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

Optimal. Leaf size=65 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (3 a c^2+2 b\right )}{3 c^4}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{3 c^2} \]

[Out]

((2*b + 3*a*c^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^4) + (b*x^2*Sqrt[-1 + c*x]*S
qrt[1 + c*x])/(3*c^2)

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Rubi [A]  time = 0.142211, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (3 a c^2+2 b\right )}{3 c^4}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{3 c^2} \]

Antiderivative was successfully verified.

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

[Out]

((2*b + 3*a*c^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^4) + (b*x^2*Sqrt[-1 + c*x]*S
qrt[1 + c*x])/(3*c^2)

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Rubi in Sympy [A]  time = 8.28265, size = 58, normalized size = 0.89 \[ \frac{b x^{2} \sqrt{c x - 1} \sqrt{c x + 1}}{3 c^{2}} + \frac{\left (3 a c^{2} + 2 b\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{3 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**2*sqrt(c*x - 1)*sqrt(c*x + 1)/(3*c**2) + (3*a*c**2 + 2*b)*sqrt(c*x - 1)*sqr
t(c*x + 1)/(3*c**4)

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Mathematica [A]  time = 0.0491001, size = 43, normalized size = 0.66 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (3 a c^2+b \left (c^2 x^2+2\right )\right )}{3 c^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3*a*c^2 + b*(2 + c^2*x^2)))/(3*c^4)

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Maple [A]  time = 0.006, size = 38, normalized size = 0.6 \[{\frac{b{x}^{2}{c}^{2}+3\,a{c}^{2}+2\,b}{3\,{c}^{4}}\sqrt{cx-1}\sqrt{cx+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.41744, size = 73, normalized size = 1.12 \[ \frac{\sqrt{c^{2} x^{2} - 1} b x^{2}}{3 \, c^{2}} + \frac{\sqrt{c^{2} x^{2} - 1} a}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1} b}{3 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.242697, size = 216, normalized size = 3.32 \[ -\frac{4 \, b c^{6} x^{6} + 3 \,{\left (4 \, a c^{6} + b c^{4}\right )} x^{4} + 3 \, a c^{2} - 3 \,{\left (5 \, a c^{4} + 3 \, b c^{2}\right )} x^{2} -{\left (4 \, b c^{5} x^{5} +{\left (12 \, a c^{5} + 5 \, b c^{3}\right )} x^{3} - 3 \,{\left (3 \, a c^{3} + 2 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 2 \, b}{3 \,{\left (4 \, c^{7} x^{3} - 3 \, c^{5} x -{\left (4 \, c^{6} x^{2} - c^{4}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(4*b*c^6*x^6 + 3*(4*a*c^6 + b*c^4)*x^4 + 3*a*c^2 - 3*(5*a*c^4 + 3*b*c^2)*x^
2 - (4*b*c^5*x^5 + (12*a*c^5 + 5*b*c^3)*x^3 - 3*(3*a*c^3 + 2*b*c)*x)*sqrt(c*x +
1)*sqrt(c*x - 1) + 2*b)/(4*c^7*x^3 - 3*c^5*x - (4*c^6*x^2 - c^4)*sqrt(c*x + 1)*s
qrt(c*x - 1))

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Sympy [A]  time = 57.3893, size = 202, normalized size = 3.11 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} + \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220132, size = 74, normalized size = 1.14 \[ \frac{{\left (3 \, a c^{11} + 3 \, b c^{9} +{\left ({\left (c x + 1\right )} b c^{9} - 2 \, b c^{9}\right )}{\left (c x + 1\right )}\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{1920 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*(3*a*c^11 + 3*b*c^9 + ((c*x + 1)*b*c^9 - 2*b*c^9)*(c*x + 1))*sqrt(c*x + 1
)*sqrt(c*x - 1)/c