3.155 \(\int \frac{a+b x+c x^2}{x \sqrt{-1+d x} \sqrt{1+d x}} \, dx\)

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]

(c*Sqrt[-1 + d*x]*Sqrt[1 + d*x])/d^2 + (b*ArcCosh[d*x])/d + a*ArcTan[Sqrt[-1 + d
*x]*Sqrt[1 + d*x]]

_______________________________________________________________________________________

Rubi [B]  time = 0.360079, 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]

-((c*(1 - d^2*x^2))/(d^2*Sqrt[-1 + d*x]*Sqrt[1 + d*x])) + (a*Sqrt[-1 + d^2*x^2]*
ArcTan[Sqrt[-1 + d^2*x^2]])/(Sqrt[-1 + d*x]*Sqrt[1 + d*x]) + (b*Sqrt[-1 + d^2*x^
2]*ArcTanh[(d*x)/Sqrt[-1 + d^2*x^2]])/(d*Sqrt[-1 + d*x]*Sqrt[1 + d*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.1791, 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]

a*atan(sqrt(d*x - 1)*sqrt(d*x + 1)) + c*sqrt(d*x - 1)*sqrt(d*x + 1)/d**2 - c*aco
sh(d*x)/d**2 + (b*d + c)*acosh(d*x)/d**2

_______________________________________________________________________________________

Mathematica [A]  time = 0.107888, 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]

(c*Sqrt[-1 + d*x]*Sqrt[1 + d*x])/d^2 - a*ArcTan[1/(Sqrt[-1 + d*x]*Sqrt[1 + d*x])
] + (b*Log[d*x + Sqrt[-1 + d*x]*Sqrt[1 + d*x]])/d

_______________________________________________________________________________________

Maple [C]  time = 0.027, 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]

(d*x-1)^(1/2)*(d*x+1)^(1/2)/d^2*(-csgn(d)*arctan(1/(d^2*x^2-1)^(1/2))*a*d^2+ln((
csgn(d)*(d^2*x^2-1)^(1/2)+d*x)*csgn(d))*b*d+csgn(d)*(d^2*x^2-1)^(1/2)*c)*csgn(d)
/(d^2*x^2-1)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.47292, 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]

-a*arcsin(1/(sqrt(d^2)*abs(x))) + b*log(2*d^2*x + 2*sqrt(d^2*x^2 - 1)*sqrt(d^2))
/sqrt(d^2) + sqrt(d^2*x^2 - 1)*c/d^2

_______________________________________________________________________________________

Fricas [A]  time = 0.234097, 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]

-(c*d^2*x^2 - sqrt(d*x + 1)*sqrt(d*x - 1)*c*d*x - 2*(a*d^3*x - sqrt(d*x + 1)*sqr
t(d*x - 1)*a*d^2)*arctan(-d*x + sqrt(d*x + 1)*sqrt(d*x - 1)) + (b*d^2*x - sqrt(d
*x + 1)*sqrt(d*x - 1)*b*d)*log(-d*x + sqrt(d*x + 1)*sqrt(d*x - 1)) - c)/(d^3*x -
sqrt(d*x + 1)*sqrt(d*x - 1)*d^2)

_______________________________________________________________________________________

Sympy [A]  time = 57.2018, 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]

-a*meijerg(((3/4, 5/4, 1), (1, 1, 3/2)), ((1/2, 3/4, 1, 5/4, 3/2), (0,)), 1/(d**
2*x**2))/(4*pi**(3/2)) + I*a*meijerg(((0, 1/4, 1/2, 3/4, 1, 1), ()), ((1/4, 3/4)
, (0, 1/2, 1/2, 0)), exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)) + b*meijerg(((
1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), 1/(d**2*x**2))/(4*
pi**(3/2)*d) - I*b*meijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ()), ((-1/4, 1/4), (-1
/2, 0, 0, 0)), exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*d) + c*meijerg(((-1/4
, 1/4), (0, 0, 1/2, 1)), ((-1/2, -1/4, 0, 1/4, 1/2, 0), ()), 1/(d**2*x**2))/(4*p
i**(3/2)*d**2) + I*c*meijerg(((-1, -3/4, -1/2, -1/4, 0, 1), ()), ((-3/4, -1/4),
(-1, -1/2, -1/2, 0)), exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*d**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225377, 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]

-2*a*arctan(1/2*(sqrt(d*x + 1) - sqrt(d*x - 1))^2) - b*ln((sqrt(d*x + 1) - sqrt(
d*x - 1))^2)/d + sqrt(d*x + 1)*sqrt(d*x - 1)*c/d^2