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

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 = 0.0937899, 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 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{a c-b c x}}{a \sqrt{c}}\right )}{a \sqrt{c}} \]

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]))

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Rubi in Sympy [A]  time = 9.53729, size = 36, normalized size = 0.86 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x} \sqrt{a c - b c x}}{a \sqrt{c}} \right )}}{a \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(a + b*x)*sqrt(a*c - b*c*x)/(a*sqrt(c)))/(a*sqrt(c))

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Mathematica [A]  time = 0.0664701, size = 54, normalized size = 1.29 \[ \frac{\sqrt{a-b x} \left (\log (x)-\log \left (\sqrt{a-b x} \sqrt{a+b x}+a\right )\right )}{a \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.042, size = 85, normalized size = 2. \[ -{1\sqrt{bx+a}\sqrt{-c \left ( bx-a \right ) }\ln \left ( 2\,{\frac{{a}^{2}c+\sqrt{{a}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}{x}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}{\frac{1}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231843, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \, \sqrt{-b c x + a c} \sqrt{b x + a} a +{\left (b^{2} x^{2} - 2 \, a^{2}\right )} \sqrt{c}}{x^{2}}\right )}{2 \, a \sqrt{c}}, -\frac{\arctan \left (\frac{a \sqrt{-c}}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right )}{a \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.7479, size = 83, normalized size = 1.98 \[ \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} a \sqrt{c}} - \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} a \sqrt{c}} \]

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/XCAS [A]  time = 0.226588, size = 88, normalized size = 2.1 \[ -\frac{2 \, \sqrt{-c} \arctan \left (\frac{{\left (\sqrt{-b c x + a c} \sqrt{-c} - \sqrt{2 \, a c^{2} +{\left (b c x - a c\right )} c}\right )}^{2}}{2 \, a c^{2}}\right )}{a{\left | c \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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