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

Optimal. Leaf size=33 \[ \frac{a \sqrt{c x-1} \sqrt{c x+1}}{x}+\frac{b \cosh ^{-1}(c x)}{c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0565493, antiderivative size = 33, 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, 52} \[ \frac{a \sqrt{c x-1} \sqrt{c x+1}}{x}+\frac{b \cosh ^{-1}(c x)}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

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

Mathematica [B]  time = 0.0228771, size = 73, normalized size = 2.21 \[ \frac{\sqrt{c^2 x^2-1} \left (\frac{a \sqrt{c^2 x^2-1}}{x}+\frac{b \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{c}\right )}{\sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.018, size = 77, normalized size = 2.3 \begin{align*}{\frac{{\it csgn} \left ( c \right ) }{cx}\sqrt{cx-1}\sqrt{cx+1} \left ({\it csgn} \left ( c \right ) c\sqrt{{c}^{2}{x}^{2}-1}a+\ln \left ( \left ( \sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) +cx \right ){\it csgn} \left ( c \right ) \right ) xb \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.47724, size = 68, normalized size = 2.06 \begin{align*} \frac{b \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} a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.56034, size = 131, normalized size = 3.97 \begin{align*} \frac{a c^{2} x + \sqrt{c x + 1} \sqrt{c x - 1} a c - b x \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 18.89, size = 148, normalized size = 4.48 \begin{align*} - \frac{a 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 a 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}}} + \frac{b{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 b{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*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*a
*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*p
i**(3/2)) + b*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*b*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)

________________________________________________________________________________________

Giac [A]  time = 1.65691, size = 78, normalized size = 2.36 \begin{align*} \frac{\frac{16 \, a c^{2}}{{\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{4} + 4} - b \log \left ({\left (\sqrt{c x + 1} - \sqrt{c x - 1}\right )}^{4}\right )}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(16*a*c^2/((sqrt(c*x + 1) - sqrt(c*x - 1))^4 + 4) - b*log((sqrt(c*x + 1) - sqrt(c*x - 1))^4))/c