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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]