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