2 b x)}{x^2}-\left (a d^2+2 c\right ) \log \left (\sqrt{1-d^2 x^2}+1\right )+\log (x) \left (a d^2+2 c\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-(((a + 2*b*x)*Sqrt[1 - d^2*x^2])/x^2) + (2*c + a*d^2)*Log[x] - (2*c + a*d^2)*L
og[1 + Sqrt[1 - d^2*x^2]])/2

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Maple [C]  time = 0., size = 108, normalized size = 1.5 \[ -{\frac{ \left ({\it csgn} \left ( d \right ) \right ) ^{2}}{2\,{x}^{2}}\sqrt{-dx+1}\sqrt{dx+1} \left ({\it Artanh} \left ({\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){x}^{2}a{d}^{2}+2\,{\it Artanh} \left ({\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){x}^{2}c+2\,bx\sqrt{-{d}^{2}{x}^{2}+1}+\sqrt{-{d}^{2}{x}^{2}+1}a \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/x^3/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x)

[Out]

-1/2*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*csgn(d)^2*(arctanh(1/(-d^2*x^2+1)^(1/2))*x^2*a
*d^2+2*arctanh(1/(-d^2*x^2+1)^(1/2))*x^2*c+2*b*x*(-d^2*x^2+1)^(1/2)+(-d^2*x^2+1)
^(1/2)*a)/(-d^2*x^2+1)^(1/2)/x^2

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Maxima [A]  time = 1.50378, size = 132, normalized size = 1.86 \[ -\frac{1}{2} \, a d^{2} \log \left (\frac{2 \, \sqrt{-d^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - c \log \left (\frac{2 \, \sqrt{-d^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{\sqrt{-d^{2} x^{2} + 1} b}{x} - \frac{\sqrt{-d^{2} x^{2} + 1} a}{2 \, x^{2}} \]

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^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.227784, size = 255, normalized size = 3.59 \[ \frac{4 \, b d^{2} x^{3} + 2 \, a d^{2} x^{2} -{\left (2 \, b d^{2} x^{3} + a d^{2} x^{2} - 4 \, b x - 2 \, a\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 4 \, b x +{\left ({\left (a d^{4} + 2 \, c d^{2}\right )} x^{4} + 2 \,{\left (a d^{2} + 2 \, c\right )} \sqrt{d x + 1} \sqrt{-d x + 1} x^{2} - 2 \,{\left (a d^{2} + 2 \, c\right )} x^{2}\right )} \log \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{x}\right ) - 2 \, a}{2 \,{\left (d^{2} x^{4} + 2 \, \sqrt{d x + 1} \sqrt{-d x + 1} x^{2} - 2 \, x^{2}\right )}} \]

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^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 81.9541, size = 218, normalized size = 3.07 \[ \frac{i a d^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{a d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i b d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{b d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i c{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{c{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)/x**3/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

I*a*d**2*meijerg(((7/4, 9/4, 1), (2, 2, 5/2)), ((3/2, 7/4, 2, 9/4, 5/2), (0,)),
1/(d**2*x**2))/(4*pi**(3/2)) - a*d**2*meijerg(((1, 5/4, 3/2, 7/4, 2, 1), ()), ((
5/4, 7/4), (1, 3/2, 3/2, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/2)) + I*
b*d*meijerg(((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7/4, 2), (0,)), 1/(d*
*2*x**2))/(4*pi**(3/2)) + b*d*meijerg(((1/2, 3/4, 1, 5/4, 3/2, 1), ()), ((3/4, 5
/4), (1/2, 1, 1, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/2)) + I*c*meijer
g(((3/4, 5/4, 1), (1, 1, 3/2)), ((1/2, 3/4, 1, 5/4, 3/2), (0,)), 1/(d**2*x**2))/
(4*pi**(3/2)) - c*meijerg(((0, 1/4, 1/2, 3/4, 1, 1), ()), ((1/4, 3/4), (0, 1/2,
1/2, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/2))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError