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

Optimal. Leaf size=42 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{a c-b c x}}{a \sqrt{c}}\right )}{a \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0202499, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {92, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{a c-b c x}}{a \sqrt{c}}\right )}{a \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.025462, size = 63, normalized size = 1.5 \[ -\frac{\sqrt{a^2-b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{a^2-b^2 x^2}}{a}\right )}{a \sqrt{a+b x} \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.03, size = 85, normalized size = 2. \begin{align*} -{\sqrt{bx+a}\sqrt{-c \left ( bx-a \right ) }\ln \left ( 2\,{\frac{{a}^{2}c+\sqrt{{a}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}{x}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}{\frac{1}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.11043, size = 255, normalized size = 6.07 \begin{align*} \left [\frac{\log \left (-\frac{b^{2} c x^{2} - 2 \, a^{2} c + 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} a \sqrt{c}}{x^{2}}\right )}{2 \, a \sqrt{c}}, -\frac{\sqrt{-c} \arctan \left (\frac{\sqrt{-b c x + a c} \sqrt{b x + a} a \sqrt{-c}}{b^{2} c x^{2} - a^{2} c}\right )}{a c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log(-(b^2*c*x^2 - 2*a^2*c + 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*a*sqrt(c))/x^2)/(a*sqrt(c)), -sqrt(-c)*arc
tan(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*a*sqrt(-c)/(b^2*c*x^2 - a^2*c))/(a*c)]

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Sympy [C]  time = 5.00701, size = 83, normalized size = 1.98 \begin{align*} \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} a \sqrt{c}} - \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} a \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 3.16143, size = 88, normalized size = 2.1 \begin{align*} -\frac{2 \, \sqrt{-c} \arctan \left (\frac{{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2}}{2 \, a c^{2}}\right )}{a{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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