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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]