3.1148 \(\int \frac{1}{\sqrt{a+b x} \sqrt{a c-b c x}} \, dx\)

Optimal. Leaf size=39 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a c-b c x}}\right )}{b \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0430537, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a c-b c x}}\right )}{b \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.54457, size = 36, normalized size = 0.92 \[ - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{a c - b c x}}{\sqrt{c} \sqrt{a + b x}} \right )}}{b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atan(sqrt(a*c - b*c*x)/(sqrt(c)*sqrt(a + b*x)))/(b*sqrt(c))

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Mathematica [A]  time = 0.0345239, size = 49, normalized size = 1.26 \[ \frac{\sqrt{a-b x} \tan ^{-1}\left (\frac{b x}{\sqrt{a-b x} \sqrt{a+b x}}\right )}{b \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 71, normalized size = 1.8 \[{1\sqrt{ \left ( bx+a \right ) \left ( -bcx+ac \right ) }\arctan \left ({x\sqrt{{b}^{2}c}{\frac{1}{\sqrt{-{b}^{2}c{x}^{2}+{a}^{2}c}}}} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{-bcx+ac}}}{\frac{1}{\sqrt{{b}^{2}c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221725, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b x +{\left (2 \, b^{2} x^{2} - a^{2}\right )} \sqrt{-c}\right )}{2 \, b \sqrt{-c}}, \frac{\arctan \left (\frac{b \sqrt{c} x}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right )}{b \sqrt{c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.8277, size = 90, normalized size = 2.31 \[ - \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{c}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out