3.104 \(\int \sqrt{(b-x) (-a+x)} \, dx\)

Optimal. Leaf size=71 \[ -\frac{1}{4} (a+b-2 x) \sqrt{x (a+b)-a b-x^2}-\frac{1}{8} (a-b)^2 \tan ^{-1}\left (\frac{a+b-2 x}{2 \sqrt{x (a+b)-a b-x^2}}\right ) \]

[Out]

-((a + b - 2*x)*Sqrt[-(a*b) + (a + b)*x - x^2])/4 - ((a - b)^2*ArcTan[(a + b - 2
*x)/(2*Sqrt[-(a*b) + (a + b)*x - x^2])])/8

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Rubi [A]  time = 0.0523851, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{1}{4} (a+b-2 x) \sqrt{x (a+b)-a b-x^2}-\frac{1}{8} (a-b)^2 \tan ^{-1}\left (\frac{a+b-2 x}{2 \sqrt{x (a+b)-a b-x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(b - x)*(-a + x)],x]

[Out]

-((a + b - 2*x)*Sqrt[-(a*b) + (a + b)*x - x^2])/4 - ((a - b)^2*ArcTan[(a + b - 2
*x)/(2*Sqrt[-(a*b) + (a + b)*x - x^2])])/8

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Rubi in Sympy [A]  time = 1.93067, size = 56, normalized size = 0.79 \[ - \frac{\left (a - b\right )^{2} \operatorname{atan}{\left (\frac{a + b - 2 x}{2 \sqrt{- a b - x^{2} + x \left (a + b\right )}} \right )}}{8} - \frac{\left (a + b - 2 x\right ) \sqrt{- a b - x^{2} + x \left (a + b\right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a - b)**2*atan((a + b - 2*x)/(2*sqrt(-a*b - x**2 + x*(a + b))))/8 - (a + b - 2
*x)*sqrt(-a*b - x**2 + x*(a + b))/4

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Mathematica [A]  time = 0.184948, size = 84, normalized size = 1.18 \[ \frac{1}{8} \sqrt{(a-x) (x-b)} \left (-2 (a+b-2 x)-\frac{(a-b)^2 \tan ^{-1}\left (\frac{a+b-2 x}{2 \sqrt{x-a} \sqrt{b-x}}\right )}{\sqrt{x-a} \sqrt{b-x}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a - x)*(-b + x)]*(-2*(a + b - 2*x) - ((a - b)^2*ArcTan[(a + b - 2*x)/(2*S
qrt[b - x]*Sqrt[-a + x])])/(Sqrt[b - x]*Sqrt[-a + x])))/8

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Maple [A]  time = 0.02, size = 122, normalized size = 1.7 \[ -{\frac{a+b-2\,x}{4}\sqrt{-ab+ \left ( a+b \right ) x-{x}^{2}}}-{\frac{ab}{4}\arctan \left ({1 \left ( x-{\frac{a}{2}}-{\frac{b}{2}} \right ){\frac{1}{\sqrt{-ab+ \left ( a+b \right ) x-{x}^{2}}}}} \right ) }+{\frac{{a}^{2}}{8}\arctan \left ({1 \left ( x-{\frac{a}{2}}-{\frac{b}{2}} \right ){\frac{1}{\sqrt{-ab+ \left ( a+b \right ) x-{x}^{2}}}}} \right ) }+{\frac{{b}^{2}}{8}\arctan \left ({1 \left ( x-{\frac{a}{2}}-{\frac{b}{2}} \right ){\frac{1}{\sqrt{-ab+ \left ( a+b \right ) x-{x}^{2}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b-x)*(-a+x))^(1/2),x)

[Out]

-1/4*(a+b-2*x)*(-a*b+(a+b)*x-x^2)^(1/2)-1/4*arctan((x-1/2*a-1/2*b)/(-a*b+(a+b)*x
-x^2)^(1/2))*a*b+1/8*arctan((x-1/2*a-1/2*b)/(-a*b+(a+b)*x-x^2)^(1/2))*a^2+1/8*ar
ctan((x-1/2*a-1/2*b)/(-a*b+(a+b)*x-x^2)^(1/2))*b^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)*(b - x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223167, size = 88, normalized size = 1.24 \[ \frac{1}{8} \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (-\frac{a + b - 2 \, x}{2 \, \sqrt{-a b +{\left (a + b\right )} x - x^{2}}}\right ) - \frac{1}{4} \, \sqrt{-a b +{\left (a + b\right )} x - x^{2}}{\left (a + b - 2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(a^2 - 2*a*b + b^2)*arctan(-1/2*(a + b - 2*x)/sqrt(-a*b + (a + b)*x - x^2))
- 1/4*sqrt(-a*b + (a + b)*x - x^2)*(a + b - 2*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (- a + x\right ) \left (b - x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((-a + x)*(b - x)), x)

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GIAC/XCAS [A]  time = 0.214958, size = 82, normalized size = 1.15 \[ \frac{1}{8} \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \arcsin \left (\frac{a + b - 2 \, x}{a - b}\right ){\rm sign}\left (-a + b\right ) - \frac{1}{4} \, \sqrt{-a b + a x + b x - x^{2}}{\left (a + b - 2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(a^2 - 2*a*b + b^2)*arcsin((a + b - 2*x)/(a - b))*sign(-a + b) - 1/4*sqrt(-a
*b + a*x + b*x - x^2)*(a + b - 2*x)