3.3.59 \(\int \sqrt {(b-x) (-a+x)} \, dx\) [259]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1976, 626, 635, 210} \begin {gather*} -\frac {1}{8} (a-b)^2 \text {ArcTan}\left (\frac {a+b-2 x}{2 \sqrt {x (a+b)-a b-x^2}}\right )-\frac {1}{4} (a+b-2 x) \sqrt {x (a+b)-a b-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[(b - x)*(-a + x)],x]

[Out]

-1/4*((a + b - 2*x)*Sqrt[-(a*b) + (a + b)*x - x^2]) - ((a - b)^2*ArcTan[(a + b - 2*x)/(2*Sqrt[-(a*b) + (a + b)
*x - x^2])])/8

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rubi steps

\begin {align*} \int \sqrt {(b-x) (-a+x)} \, dx &=\int \sqrt {-a b+(a+b) x-x^2} \, dx\\ &=-\frac {1}{4} (a+b-2 x) \sqrt {-a b+(a+b) x-x^2}+\frac {1}{8} (a-b)^2 \int \frac {1}{\sqrt {-a b+(a+b) x-x^2}} \, dx\\ &=-\frac {1}{4} (a+b-2 x) \sqrt {-a b+(a+b) x-x^2}+\frac {1}{4} (a-b)^2 \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {a+b-2 x}{\sqrt {-a b+(a+b) x-x^2}}\right )\\ &=-\frac {1}{4} (a+b-2 x) \sqrt {-a b+(a+b) x-x^2}-\frac {1}{8} (a-b)^2 \tan ^{-1}\left (\frac {a+b-2 x}{2 \sqrt {-a b+(a+b) x-x^2}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 76, normalized size = 1.07 \begin {gather*} \frac {1}{4} \sqrt {(a-x) (-b+x)} \left (-a-b+2 x-\frac {(a-b)^2 \tan ^{-1}\left (\frac {\sqrt {b-x}}{\sqrt {-a+x}}\right )}{\sqrt {b-x} \sqrt {-a+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(b - x)*(-a + x)],x]

[Out]

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

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Maple [A]
time = 0.26, size = 68, normalized size = 0.96

method	result	size
default	\(-\frac {\left (a +b -2 x \right ) \sqrt {-a b +\left (a +b \right ) x -x^{2}}}{4}-\frac {\left (4 a b -\left (a +b \right )^{2}\right ) \arctan \left (\frac {x -\frac {a}{2}-\frac {b}{2}}{\sqrt {-a b +\left (a +b \right ) x -x^{2}}}\right )}{8}\)	\(68\)
risch	\(\frac {\left (b -x \right ) \left (a -x \right ) \left (a +b -2 x \right )}{4 \sqrt {-\left (-b +x \right ) \left (-a +x \right )}}-\frac {\arctan \left (\frac {x -\frac {a}{2}-\frac {b}{2}}{\sqrt {-a b +\left (a +b \right ) x -x^{2}}}\right ) a b}{4}+\frac {\arctan \left (\frac {x -\frac {a}{2}-\frac {b}{2}}{\sqrt {-a b +\left (a +b \right ) x -x^{2}}}\right ) a^{2}}{8}+\frac {\arctan \left (\frac {x -\frac {a}{2}-\frac {b}{2}}{\sqrt {-a b +\left (a +b \right ) x -x^{2}}}\right ) b^{2}}{8}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b-x)*(-a+x))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b-x)*(-a+x))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

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Fricas [A]
time = 0.37, size = 80, normalized size = 1.13 \begin {gather*} -\frac {1}{8} \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (-\frac {\sqrt {-a b + {\left (a + b\right )} x - x^{2}} {\left (a + b - 2 \, x\right )}}{2 \, {\left (a b - {\left (a + b\right )} x + x^{2}\right )}}\right ) - \frac {1}{4} \, \sqrt {-a b + {\left (a + b\right )} x - x^{2}} {\left (a + b - 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b-x)*(-a+x))^(1/2),x, algorithm="fricas")

[Out]

-1/8*(a^2 - 2*a*b + b^2)*arctan(-1/2*sqrt(-a*b + (a + b)*x - x^2)*(a + b - 2*x)/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\left (- a + x\right ) \left (b - x\right )}\, dx \end {gather*}

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 [A]
time = 5.45, size = 61, normalized size = 0.86 \begin {gather*} \frac {1}{8} \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \arcsin \left (\frac {a + b - 2 \, x}{a - b}\right ) \mathrm {sgn}\left (-a + b\right ) - \frac {1}{4} \, \sqrt {-a b + a x + b x - x^{2}} {\left (a + b - 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b-x)*(-a+x))^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {-\left (a-x\right )\,\left (b-x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-(a - x)*(b - x))^(1/2),x)

[Out]

int((-(a - x)*(b - x))^(1/2), x)

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