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

Optimal. Leaf size=46 \[ a \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c^2} \]

[Out]

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

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Rubi [A]  time = 0.211028, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ a \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.71313, size = 41, normalized size = 0.89 \[ a \operatorname{atan}{\left (\sqrt{c x - 1} \sqrt{c x + 1} \right )} + \frac{b \sqrt{c x - 1} \sqrt{c x + 1}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*atan(sqrt(c*x - 1)*sqrt(c*x + 1)) + b*sqrt(c*x - 1)*sqrt(c*x + 1)/c**2

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Mathematica [A]  time = 0.0651869, size = 47, normalized size = 1.02 \[ \frac{b \sqrt{c x-1} \sqrt{c x+1}}{c^2}-a \tan ^{-1}\left (\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 62, normalized size = 1.4 \[{\frac{1}{{c}^{2}} \left ( -a\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ){c}^{2}+b\sqrt{{c}^{2}{x}^{2}-1} \right ) \sqrt{cx-1}\sqrt{cx+1}{\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-a*arctan(1/(c^2*x^2-1)^(1/2))*c^2+b*(c^2*x^2-1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(
1/2)/(c^2*x^2-1)^(1/2)/c^2

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Maxima [A]  time = 1.70736, size = 42, normalized size = 0.91 \[ -a \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{\sqrt{c^{2} x^{2} - 1} b}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.242114, size = 149, normalized size = 3.24 \[ -\frac{b c^{2} x^{2} - \sqrt{c x + 1} \sqrt{c x - 1} b c x - 2 \,{\left (a c^{3} x - \sqrt{c x + 1} \sqrt{c x - 1} a c^{2}\right )} \arctan \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) - b}{c^{3} x - \sqrt{c x + 1} \sqrt{c x - 1} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*c^2*x^2 - sqrt(c*x + 1)*sqrt(c*x - 1)*b*c*x - 2*(a*c^3*x - sqrt(c*x + 1)*sqr
t(c*x - 1)*a*c^2)*arctan(-c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) - b)/(c^3*x - sqrt(
c*x + 1)*sqrt(c*x - 1)*c^2)

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Sympy [A]  time = 36.4893, size = 162, normalized size = 3.52 \[ - \frac{a{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} + \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218052, size = 61, normalized size = 1.33 \[ -2 \, a \arctan \left (\frac{1}{2} \,{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{2}\right ) + \frac{\sqrt{c x + 1} \sqrt{c x - 1} b}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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