3.1139 \(\int \sqrt{a+a x} \sqrt{c-c x} \, dx\)

Optimal. Leaf size=67 \[ \frac{1}{2} x \sqrt{a x+a} \sqrt{c-c x}+\sqrt{a} \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right ) \]

[Out]

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

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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]

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

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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]

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

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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]

[Out]

(Sqrt[a*(1 + x)]*(x*Sqrt[1 + x]*Sqrt[c - c*x] - 2*Sqrt[c]*ArcTan[(Sqrt[1 + x]*Sq
rt[c - c*x])/(Sqrt[c]*(-1 + x))]))/(2*Sqrt[1 + 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]

-1/2/c*(a*x+a)^(1/2)*(-c*x+c)^(3/2)+1/2*(a*x+a)^(1/2)*(-c*x+c)^(1/2)+1/2*a*c*((-
c*x+c)*(a*x+a))^(1/2)/(-c*x+c)^(1/2)/(a*x+a)^(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(sqrt(a*x + a)*sqrt(-c*x + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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]

[1/2*sqrt(a*x + a)*sqrt(-c*x + c)*x + 1/4*sqrt(-a*c)*log(2*a*c*x^2 + 2*sqrt(-a*c
)*sqrt(a*x + a)*sqrt(-c*x + c)*x - a*c), 1/2*sqrt(a*x + a)*sqrt(-c*x + c)*x + 1/
2*sqrt(a*c)*arctan(a*c*x/(sqrt(a*c)*sqrt(a*x + a)*sqrt(-c*x + c)))]

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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)

[Out]

Integral(sqrt(a*(x + 1))*sqrt(-c*(x - 1)), 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]

-1/2*(2*a^3*c*ln(abs(-sqrt(-a*c)*sqrt(a*x + a) + sqrt(-(a*x + a)*a*c + 2*a^2*c))
)/sqrt(-a*c) - sqrt(-(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a)*a*x)*abs(a)/a^3