3.1140 \(\int \frac{1}{\sqrt{a+a x} \sqrt{c-c x}} \, dx\)

Optimal. Leaf size=43 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )}{\sqrt{a} \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0388987, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )}{\sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.48174, size = 41, normalized size = 0.95 \[ - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{- c x + c}}{\sqrt{c} \sqrt{a x + a}} \right )}}{\sqrt{a} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0317823, size = 52, normalized size = 1.21 \[ -\frac{2 \sqrt{x+1} \tan ^{-1}\left (\frac{\sqrt{x+1} \sqrt{-c (x-1)}}{\sqrt{c} (x-1)}\right )}{\sqrt{c} \sqrt{a (x+1)}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 + x]*ArcTan[(Sqrt[-(c*(-1 + x))]*Sqrt[1 + x])/(Sqrt[c]*(-1 + x))])/(S
qrt[c]*Sqrt[a*(1 + x)])

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Maple [A]  time = 0.007, size = 57, normalized size = 1.3 \[{1\sqrt{ \left ( -cx+c \right ) \left ( ax+a \right ) }\arctan \left ({x\sqrt{ac}{\frac{1}{\sqrt{-ac{x}^{2}+ac}}}} \right ){\frac{1}{\sqrt{ax+a}}}{\frac{1}{\sqrt{-cx+c}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22343, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (2 \, \sqrt{a x + a} \sqrt{-c x + c} x + \sqrt{-a c}{\left (2 \, x^{2} - 1\right )}\right )}{2 \, \sqrt{-a c}}, \frac{\arctan \left (\frac{\sqrt{a c} x}{\sqrt{a x + a} \sqrt{-c x + c}}\right )}{\sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.65252, size = 85, normalized size = 1.98 \[ - \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{a} \sqrt{c}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{a} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x+a)**(1/2)/(-c*x+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), ()), x**(-
2))/(4*pi**(3/2)*sqrt(a)*sqrt(c)) + 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)/x**2)/(4*pi**(3/2)*sqrt(a)*sq
rt(c))

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GIAC/XCAS [A]  time = 0.246815, size = 66, normalized size = 1.53 \[ -\frac{2 \, a{\rm ln}\left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*ln(abs(-sqrt(-a*c)*sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a^2*c)))/(sqrt(-
a*c)*abs(a))