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

Optimal. Leaf size=60 \[ \frac{1}{2} \left (a c^2+2 b\right ) \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.215672, antiderivative size = 60, 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 \[ \frac{1}{2} \left (a c^2+2 b\right ) \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.67229, size = 49, normalized size = 0.82 \[ \frac{a \sqrt{c x - 1} \sqrt{c x + 1}}{2 x^{2}} + \left (\frac{a c^{2}}{2} + b\right ) \operatorname{atan}{\left (\sqrt{c x - 1} \sqrt{c x + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0891357, size = 61, normalized size = 1.02 \[ \frac{1}{2} \left (-a c^2-2 b\right ) \tan ^{-1}\left (\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right )+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.028, size = 84, normalized size = 1.4 \[ -{\frac{1}{2\,{x}^{2}}\sqrt{cx-1}\sqrt{cx+1} \left ( \arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ) a{c}^{2}{x}^{2}+2\,\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ) b{x}^{2}-a\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.54408, size = 66, normalized size = 1.1 \[ -\frac{1}{2} \, a c^{2} \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) - b \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{\sqrt{c^{2} x^{2} - 1} a}{2 \, x^{2}} \]

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^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.242027, size = 223, normalized size = 3.72 \[ -\frac{2 \, a c^{3} x^{3} - 2 \, a c x -{\left (2 \, a c^{2} x^{2} - a\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 2 \,{\left (2 \,{\left (a c^{3} + 2 \, b c\right )} \sqrt{c x + 1} \sqrt{c x - 1} x^{3} - 2 \,{\left (a c^{4} + 2 \, b c^{2}\right )} x^{4} +{\left (a c^{2} + 2 \, b\right )} x^{2}\right )} \arctan \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{2 \,{\left (2 \, c^{2} x^{4} - 2 \, \sqrt{c x + 1} \sqrt{c x - 1} c x^{3} - x^{2}\right )}} \]

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^3),x, algorithm="fricas")

[Out]

-1/2*(2*a*c^3*x^3 - 2*a*c*x - (2*a*c^2*x^2 - a)*sqrt(c*x + 1)*sqrt(c*x - 1) + 2*
(2*(a*c^3 + 2*b*c)*sqrt(c*x + 1)*sqrt(c*x - 1)*x^3 - 2*(a*c^4 + 2*b*c^2)*x^4 + (
a*c^2 + 2*b)*x^2)*arctan(-c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(2*c^2*x^4 - 2*sqr
t(c*x + 1)*sqrt(c*x - 1)*c*x^3 - x^2)

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Sympy [A]  time = 56.1081, size = 141, normalized size = 2.35 \[ - \frac{a c^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i a c^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{b{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22855, size = 154, normalized size = 2.57 \[ -\frac{{\left (a c^{3} + 2 \, b c\right )} \arctan \left (\frac{1}{2} \,{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{2}\right ) + \frac{2 \,{\left (a c^{3}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{6} - 4 \, a c^{3}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{2}\right )}}{{\left ({\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{4} + 4\right )}^{2}}}{c} \]

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^3),x, algorithm="giac")

[Out]

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