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

Optimal. Leaf size=33 \[ \frac{a \sqrt{c x-1} \sqrt{c x+1}}{x}+\frac{b \cosh ^{-1}(c x)}{c} \]

[Out]

(a*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/x + (b*ArcCosh[c*x])/c

_______________________________________________________________________________________

Rubi [A]  time = 0.183341, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{a \sqrt{c x-1} \sqrt{c x+1}}{x}+\frac{b \cosh ^{-1}(c x)}{c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]

[Out]

(a*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/x + (b*ArcCosh[c*x])/c

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.5764, size = 27, normalized size = 0.82 \[ \frac{a \sqrt{c x - 1} \sqrt{c x + 1}}{x} + \frac{b \operatorname{acosh}{\left (c x \right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/x**2/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)

[Out]

a*sqrt(c*x - 1)*sqrt(c*x + 1)/x + b*acosh(c*x)/c

_______________________________________________________________________________________

Mathematica [A]  time = 0.0546701, size = 53, normalized size = 1.61 \[ \frac{a \sqrt{c x-1} \sqrt{c x+1}}{x}+\frac{b \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )}{c} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]

[Out]

(a*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/x + (b*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])
/c

_______________________________________________________________________________________

Maple [C]  time = 0.029, size = 77, normalized size = 2.3 \[{\frac{{\it csgn} \left ( c \right ) }{cx}\sqrt{cx-1}\sqrt{cx+1} \left ( a\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) c+b\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ) x \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.69408, size = 68, normalized size = 2.06 \[ \frac{b \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}}} + \frac{\sqrt{c^{2} x^{2} - 1} a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(sqrt(c*x + 1)*sqrt(c*x - 1)*x^2),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.239542, size = 109, normalized size = 3.3 \[ \frac{a c -{\left (b c x^{2} - \sqrt{c x + 1} \sqrt{c x - 1} b x\right )} \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{c^{2} x^{2} - \sqrt{c x + 1} \sqrt{c x - 1} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(sqrt(c*x + 1)*sqrt(c*x - 1)*x^2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 42.4737, size = 148, normalized size = 4.48 \[ - \frac{a c{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i a c{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/x**2/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223461, size = 78, normalized size = 2.36 \[ \frac{\frac{16 \, a c^{2}}{{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{4} + 4} - b{\rm ln}\left ({\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{4}\right )}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(sqrt(c*x + 1)*sqrt(c*x - 1)*x^2),x, algorithm="giac")

[Out]

1/2*(16*a*c^2/((sqrt(c*x + 1) - sqrt(c*x - 1))^4 + 4) - b*ln((sqrt(c*x + 1) - sq
rt(c*x - 1))^4))/c