Optimal. Leaf size=83 \[ \frac{1}{2} \left (a d^2+2 c\right ) \tan ^{-1}\left (\sqrt{d x-1} \sqrt{d x+1}\right )+\frac{a \sqrt{d x-1} \sqrt{d x+1}}{2 x^2}+\frac{b \sqrt{d x-1} \sqrt{d x+1}}{x} \]
[Out]
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Rubi [A] time = 0.348408, antiderivative size = 129, normalized size of antiderivative = 1.55, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{\sqrt{d^2 x^2-1} \left (a d^2+2 c\right ) \tan ^{-1}\left (\sqrt{d^2 x^2-1}\right )}{2 \sqrt{d x-1} \sqrt{d x+1}}-\frac{a \left (1-d^2 x^2\right )}{2 x^2 \sqrt{d x-1} \sqrt{d x+1}}-\frac{b \left (1-d^2 x^2\right )}{x \sqrt{d x-1} \sqrt{d x+1}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)/(x^3*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 20.5309, size = 88, normalized size = 1.06 \[ \frac{a d^{2} \operatorname{atan}{\left (\sqrt{d x - 1} \sqrt{d x + 1} \right )}}{2} + \frac{a \sqrt{d x - 1} \sqrt{d x + 1}}{2 x^{2}} + \frac{b \sqrt{d x - 1} \sqrt{d x + 1}}{x} + c \operatorname{atan}{\left (\sqrt{d x - 1} \sqrt{d x + 1} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)/x**3/(d*x-1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.111744, size = 64, normalized size = 0.77 \[ \frac{1}{2} \left (\frac{\sqrt{d x-1} \sqrt{d x+1} (a+2 b x)}{x^2}-\left (a d^2+2 c\right ) \tan ^{-1}\left (\frac{1}{\sqrt{d x-1} \sqrt{d x+1}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)/(x^3*Sqrt[-1 + d*x]*Sqrt[1 + d*x]),x]
[Out]
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Maple [C] time = 0., size = 103, normalized size = 1.2 \[ -{\frac{ \left ({\it csgn} \left ( d \right ) \right ) ^{2}}{2\,{x}^{2}}\sqrt{dx-1}\sqrt{dx+1} \left ( \arctan \left ({\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}} \right ){x}^{2}a{d}^{2}+2\,\arctan \left ({\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}} \right ){x}^{2}c-2\,bx\sqrt{{d}^{2}{x}^{2}-1}-a\sqrt{{d}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)/x^3/(d*x-1)^(1/2)/(d*x+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.49797, size = 88, normalized size = 1.06 \[ -\frac{1}{2} \, a d^{2} \arcsin \left (\frac{1}{\sqrt{d^{2}}{\left | x \right |}}\right ) - c \arcsin \left (\frac{1}{\sqrt{d^{2}}{\left | x \right |}}\right ) + \frac{\sqrt{d^{2} x^{2} - 1} b}{x} + \frac{\sqrt{d^{2} x^{2} - 1} a}{2 \, x^{2}} \]
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^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237194, size = 238, normalized size = 2.87 \[ -\frac{2 \, a d^{3} x^{3} - 2 \, b d x^{2} - 2 \, a d x -{\left (2 \, a d^{2} x^{2} - 2 \, b x - a\right )} \sqrt{d x + 1} \sqrt{d x - 1} + 2 \,{\left (2 \,{\left (a d^{3} + 2 \, c d\right )} \sqrt{d x + 1} \sqrt{d x - 1} x^{3} - 2 \,{\left (a d^{4} + 2 \, c d^{2}\right )} x^{4} +{\left (a d^{2} + 2 \, c\right )} x^{2}\right )} \arctan \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right )}{2 \,{\left (2 \, d^{2} x^{4} - 2 \, \sqrt{d x + 1} \sqrt{d x - 1} d x^{3} - x^{2}\right )}} \]
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^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 82.1259, size = 212, normalized size = 2.55 \[ - \frac{a d^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i a d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{b d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i b d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{c{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)/x**3/(d*x-1)**(1/2)/(d*x+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.235281, size = 196, normalized size = 2.36 \[ -\frac{{\left (a d^{3} + 2 \, c d\right )} \arctan \left (\frac{1}{2} \,{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2}\right ) + \frac{2 \,{\left (a d^{3}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{6} - 4 \, b d^{2}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{4} - 4 \, a d^{3}{\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{2} - 16 \, b d^{2}\right )}}{{\left ({\left (\sqrt{d x + 1} - \sqrt{d x - 1}\right )}^{4} + 4\right )}^{2}}}{d} \]
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^3),x, algorithm="giac")
[Out]