Optimal. Leaf size=52 \[ \frac{\left (2 a d^2+c\right ) \cosh ^{-1}(d x)}{2 d^3}+\frac{\sqrt{d x-1} \sqrt{d x+1} (2 b+c x)}{2 d^2} \]
[Out]
_______________________________________________________________________________________
Rubi [B] time = 0.163503, antiderivative size = 135, normalized size of antiderivative = 2.6, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{\sqrt{d^2 x^2-1} \left (2 a d^2+c\right ) \tanh ^{-1}\left (\frac{d x}{\sqrt{d^2 x^2-1}}\right )}{2 d^3 \sqrt{d x-1} \sqrt{d x+1}}-\frac{b \left (1-d^2 x^2\right )}{d^2 \sqrt{d x-1} \sqrt{d x+1}}-\frac{c x \left (1-d^2 x^2\right )}{2 d^2 \sqrt{d x-1} \sqrt{d x+1}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)/(Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.1683, size = 87, normalized size = 1.67 \[ \frac{\left (2 b + c x\right ) \sqrt{d x - 1} \sqrt{d x + 1}}{2 d^{2}} + \frac{\left (2 a d^{2} + c\right ) \sqrt{d x - 1} \sqrt{d x + 1} \operatorname{atanh}{\left (\frac{d x}{\sqrt{d^{2} x^{2} - 1}} \right )}}{2 d^{3} \sqrt{d^{2} x^{2} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)/(d*x-1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0852009, size = 68, normalized size = 1.31 \[ \frac{\left (2 a d^2+c\right ) \log \left (d x+\sqrt{d x-1} \sqrt{d x+1}\right )+d \sqrt{d x-1} \sqrt{d x+1} (2 b+c x)}{2 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)/(Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0., size = 120, normalized size = 2.3 \[{\frac{{\it csgn} \left ( d \right ) }{2\,{d}^{3}}\sqrt{dx-1}\sqrt{dx+1} \left ( cx\sqrt{{d}^{2}{x}^{2}-1}{\it csgn} \left ( d \right ) d+2\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}+dx \right ){\it csgn} \left ( d \right ) \right ) a{d}^{2}+2\,\sqrt{{d}^{2}{x}^{2}-1}b{\it csgn} \left ( d \right ) d+c\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}+dx \right ){\it csgn} \left ( d \right ) \right ) \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)/(d*x-1)^(1/2)/(d*x+1)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.36094, size = 142, normalized size = 2.73 \[ \frac{a \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - 1} \sqrt{d^{2}}\right )}{\sqrt{d^{2}}} + \frac{\sqrt{d^{2} x^{2} - 1} c x}{2 \, d^{2}} + \frac{\sqrt{d^{2} x^{2} - 1} b}{d^{2}} + \frac{c \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - 1} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}} d^{2}} \]
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, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.236151, size = 259, normalized size = 4.98 \[ -\frac{2 \, c d^{4} x^{4} + 4 \, b d^{4} x^{3} - 2 \, c d^{2} x^{2} - 4 \, b d^{2} x -{\left (2 \, c d^{3} x^{3} + 4 \, b d^{3} x^{2} - c d x - 2 \, b d\right )} \sqrt{d x + 1} \sqrt{d x - 1} -{\left (2 \, a d^{2} + 2 \,{\left (2 \, a d^{3} + c d\right )} \sqrt{d x + 1} \sqrt{d x - 1} x - 2 \,{\left (2 \, a d^{4} + c d^{2}\right )} x^{2} + c\right )} \log \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right )}{2 \,{\left (2 \, d^{5} x^{2} - 2 \, \sqrt{d x + 1} \sqrt{d x - 1} d^{4} x - d^{3}\right )}} \]
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, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 54.7001, size = 277, normalized size = 5.33 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)/(d*x-1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.260496, size = 104, normalized size = 2. \[ \frac{{\left ({\left (d x + 1\right )} c d^{4} + 2 \, b d^{5} - c d^{4}\right )} \sqrt{d x + 1} \sqrt{d x - 1} - 2 \,{\left (2 \, a d^{6} + c d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{d x + 1} + \sqrt{d x - 1} \right |}\right )}{192 \, d} \]
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, algorithm="giac")
[Out]