Optimal. Leaf size=55 \[ a \tan ^{-1}\left (\sqrt{d x-1} \sqrt{d x+1}\right )+\frac{b \cosh ^{-1}(d x)}{d}+\frac{c \sqrt{d x-1} \sqrt{d x+1}}{d^2} \]
[Out]
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Rubi [B] time = 0.354064, antiderivative size = 135, normalized size of antiderivative = 2.45, number of steps used = 8, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{a \sqrt{d^2 x^2-1} \tan ^{-1}\left (\sqrt{d^2 x^2-1}\right )}{\sqrt{d x-1} \sqrt{d x+1}}+\frac{b \sqrt{d^2 x^2-1} \tanh ^{-1}\left (\frac{d x}{\sqrt{d^2 x^2-1}}\right )}{d \sqrt{d x-1} \sqrt{d x+1}}-\frac{c \left (1-d^2 x^2\right )}{d^2 \sqrt{d x-1} \sqrt{d x+1}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)/(x*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 21.2297, size = 65, normalized size = 1.18 \[ a \operatorname{atan}{\left (\sqrt{d x - 1} \sqrt{d x + 1} \right )} + \frac{c \sqrt{d x - 1} \sqrt{d x + 1}}{d^{2}} - \frac{c \operatorname{acosh}{\left (d x \right )}}{d^{2}} + \frac{\left (b d + c\right ) \operatorname{acosh}{\left (d x \right )}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)/x/(d*x-1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.107443, size = 76, normalized size = 1.38 \[ -a \tan ^{-1}\left (\frac{1}{\sqrt{d x-1} \sqrt{d x+1}}\right )+\frac{b \log \left (d x+\sqrt{d x-1} \sqrt{d x+1}\right )}{d}+\frac{c \sqrt{d x-1} \sqrt{d x+1}}{d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)/(x*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]
[Out]
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Maple [C] time = 0., size = 93, normalized size = 1.7 \[{\frac{{\it csgn} \left ( d \right ) }{{d}^{2}}\sqrt{dx-1}\sqrt{dx+1} \left ( -{\it csgn} \left ( d \right ) \arctan \left ({\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}} \right ) a{d}^{2}+\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}+dx \right ){\it csgn} \left ( d \right ) \right ) bd+{\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}c \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)/x/(d*x-1)^(1/2)/(d*x+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.49751, size = 86, normalized size = 1.56 \[ -a \arcsin \left (\frac{1}{\sqrt{d^{2}}{\left | x \right |}}\right ) + \frac{b \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - 1} \sqrt{d^{2}}\right )}{\sqrt{d^{2}}} + \frac{\sqrt{d^{2} x^{2} - 1} c}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(sqrt(d*x + 1)*sqrt(d*x - 1)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.246562, size = 212, normalized size = 3.85 \[ -\frac{c d^{2} x^{2} - \sqrt{d x + 1} \sqrt{d x - 1} c d x - 2 \,{\left (a d^{3} x - \sqrt{d x + 1} \sqrt{d x - 1} a d^{2}\right )} \arctan \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right ) +{\left (b d^{2} x - \sqrt{d x + 1} \sqrt{d x - 1} b d\right )} \log \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right ) - c}{d^{3} x - \sqrt{d x + 1} \sqrt{d x - 1} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(sqrt(d*x + 1)*sqrt(d*x - 1)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 57.3745, size = 240, normalized size = 4.36 \[ - \frac{a{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} + \frac{c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)/x/(d*x-1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.22993, size = 96, normalized size = 1.75 \[ -2 \, a \arctan \left (\frac{1}{2} \,{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2}\right ) - \frac{b{\rm ln}\left ({\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2}\right )}{d} + \frac{\sqrt{d x + 1} \sqrt{d x - 1} c}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(sqrt(d*x + 1)*sqrt(d*x - 1)*x),x, algorithm="giac")
[Out]