3.258 \(\int \frac{a+b x^2}{x^5 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\)

Optimal. Leaf size=99 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (3 a c^2+4 b\right )}{8 x^2}+\frac{1}{8} c^2 \left (3 a c^2+4 b\right ) \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{4 x^4} \]

[Out]

(a*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*x^4) + ((4*b + 3*a*c^2)*Sqrt[-1 + c*x]*Sqrt[
1 + c*x])/(8*x^2) + (c^2*(4*b + 3*a*c^2)*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/8

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Rubi [A]  time = 0.285726, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (3 a c^2+4 b\right )}{8 x^2}+\frac{1}{8} c^2 \left (3 a c^2+4 b\right ) \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/(x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]

[Out]

(a*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*x^4) + ((4*b + 3*a*c^2)*Sqrt[-1 + c*x]*Sqrt[
1 + c*x])/(8*x^2) + (c^2*(4*b + 3*a*c^2)*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/8

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Rubi in Sympy [A]  time = 13.7745, size = 88, normalized size = 0.89 \[ \frac{a \sqrt{c x - 1} \sqrt{c x + 1}}{4 x^{4}} + \frac{c^{2} \left (\frac{3 a c^{2}}{4} + b\right ) \operatorname{atan}{\left (\sqrt{c x - 1} \sqrt{c x + 1} \right )}}{2} + \frac{\left (\frac{3 a c^{2}}{8} + \frac{b}{2}\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/x**5/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)

[Out]

a*sqrt(c*x - 1)*sqrt(c*x + 1)/(4*x**4) + c**2*(3*a*c**2/4 + b)*atan(sqrt(c*x - 1
)*sqrt(c*x + 1))/2 + (3*a*c**2/8 + b/2)*sqrt(c*x - 1)*sqrt(c*x + 1)/x**2

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Mathematica [A]  time = 0.132844, size = 80, normalized size = 0.81 \[ \frac{1}{8} \left (\frac{\sqrt{c x-1} \sqrt{c x+1} \left (x^2 \left (3 a c^2+4 b\right )+2 a\right )}{x^4}-c^2 \left (3 a c^2+4 b\right ) \tan ^{-1}\left (\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)/(x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]

[Out]

((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*a + (4*b + 3*a*c^2)*x^2))/x^4 - c^2*(4*b + 3*a
*c^2)*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])])/8

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Maple [A]  time = 0.029, size = 125, normalized size = 1.3 \[ -{\frac{1}{8\,{x}^{4}}\sqrt{cx-1}\sqrt{cx+1} \left ( 3\,{c}^{4}\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ) a{x}^{4}+4\,{c}^{2}\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ) b{x}^{4}-3\,\sqrt{{c}^{2}{x}^{2}-1}a{c}^{2}{x}^{2}-4\,\sqrt{{c}^{2}{x}^{2}-1}b{x}^{2}-2\,a\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/x^5/(c*x-1)^(1/2)/(c*x+1)^(1/2),x)

[Out]

-1/8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(3*c^4*arctan(1/(c^2*x^2-1)^(1/2))*a*x^4+4*c^2*
arctan(1/(c^2*x^2-1)^(1/2))*b*x^4-3*(c^2*x^2-1)^(1/2)*a*c^2*x^2-4*(c^2*x^2-1)^(1
/2)*b*x^2-2*a*(c^2*x^2-1)^(1/2))/(c^2*x^2-1)^(1/2)/x^4

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Maxima [A]  time = 1.58707, size = 120, normalized size = 1.21 \[ -\frac{3}{8} \, a c^{4} \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) - \frac{1}{2} \, b c^{2} \arcsin \left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{3 \, \sqrt{c^{2} x^{2} - 1} a c^{2}}{8 \, x^{2}} + \frac{\sqrt{c^{2} x^{2} - 1} b}{2 \, x^{2}} + \frac{\sqrt{c^{2} x^{2} - 1} a}{4 \, x^{4}} \]

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^5),x, algorithm="maxima")

[Out]

-3/8*a*c^4*arcsin(1/(sqrt(c^2)*abs(x))) - 1/2*b*c^2*arcsin(1/(sqrt(c^2)*abs(x)))
+ 3/8*sqrt(c^2*x^2 - 1)*a*c^2/x^2 + 1/2*sqrt(c^2*x^2 - 1)*b/x^2 + 1/4*sqrt(c^2*
x^2 - 1)*a/x^4

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Fricas [A]  time = 0.244998, size = 423, normalized size = 4.27 \[ -\frac{8 \,{\left (3 \, a c^{7} + 4 \, b c^{5}\right )} x^{7} - 4 \,{\left (5 \, a c^{5} + 12 \, b c^{3}\right )} x^{5} - 4 \,{\left (3 \, a c^{3} - 4 \, b c\right )} x^{3} + 8 \, a c x -{\left (8 \,{\left (3 \, a c^{6} + 4 \, b c^{4}\right )} x^{6} - 8 \,{\left (a c^{4} + 4 \, b c^{2}\right )} x^{4} -{\left (13 \, a c^{2} - 4 \, b\right )} x^{2} + 2 \, a\right )} \sqrt{c x + 1} \sqrt{c x - 1} - 2 \,{\left (8 \,{\left (3 \, a c^{8} + 4 \, b c^{6}\right )} x^{8} - 8 \,{\left (3 \, a c^{6} + 4 \, b c^{4}\right )} x^{6} +{\left (3 \, a c^{4} + 4 \, b c^{2}\right )} x^{4} - 4 \,{\left (2 \,{\left (3 \, a c^{7} + 4 \, b c^{5}\right )} x^{7} -{\left (3 \, a c^{5} + 4 \, b c^{3}\right )} x^{5}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \arctan \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{8 \,{\left (8 \, c^{4} x^{8} - 8 \, c^{2} x^{6} + x^{4} - 4 \,{\left (2 \, c^{3} x^{7} - c x^{5}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}} \]

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^5),x, algorithm="fricas")

[Out]

-1/8*(8*(3*a*c^7 + 4*b*c^5)*x^7 - 4*(5*a*c^5 + 12*b*c^3)*x^5 - 4*(3*a*c^3 - 4*b*
c)*x^3 + 8*a*c*x - (8*(3*a*c^6 + 4*b*c^4)*x^6 - 8*(a*c^4 + 4*b*c^2)*x^4 - (13*a*
c^2 - 4*b)*x^2 + 2*a)*sqrt(c*x + 1)*sqrt(c*x - 1) - 2*(8*(3*a*c^8 + 4*b*c^6)*x^8
- 8*(3*a*c^6 + 4*b*c^4)*x^6 + (3*a*c^4 + 4*b*c^2)*x^4 - 4*(2*(3*a*c^7 + 4*b*c^5
)*x^7 - (3*a*c^5 + 4*b*c^3)*x^5)*sqrt(c*x + 1)*sqrt(c*x - 1))*arctan(-c*x + sqrt
(c*x + 1)*sqrt(c*x - 1)))/(8*c^4*x^8 - 8*c^2*x^6 + x^4 - 4*(2*c^3*x^7 - c*x^5)*s
qrt(c*x + 1)*sqrt(c*x - 1))

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Sympy [A]  time = 124.583, size = 148, normalized size = 1.49 \[ - \frac{a c^{4}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{11}{4}, \frac{13}{4}, 1 & 3, 3, \frac{7}{2} \\\frac{5}{2}, \frac{11}{4}, 3, \frac{13}{4}, \frac{7}{2} & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i a c^{4}{G_{6, 6}^{2, 6}\left (\begin{matrix} 2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4}, 3, 1 & \\\frac{9}{4}, \frac{11}{4} & 2, \frac{5}{2}, \frac{5}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{b c^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i b c^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/x**5/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)

[Out]

-a*c**4*meijerg(((11/4, 13/4, 1), (3, 3, 7/2)), ((5/2, 11/4, 3, 13/4, 7/2), (0,)
), 1/(c**2*x**2))/(4*pi**(3/2)) + I*a*c**4*meijerg(((2, 9/4, 5/2, 11/4, 3, 1), (
)), ((9/4, 11/4), (2, 5/2, 5/2, 0)), exp_polar(2*I*pi)/(c**2*x**2))/(4*pi**(3/2)
) - b*c**2*meijerg(((7/4, 9/4, 1), (2, 2, 5/2)), ((3/2, 7/4, 2, 9/4, 5/2), (0,))
, 1/(c**2*x**2))/(4*pi**(3/2)) + I*b*c**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)/(c**2*x**2))/(4*pi**(3/2))

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GIAC/XCAS [A]  time = 0.229307, size = 362, normalized size = 3.66 \[ -\frac{{\left (3 \, a c^{5} + 4 \, b c^{3}\right )} \arctan \left (\frac{1}{2} \,{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{2}\right ) + \frac{2 \,{\left (3 \, a c^{5}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{14} + 4 \, b c^{3}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{14} + 44 \, a c^{5}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{10} + 16 \, b c^{3}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{10} - 176 \, a c^{5}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{6} - 64 \, b c^{3}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{6} - 192 \, a c^{5}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{2} - 256 \, b c^{3}{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{2}\right )}}{{\left ({\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{4} + 4\right )}^{4}}}{4 \, c} \]

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^5),x, algorithm="giac")

[Out]

-1/4*((3*a*c^5 + 4*b*c^3)*arctan(1/2*(sqrt(c*x + 1) - sqrt(c*x - 1))^2) + 2*(3*a
*c^5*(sqrt(c*x + 1) - sqrt(c*x - 1))^14 + 4*b*c^3*(sqrt(c*x + 1) - sqrt(c*x - 1)
)^14 + 44*a*c^5*(sqrt(c*x + 1) - sqrt(c*x - 1))^10 + 16*b*c^3*(sqrt(c*x + 1) - s
qrt(c*x - 1))^10 - 176*a*c^5*(sqrt(c*x + 1) - sqrt(c*x - 1))^6 - 64*b*c^3*(sqrt(
c*x + 1) - sqrt(c*x - 1))^6 - 192*a*c^5*(sqrt(c*x + 1) - sqrt(c*x - 1))^2 - 256*
b*c^3*(sqrt(c*x + 1) - sqrt(c*x - 1))^2)/((sqrt(c*x + 1) - sqrt(c*x - 1))^4 + 4)
^4)/c