3.72 \(\int \frac{a+b x+c x^2}{x^3 \sqrt{-1+d x} \sqrt{1+d x}} \, dx\)

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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

-1/2*(2*a*d^3*x^3 - 2*b*d*x^2 - 2*a*d*x - (2*a*d^2*x^2 - 2*b*x - a)*sqrt(d*x + 1
)*sqrt(d*x - 1) + 2*(2*(a*d^3 + 2*c*d)*sqrt(d*x + 1)*sqrt(d*x - 1)*x^3 - 2*(a*d^
4 + 2*c*d^2)*x^4 + (a*d^2 + 2*c)*x^2)*arctan(-d*x + sqrt(d*x + 1)*sqrt(d*x - 1))
)/(2*d^2*x^4 - 2*sqrt(d*x + 1)*sqrt(d*x - 1)*d*x^3 - x^2)

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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]

-a*d**2*meijerg(((7/4, 9/4, 1), (2, 2, 5/2)), ((3/2, 7/4, 2, 9/4, 5/2), (0,)), 1
/(d**2*x**2))/(4*pi**(3/2)) + I*a*d**2*meijerg(((1, 5/4, 3/2, 7/4, 2, 1), ()), (
(5/4, 7/4), (1, 3/2, 3/2, 0)), exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)) - b*
d*meijerg(((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7/4, 2), (0,)), 1/(d**2
*x**2))/(4*pi**(3/2)) - I*b*d*meijerg(((1/2, 3/4, 1, 5/4, 3/2, 1), ()), ((3/4, 5
/4), (1/2, 1, 1, 0)), exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)) - c*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*c*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))

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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]

-((a*d^3 + 2*c*d)*arctan(1/2*(sqrt(d*x + 1) - sqrt(d*x - 1))^2) + 2*(a*d^3*(sqrt
(d*x + 1) - sqrt(d*x - 1))^6 - 4*b*d^2*(sqrt(d*x + 1) - sqrt(d*x - 1))^4 - 4*a*d
^3*(sqrt(d*x + 1) - sqrt(d*x - 1))^2 - 16*b*d^2)/((sqrt(d*x + 1) - sqrt(d*x - 1)
)^4 + 4)^2)/d