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

Optimal. Leaf size=77 \[ \frac{a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{a-b x}-\frac{a \sqrt{x} \sqrt{a-b x}}{4 b} \]

[Out]

-(a*Sqrt[x]*Sqrt[a - b*x])/(4*b) + (x^(3/2)*Sqrt[a - b*x])/2 + (a^2*ArcTan[(Sqrt
[b]*Sqrt[x])/Sqrt[a - b*x]])/(4*b^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0536221, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{a-b x}-\frac{a \sqrt{x} \sqrt{a-b x}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*Sqrt[a - b*x],x]

[Out]

-(a*Sqrt[x]*Sqrt[a - b*x])/(4*b) + (x^(3/2)*Sqrt[a - b*x])/2 + (a^2*ArcTan[(Sqrt
[b]*Sqrt[x])/Sqrt[a - b*x]])/(4*b^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.71018, size = 65, normalized size = 0.84 \[ \frac{a^{2} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a - b x}} \right )}}{4 b^{\frac{3}{2}}} + \frac{a \sqrt{x} \sqrt{a - b x}}{4 b} - \frac{\sqrt{x} \left (a - b x\right )^{\frac{3}{2}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*atan(sqrt(b)*sqrt(x)/sqrt(a - b*x))/(4*b**(3/2)) + a*sqrt(x)*sqrt(a - b*x)/
(4*b) - sqrt(x)*(a - b*x)**(3/2)/(2*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0477632, size = 65, normalized size = 0.84 \[ \frac{a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )+\sqrt{b} \sqrt{x} \sqrt{a-b x} (2 b x-a)}{4 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*Sqrt[a - b*x],x]

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a - b*x]*(-a + 2*b*x) + a^2*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[
a - b*x]])/(4*b^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 86, normalized size = 1.1 \[{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{-bx+a}}-{\frac{a}{4\,b}\sqrt{x}\sqrt{-bx+a}}+{\frac{{a}^{2}}{8}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(-b*x+a)^(1/2),x)

[Out]

1/2*x^(3/2)*(-b*x+a)^(1/2)-1/4*a*x^(1/2)*(-b*x+a)^(1/2)/b+1/8*a^2/b^(3/2)*(x*(-b
*x+a))^(1/2)/x^(1/2)/(-b*x+a)^(1/2)*arctan(b^(1/2)*(x-1/2*a/b)/(-b*x^2+a*x)^(1/2
))

_______________________________________________________________________________________

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(-b*x + a)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.224612, size = 1, normalized size = 0.01 \[ \left [\frac{a^{2} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) + 2 \,{\left (2 \, b x - a\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{8 \, \sqrt{-b} b}, -\frac{a^{2} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (2 \, b x - a\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{4 \, b^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(a^2*log(-2*sqrt(-b*x + a)*b*sqrt(x) - (2*b*x - a)*sqrt(-b)) + 2*(2*b*x - a
)*sqrt(-b*x + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b), -1/4*(a^2*arctan(sqrt(-b*x + a)
/(sqrt(b)*sqrt(x))) - (2*b*x - a)*sqrt(-b*x + a)*sqrt(b)*sqrt(x))/b^(3/2)]

_______________________________________________________________________________________

Sympy [A]  time = 11.8246, size = 207, normalized size = 2.69 \[ \begin{cases} \frac{i a^{\frac{3}{2}} \sqrt{x}}{4 b \sqrt{-1 + \frac{b x}{a}}} - \frac{3 i \sqrt{a} x^{\frac{3}{2}}}{4 \sqrt{-1 + \frac{b x}{a}}} - \frac{i a^{2} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} + \frac{i b x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{a^{\frac{3}{2}} \sqrt{x}}{4 b \sqrt{1 - \frac{b x}{a}}} + \frac{3 \sqrt{a} x^{\frac{3}{2}}}{4 \sqrt{1 - \frac{b x}{a}}} + \frac{a^{2} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} - \frac{b x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*a**(3/2)*sqrt(x)/(4*b*sqrt(-1 + b*x/a)) - 3*I*sqrt(a)*x**(3/2)/(4*s
qrt(-1 + b*x/a)) - I*a**2*acosh(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(3/2)) + I*b*x**(
5/2)/(2*sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (-a**(3/2)*sqrt(x)/(4*b*sqrt
(1 - b*x/a)) + 3*sqrt(a)*x**(3/2)/(4*sqrt(1 - b*x/a)) + a**2*asin(sqrt(b)*sqrt(x
)/sqrt(a))/(4*b**(3/2)) - b*x**(5/2)/(2*sqrt(a)*sqrt(1 - b*x/a)), True))

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError