Integrand size = 19, antiderivative size = 92 \[ \int x \sqrt {\frac {-a+x}{b-x}} \, dx=\frac {1}{4} (a-5 b) (b-x) \sqrt {\frac {-a+x}{b-x}}+\frac {1}{2} (b-x)^2 \sqrt {\frac {-a+x}{b-x}}-\frac {1}{4} (a-b) (a+3 b) \arctan \left (\sqrt {\frac {-a+x}{b-x}}\right ) \] Output:
-1/4*(a-b)*(a+3*b)*arctan(((-a+x)/(b-x))^(1/2))+1/4*(a-5*b)*(b-x)*((-a+x)/ (b-x))^(1/2)+1/2*(b-x)^2*((-a+x)/(b-x))^(1/2)
Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08 \[ \int x \sqrt {\frac {-a+x}{b-x}} \, dx=\frac {\sqrt {\frac {-a+x}{b-x}} \left ((a-3 b-2 x) (b-x) \sqrt {-a+x}+\left (-a^2-2 a b+3 b^2\right ) \sqrt {b-x} \arctan \left (\frac {\sqrt {-a+x}}{\sqrt {b-x}}\right )\right )}{4 \sqrt {-a+x}} \] Input:
Integrate[x*Sqrt[(-a + x)/(b - x)],x]
Output:
(Sqrt[(-a + x)/(b - x)]*((a - 3*b - 2*x)*(b - x)*Sqrt[-a + x] + (-a^2 - 2* a*b + 3*b^2)*Sqrt[b - x]*ArcTan[Sqrt[-a + x]/Sqrt[b - x]]))/(4*Sqrt[-a + x ])
Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.42, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2052, 360, 25, 298, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \sqrt {\frac {x-a}{b-x}} \, dx\) |
\(\Big \downarrow \) 2052 |
\(\displaystyle -2 (a-b) \int -\frac {\left (a-\frac {b (a-x)}{b-x}\right ) (a-x)}{\left (1-\frac {a-x}{b-x}\right )^3 (b-x)}d\sqrt {-\frac {a-x}{b-x}}\) |
\(\Big \downarrow \) 360 |
\(\displaystyle -2 (a-b) \left (-\frac {1}{4} \int -\frac {a-b-\frac {4 b (a-x)}{b-x}}{\left (1-\frac {a-x}{b-x}\right )^2}d\sqrt {-\frac {a-x}{b-x}}-\frac {(a-b) \sqrt {-\frac {a-x}{b-x}}}{4 \left (1-\frac {a-x}{b-x}\right )^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 (a-b) \left (\frac {1}{4} \int \frac {a-b-\frac {4 b (a-x)}{b-x}}{\left (1-\frac {a-x}{b-x}\right )^2}d\sqrt {-\frac {a-x}{b-x}}-\frac {(a-b) \sqrt {-\frac {a-x}{b-x}}}{4 \left (1-\frac {a-x}{b-x}\right )^2}\right )\) |
\(\Big \downarrow \) 298 |
\(\displaystyle -2 (a-b) \left (\frac {1}{4} \left (\frac {1}{2} (a+3 b) \int \frac {1}{1-\frac {a-x}{b-x}}d\sqrt {-\frac {a-x}{b-x}}+\frac {(a-5 b) \sqrt {-\frac {a-x}{b-x}}}{2 \left (1-\frac {a-x}{b-x}\right )}\right )-\frac {(a-b) \sqrt {-\frac {a-x}{b-x}}}{4 \left (1-\frac {a-x}{b-x}\right )^2}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -2 (a-b) \left (\frac {1}{4} \left (\frac {1}{2} (a+3 b) \arctan \left (\sqrt {-\frac {a-x}{b-x}}\right )+\frac {(a-5 b) \sqrt {-\frac {a-x}{b-x}}}{2 \left (1-\frac {a-x}{b-x}\right )}\right )-\frac {(a-b) \sqrt {-\frac {a-x}{b-x}}}{4 \left (1-\frac {a-x}{b-x}\right )^2}\right )\) |
Input:
Int[x*Sqrt[(-a + x)/(b - x)],x]
Output:
-2*(a - b)*(-1/4*((a - b)*Sqrt[-((a - x)/(b - x))])/(1 - (a - x)/(b - x))^ 2 + (((a - 5*b)*Sqrt[-((a - x)/(b - x))])/(2*(1 - (a - x)/(b - x))) + ((a + 3*b)*ArcTan[Sqrt[-((a - x)/(b - x))]])/2)/4)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Time = 0.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.53
method | result | size |
risch | \(\frac {\left (a -3 b -2 x \right ) \left (b -x \right ) \sqrt {-\frac {a -x}{b -x}}\, \sqrt {-\left (b -x \right ) \left (a -x \right )}}{4 \sqrt {-\left (-b +x \right ) \left (-a +x \right )}}+\frac {\left (\frac {1}{4} a b -\frac {3}{8} b^{2}+\frac {1}{8} a^{2}\right ) \arctan \left (\frac {x -\frac {b}{2}-\frac {a}{2}}{\sqrt {-a b +\left (a +b \right ) x -x^{2}}}\right ) \sqrt {-\frac {a -x}{b -x}}\, \sqrt {-\left (b -x \right ) \left (a -x \right )}}{a -x}\) | \(141\) |
default | \(\frac {\sqrt {-\frac {a -x}{b -x}}\, \left (b -x \right ) \left (\arctan \left (\frac {a +b -2 x}{2 \sqrt {-a b +a x +b x -x^{2}}}\right ) a^{2}+2 b \arctan \left (\frac {a +b -2 x}{2 \sqrt {-a b +a x +b x -x^{2}}}\right ) a -3 \arctan \left (\frac {a +b -2 x}{2 \sqrt {-a b +a x +b x -x^{2}}}\right ) b^{2}+2 \sqrt {-a b +a x +b x -x^{2}}\, a -6 \sqrt {-a b +a x +b x -x^{2}}\, b -4 \sqrt {-a b +a x +b x -x^{2}}\, x \right )}{8 \sqrt {-\left (b -x \right ) \left (a -x \right )}}\) | \(196\) |
Input:
int(x*((-a+x)/(b-x))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/4*(a-3*b-2*x)*(b-x)/(-(-b+x)*(-a+x))^(1/2)*(-(a-x)/(b-x))^(1/2)*(-(b-x)* (a-x))^(1/2)+(1/4*a*b-3/8*b^2+1/8*a^2)*arctan((x-1/2*b-1/2*a)/(-a*b+(a+b)* x-x^2)^(1/2))*(-(a-x)/(b-x))^(1/2)*(-(b-x)*(a-x))^(1/2)/(a-x)
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.79 \[ \int x \sqrt {\frac {-a+x}{b-x}} \, dx=-\frac {1}{4} \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \arctan \left (\sqrt {-\frac {a - x}{b - x}}\right ) + \frac {1}{4} \, {\left (a b - 3 \, b^{2} - {\left (a - b\right )} x + 2 \, x^{2}\right )} \sqrt {-\frac {a - x}{b - x}} \] Input:
integrate(x*((-a+x)/(b-x))^(1/2),x, algorithm="fricas")
Output:
-1/4*(a^2 + 2*a*b - 3*b^2)*arctan(sqrt(-(a - x)/(b - x))) + 1/4*(a*b - 3*b ^2 - (a - b)*x + 2*x^2)*sqrt(-(a - x)/(b - x))
\[ \int x \sqrt {\frac {-a+x}{b-x}} \, dx=\int x \sqrt {\frac {- a + x}{b - x}}\, dx \] Input:
integrate(x*((-a+x)/(b-x))**(1/2),x)
Output:
Integral(x*sqrt((-a + x)/(b - x)), x)
Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.41 \[ \int x \sqrt {\frac {-a+x}{b-x}} \, dx=-\frac {1}{4} \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \arctan \left (\sqrt {-\frac {a - x}{b - x}}\right ) - \frac {{\left (a^{2} - 6 \, a b + 5 \, b^{2}\right )} \left (-\frac {a - x}{b - x}\right )^{\frac {3}{2}} - {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \sqrt {-\frac {a - x}{b - x}}}{4 \, {\left (\frac {{\left (a - x\right )}^{2}}{{\left (b - x\right )}^{2}} - \frac {2 \, {\left (a - x\right )}}{b - x} + 1\right )}} \] Input:
integrate(x*((-a+x)/(b-x))^(1/2),x, algorithm="maxima")
Output:
-1/4*(a^2 + 2*a*b - 3*b^2)*arctan(sqrt(-(a - x)/(b - x))) - 1/4*((a^2 - 6* a*b + 5*b^2)*(-(a - x)/(b - x))^(3/2) - (a^2 + 2*a*b - 3*b^2)*sqrt(-(a - x )/(b - x)))/((a - x)^2/(b - x)^2 - 2*(a - x)/(b - x) + 1)
Time = 0.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.12 \[ \int x \sqrt {\frac {-a+x}{b-x}} \, dx=\frac {1}{8} \, {\left (a^{2} \mathrm {sgn}\left (-b + x\right ) + 2 \, a b \mathrm {sgn}\left (-b + x\right ) - 3 \, b^{2} \mathrm {sgn}\left (-b + x\right )\right )} \arcsin \left (\frac {a + b - 2 \, x}{a - b}\right ) \mathrm {sgn}\left (-a + b\right ) - \frac {1}{4} \, \sqrt {-a b + a x + b x - x^{2}} {\left (a \mathrm {sgn}\left (-b + x\right ) - 3 \, b \mathrm {sgn}\left (-b + x\right ) - 2 \, x \mathrm {sgn}\left (-b + x\right )\right )} \] Input:
integrate(x*((-a+x)/(b-x))^(1/2),x, algorithm="giac")
Output:
1/8*(a^2*sgn(-b + x) + 2*a*b*sgn(-b + x) - 3*b^2*sgn(-b + x))*arcsin((a + b - 2*x)/(a - b))*sgn(-a + b) - 1/4*sqrt(-a*b + a*x + b*x - x^2)*(a*sgn(-b + x) - 3*b*sgn(-b + x) - 2*x*sgn(-b + x))
Time = 0.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.52 \[ \int x \sqrt {\frac {-a+x}{b-x}} \, dx=-\frac {\sqrt {-\frac {a-x}{b-x}}\,\left (\frac {a^2\,1{}\mathrm {i}}{4}+\frac {a\,b\,1{}\mathrm {i}}{2}-\frac {b^2\,3{}\mathrm {i}}{4}\right )\,1{}\mathrm {i}-{\left (-\frac {a-x}{b-x}\right )}^{3/2}\,\left (\frac {a^2\,1{}\mathrm {i}}{4}-\frac {a\,b\,3{}\mathrm {i}}{2}+\frac {b^2\,5{}\mathrm {i}}{4}\right )\,1{}\mathrm {i}}{\frac {{\left (a-x\right )}^2}{{\left (b-x\right )}^2}-\frac {2\,\left (a-x\right )}{b-x}+1}-\frac {\mathrm {atan}\left (\sqrt {-\frac {a-x}{b-x}}\right )\,\left (a-b\right )\,\left (a+3\,b\right )}{4} \] Input:
int(x*(-(a - x)/(b - x))^(1/2),x)
Output:
- ((-(a - x)/(b - x))^(1/2)*((a*b*1i)/2 + (a^2*1i)/4 - (b^2*3i)/4)*1i - (- (a - x)/(b - x))^(3/2)*((a^2*1i)/4 - (a*b*3i)/2 + (b^2*5i)/4)*1i)/((a - x) ^2/(b - x)^2 - (2*(a - x))/(b - x) + 1) - (atan((-(a - x)/(b - x))^(1/2))* (a - b)*(a + 3*b))/4
Time = 0.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.96 \[ \int x \sqrt {\frac {-a+x}{b-x}} \, dx=\frac {i \left (\mathit {asinh} \left (\frac {\sqrt {-a +x}\, i}{\sqrt {-a +b}}\right ) a^{3}+\mathit {asinh} \left (\frac {\sqrt {-a +x}\, i}{\sqrt {-a +b}}\right ) a^{2} b -5 \mathit {asinh} \left (\frac {\sqrt {-a +x}\, i}{\sqrt {-a +b}}\right ) a \,b^{2}+3 \mathit {asinh} \left (\frac {\sqrt {-a +x}\, i}{\sqrt {-a +b}}\right ) b^{3}+\sqrt {b -x}\, \sqrt {a -b}\, \sqrt {-a +x}\, \sqrt {-a +b}\, a -3 \sqrt {b -x}\, \sqrt {a -b}\, \sqrt {-a +x}\, \sqrt {-a +b}\, b -2 \sqrt {b -x}\, \sqrt {a -b}\, \sqrt {-a +x}\, \sqrt {-a +b}\, x \right )}{4 a -4 b} \] Input:
int(x*((-a+x)/(b-x))^(1/2),x)
Output:
(i*(asinh((sqrt( - a + x)*i)/sqrt( - a + b))*a**3 + asinh((sqrt( - a + x)* i)/sqrt( - a + b))*a**2*b - 5*asinh((sqrt( - a + x)*i)/sqrt( - a + b))*a*b **2 + 3*asinh((sqrt( - a + x)*i)/sqrt( - a + b))*b**3 + sqrt(b - x)*sqrt(a - b)*sqrt( - a + x)*sqrt( - a + b)*a - 3*sqrt(b - x)*sqrt(a - b)*sqrt( - a + x)*sqrt( - a + b)*b - 2*sqrt(b - x)*sqrt(a - b)*sqrt( - a + x)*sqrt( - a + b)*x))/(4*(a - b))