Optimal. Leaf size=47 \[ \frac{\left (2 a c^2+b\right ) \cosh ^{-1}(c x)}{2 c^3}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{2 c^2} \]
[Out]
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Rubi [A] time = 0.0904182, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\left (2 a c^2+b\right ) \cosh ^{-1}(c x)}{2 c^3}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{2 c^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]
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Rubi in Sympy [A] time = 8.93401, size = 44, normalized size = 0.94 \[ \frac{a \operatorname{acosh}{\left (c x \right )}}{c} + \frac{b x \sqrt{c x - 1} \sqrt{c x + 1}}{2 c^{2}} + \frac{b \operatorname{acosh}{\left (c x \right )}}{2 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0543974, size = 63, normalized size = 1.34 \[ \frac{\left (2 a c^2+b\right ) \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )+b c x \sqrt{c x-1} \sqrt{c x+1}}{2 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]
[Out]
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Maple [C] time = 0.023, size = 103, normalized size = 2.2 \[{\frac{{\it csgn} \left ( c \right ) }{2\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1} \left ( bx\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) c+2\,a\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ){c}^{2}+b\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)/(c*x-1)^(1/2)/(c*x+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.4254, size = 120, normalized size = 2.55 \[ \frac{a \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}}} + \frac{\sqrt{c^{2} x^{2} - 1} b x}{2 \, c^{2}} + \frac{b \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{2 \, \sqrt{c^{2}} c^{2}} \]
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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242437, size = 220, normalized size = 4.68 \[ -\frac{2 \, b c^{4} x^{4} - 2 \, b c^{2} x^{2} -{\left (2 \, b c^{3} x^{3} - b c x\right )} \sqrt{c x + 1} \sqrt{c x - 1} -{\left (2 \, a c^{2} + 2 \,{\left (2 \, a c^{3} + b c\right )} \sqrt{c x + 1} \sqrt{c x - 1} x - 2 \,{\left (2 \, a c^{4} + b c^{2}\right )} x^{2} + b\right )} \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{2 \,{\left (2 \, c^{5} x^{2} - 2 \, \sqrt{c x + 1} \sqrt{c x - 1} c^{4} x - c^{3}\right )}} \]
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, algorithm="fricas")
[Out]
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Sympy [A] time = 37.0999, size = 182, normalized size = 3.87 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} - \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.246122, size = 96, normalized size = 2.04 \[ \frac{{\left ({\left (c x + 1\right )} b c^{4} - b c^{4}\right )} \sqrt{c x + 1} \sqrt{c x - 1} - 2 \,{\left (2 \, a c^{6} + b c^{4}\right )}{\rm ln}\left ({\left | -\sqrt{c x + 1} + \sqrt{c x - 1} \right |}\right )}{384 \, c} \]
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, algorithm="giac")
[Out]