3.257 \(\int \frac{a+b x^2}{x^4 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\)

Optimal. Leaf size=62 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (2 a c^2+3 b\right )}{3 x}+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.217287, antiderivative size = 62, 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 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (2 a c^2+3 b\right )}{3 x}+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/(x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.31846, size = 51, normalized size = 0.82 \[ \frac{a \sqrt{c x - 1} \sqrt{c x + 1}}{3 x^{3}} + \frac{\left (\frac{2 a c^{2}}{3} + b\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/x**4/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0517534, size = 42, normalized size = 0.68 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (2 a c^2 x^2+a+3 b x^2\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)/(x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]

[Out]

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

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Maple [A]  time = 0.008, size = 37, normalized size = 0.6 \[{\frac{2\,a{c}^{2}{x}^{2}+3\,b{x}^{2}+a}{3\,{x}^{3}}\sqrt{cx-1}\sqrt{cx+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.61251, size = 73, normalized size = 1.18 \[ \frac{2 \, \sqrt{c^{2} x^{2} - 1} a c^{2}}{3 \, x} + \frac{\sqrt{c^{2} x^{2} - 1} b}{x} + \frac{\sqrt{c^{2} x^{2} - 1} a}{3 \, x^{3}} \]

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^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.23031, size = 140, normalized size = 2.26 \[ \frac{6 \, b c^{2} x^{4} + 3 \,{\left (a c^{2} - b\right )} x^{2} - 3 \,{\left (2 \, b c x^{3} + a c x\right )} \sqrt{c x + 1} \sqrt{c x - 1} - a}{3 \,{\left (4 \, c^{3} x^{6} - 3 \, c x^{4} -{\left (4 \, c^{2} x^{5} - x^{3}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}} \]

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^4),x, algorithm="fricas")

[Out]

1/3*(6*b*c^2*x^4 + 3*(a*c^2 - b)*x^2 - 3*(2*b*c*x^3 + a*c*x)*sqrt(c*x + 1)*sqrt(
c*x - 1) - a)/(4*c^3*x^6 - 3*c*x^4 - (4*c^2*x^5 - x^3)*sqrt(c*x + 1)*sqrt(c*x -
1))

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Sympy [A]  time = 84.0966, size = 146, normalized size = 2.35 \[ - \frac{a c^{3}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{9}{4}, \frac{11}{4}, 1 & \frac{5}{2}, \frac{5}{2}, 3 \\2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4}, 3 & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i a c^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2}, 1 & \\\frac{7}{4}, \frac{9}{4} & \frac{3}{2}, 2, 2, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{b c{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i b c{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/x**4/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)

[Out]

-a*c**3*meijerg(((9/4, 11/4, 1), (5/2, 5/2, 3)), ((2, 9/4, 5/2, 11/4, 3), (0,)),
1/(c**2*x**2))/(4*pi**(3/2)) - I*a*c**3*meijerg(((3/2, 7/4, 2, 9/4, 5/2, 1), ()
), ((7/4, 9/4), (3/2, 2, 2, 0)), exp_polar(2*I*pi)/(c**2*x**2))/(4*pi**(3/2)) -
b*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*b*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))

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GIAC/XCAS [A]  time = 0.224913, size = 157, normalized size = 2.53 \[ \frac{8 \,{\left (3 \, b c^{2}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{8} + 24 \, a c^{4}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{4} + 24 \, b c^{2}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{4} + 32 \, a c^{4} + 48 \, b c^{2}\right )}}{3 \,{\left ({\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{4} + 4\right )}^{3} c} \]

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^4),x, algorithm="giac")

[Out]

8/3*(3*b*c^2*(sqrt(c*x + 1) - sqrt(c*x - 1))^8 + 24*a*c^4*(sqrt(c*x + 1) - sqrt(
c*x - 1))^4 + 24*b*c^2*(sqrt(c*x + 1) - sqrt(c*x - 1))^4 + 32*a*c^4 + 48*b*c^2)/
(((sqrt(c*x + 1) - sqrt(c*x - 1))^4 + 4)^3*c)