Optimal. Leaf size=67 \[ \frac{1}{2} x \sqrt{a x+a} \sqrt{c-c x}+\sqrt{a} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right ) \]
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Rubi [A] time = 0.0605251, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{1}{2} x \sqrt{a x+a} \sqrt{c-c x}+\sqrt{a} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + a*x]*Sqrt[c - c*x],x]
[Out]
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Rubi in Sympy [A] time = 9.91485, size = 58, normalized size = 0.87 \[ \sqrt{a} \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{a x + a}}{\sqrt{a} \sqrt{- c x + c}} \right )} + \frac{x \sqrt{a x + a} \sqrt{- c x + c}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*x+a)**(1/2)*(-c*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0613148, size = 76, normalized size = 1.13 \[ \frac{\sqrt{a (x+1)} \left (x \sqrt{x+1} \sqrt{c-c x}-2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{x+1} \sqrt{c-c x}}{\sqrt{c} (x-1)}\right )\right )}{2 \sqrt{x+1}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + a*x]*Sqrt[c - c*x],x]
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Maple [A] time = 0.007, size = 98, normalized size = 1.5 \[ -{\frac{1}{2\,c}\sqrt{ax+a} \left ( -cx+c \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}\sqrt{ax+a}\sqrt{-cx+c}}+{\frac{ac}{2}\sqrt{ \left ( -cx+c \right ) \left ( ax+a \right ) }\arctan \left ({x\sqrt{ac}{\frac{1}{\sqrt{-ac{x}^{2}+ac}}}} \right ){\frac{1}{\sqrt{ax+a}}}{\frac{1}{\sqrt{-cx+c}}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*x+a)^(1/2)*(-c*x+c)^(1/2),x)
[Out]
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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(sqrt(a*x + a)*sqrt(-c*x + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216058, size = 1, normalized size = 0.01 \[ \left [\frac{1}{2} \, \sqrt{a x + a} \sqrt{-c x + c} x + \frac{1}{4} \, \sqrt{-a c} \log \left (2 \, a c x^{2} + 2 \, \sqrt{-a c} \sqrt{a x + a} \sqrt{-c x + c} x - a c\right ), \frac{1}{2} \, \sqrt{a x + a} \sqrt{-c x + c} x + \frac{1}{2} \, \sqrt{a c} \arctan \left (\frac{a c x}{\sqrt{a c} \sqrt{a x + a} \sqrt{-c x + c}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x + a)*sqrt(-c*x + c),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a \left (x + 1\right )} \sqrt{- c \left (x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x+a)**(1/2)*(-c*x+c)**(1/2),x)
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GIAC/XCAS [A] time = 0.24585, size = 115, normalized size = 1.72 \[ -\frac{{\left (\frac{2 \, a^{3} c{\rm ln}\left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a} a x\right )}{\left | a \right |}}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x + a)*sqrt(-c*x + c),x, algorithm="giac")
[Out]