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

Optimal. Leaf size=47 \[ \frac{\left (2 a c^2+b\right ) \cosh ^{-1}(c x)}{2 c^3}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{2 c^2} \]

[Out]

(b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ((b + 2*a*c^2)*ArcCosh[c*x])/(2*c^3
)

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Rubi [A]  time = 0.0904182, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\left (2 a c^2+b\right ) \cosh ^{-1}(c x)}{2 c^3}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ((b + 2*a*c^2)*ArcCosh[c*x])/(2*c^3
)

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Rubi in Sympy [A]  time = 8.93401, size = 44, normalized size = 0.94 \[ \frac{a \operatorname{acosh}{\left (c x \right )}}{c} + \frac{b x \sqrt{c x - 1} \sqrt{c x + 1}}{2 c^{2}} + \frac{b \operatorname{acosh}{\left (c x \right )}}{2 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*acosh(c*x)/c + b*x*sqrt(c*x - 1)*sqrt(c*x + 1)/(2*c**2) + b*acosh(c*x)/(2*c**3
)

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Mathematica [A]  time = 0.0543974, size = 63, normalized size = 1.34 \[ \frac{\left (2 a c^2+b\right ) \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )+b c x \sqrt{c x-1} \sqrt{c x+1}}{2 c^3} \]

Antiderivative was successfully verified.

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

[Out]

(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + (b + 2*a*c^2)*Log[c*x + Sqrt[-1 + c*x]*Sqr
t[1 + c*x]])/(2*c^3)

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Maple [C]  time = 0.023, size = 103, normalized size = 2.2 \[{\frac{{\it csgn} \left ( c \right ) }{2\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1} \left ( bx\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) c+2\,a\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ){c}^{2}+b\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(b*x*(c^2*x^2-1)^(1/2)*csgn(c)*c+2*a*ln((csgn(c)
*(c^2*x^2-1)^(1/2)+c*x)*csgn(c))*c^2+b*ln((csgn(c)*(c^2*x^2-1)^(1/2)+c*x)*csgn(c
)))/(c^2*x^2-1)^(1/2)/c^3*csgn(c)

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Maxima [A]  time = 1.4254, size = 120, normalized size = 2.55 \[ \frac{a \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}}} + \frac{\sqrt{c^{2} x^{2} - 1} b x}{2 \, c^{2}} + \frac{b \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{2 \, \sqrt{c^{2}} c^{2}} \]

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.242437, size = 220, normalized size = 4.68 \[ -\frac{2 \, b c^{4} x^{4} - 2 \, b c^{2} x^{2} -{\left (2 \, b c^{3} x^{3} - b c x\right )} \sqrt{c x + 1} \sqrt{c x - 1} -{\left (2 \, a c^{2} + 2 \,{\left (2 \, a c^{3} + b c\right )} \sqrt{c x + 1} \sqrt{c x - 1} x - 2 \,{\left (2 \, a c^{4} + b c^{2}\right )} x^{2} + b\right )} \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{2 \,{\left (2 \, c^{5} x^{2} - 2 \, \sqrt{c x + 1} \sqrt{c x - 1} c^{4} x - c^{3}\right )}} \]

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, algorithm="fricas")

[Out]

-1/2*(2*b*c^4*x^4 - 2*b*c^2*x^2 - (2*b*c^3*x^3 - b*c*x)*sqrt(c*x + 1)*sqrt(c*x -
1) - (2*a*c^2 + 2*(2*a*c^3 + b*c)*sqrt(c*x + 1)*sqrt(c*x - 1)*x - 2*(2*a*c^4 +
b*c^2)*x^2 + b)*log(-c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(2*c^5*x^2 - 2*sqrt(c*x
+ 1)*sqrt(c*x - 1)*c^4*x - c^3)

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Sympy [A]  time = 37.0999, size = 182, normalized size = 3.87 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} - \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.246122, size = 96, normalized size = 2.04 \[ \frac{{\left ({\left (c x + 1\right )} b c^{4} - b c^{4}\right )} \sqrt{c x + 1} \sqrt{c x - 1} - 2 \,{\left (2 \, a c^{6} + b c^{4}\right )}{\rm ln}\left ({\left | -\sqrt{c x + 1} + \sqrt{c x - 1} \right |}\right )}{384 \, c} \]

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, algorithm="giac")

[Out]

1/384*(((c*x + 1)*b*c^4 - b*c^4)*sqrt(c*x + 1)*sqrt(c*x - 1) - 2*(2*a*c^6 + b*c^
4)*ln(abs(-sqrt(c*x + 1) + sqrt(c*x - 1))))/c