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

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 = 0.0623655, 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, Rules used = {454, 95} \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (2 a c^2+3 b\right )}{3 x}+\frac{a \sqrt{c x-1} \sqrt{c x+1}}{3 x^3} \]

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)

Rule 454

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(a1*a2*e*
(m + 1)), x] + Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*e^n*(m + 1)), Int[(e*x)^(m + n)*(a1
+ b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, p}, x] && EqQ[non2, n/2] && Eq
Q[a2*b1 + a1*b2, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1
])) &&  !ILtQ[p, -1]

Rule 95

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{a+b x^2}{x^4 \sqrt{-1+c x} \sqrt{1+c x}} \, dx &=\frac{a \sqrt{-1+c x} \sqrt{1+c x}}{3 x^3}+\frac{1}{3} \left (3 b+2 a c^2\right ) \int \frac{1}{x^2 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{a \sqrt{-1+c x} \sqrt{1+c x}}{3 x^3}+\frac{\left (3 b+2 a c^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}{3 x}\\ \end{align*}

Mathematica [A]  time = 0.0171105, size = 51, normalized size = 0.82 \[ \frac{\left (c^2 x^2-1\right ) \left (2 a c^2 x^2+a+3 b x^2\right )}{3 x^3 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 37, normalized size = 0.6 \begin{align*}{\frac{2\,a{c}^{2}{x}^{2}+3\,b{x}^{2}+a}{3\,{x}^{3}}\sqrt{cx-1}\sqrt{cx+1}} \end{align*}

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 = 1.46325, size = 73, normalized size = 1.18 \begin{align*} \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}} \end{align*}

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, 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 = 1.5221, size = 120, normalized size = 1.94 \begin{align*} \frac{{\left (2 \, a c^{3} + 3 \, b c\right )} x^{3} +{\left ({\left (2 \, a c^{2} + 3 \, b\right )} x^{2} + a\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{3 \, x^{3}} \end{align*}

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, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 37.4068, size = 146, normalized size = 2.35 \begin{align*} - \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}}} \end{align*}

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 [B]  time = 1.189, size = 157, normalized size = 2.53 \begin{align*} \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} \end{align*}

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, 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)