3.224 \(\int \sqrt{x} \sqrt{1-a x} \, dx\)

Optimal. Leaf size=63 \[ \frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{1-a x}-\frac{\sqrt{x} \sqrt{1-a x}}{4 a} \]

[Out]

-(Sqrt[x]*Sqrt[1 - a*x])/(4*a) + (x^(3/2)*Sqrt[1 - a*x])/2 + ArcSin[Sqrt[a]*Sqrt
[x]]/(4*a^(3/2))

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Rubi [A]  time = 0.0490371, antiderivative size = 63, 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 \[ \frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{1-a x}-\frac{\sqrt{x} \sqrt{1-a x}}{4 a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*Sqrt[1 - a*x],x]

[Out]

-(Sqrt[x]*Sqrt[1 - a*x])/(4*a) + (x^(3/2)*Sqrt[1 - a*x])/2 + ArcSin[Sqrt[a]*Sqrt
[x]]/(4*a^(3/2))

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Rubi in Sympy [A]  time = 8.20873, size = 51, normalized size = 0.81 \[ - \frac{\sqrt{x} \left (- a x + 1\right )^{\frac{3}{2}}}{2 a} + \frac{\sqrt{x} \sqrt{- a x + 1}}{4 a} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*(-a*x + 1)**(3/2)/(2*a) + sqrt(x)*sqrt(-a*x + 1)/(4*a) + asin(sqrt(a)*s
qrt(x))/(4*a**(3/2))

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Mathematica [A]  time = 0.0406145, size = 49, normalized size = 0.78 \[ \frac{\sqrt{a} \sqrt{x} \sqrt{1-a x} (2 a x-1)+\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*Sqrt[1 - a*x],x]

[Out]

(Sqrt[a]*Sqrt[x]*Sqrt[1 - a*x]*(-1 + 2*a*x) + ArcSin[Sqrt[a]*Sqrt[x]])/(4*a^(3/2
))

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Maple [A]  time = 0.007, size = 79, normalized size = 1.3 \[{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{-ax+1}}-{\frac{1}{4\,a}\sqrt{x}\sqrt{-ax+1}}+{\frac{1}{8}\sqrt{ \left ( -ax+1 \right ) x}\arctan \left ({1\sqrt{a} \left ( x-{\frac{1}{2\,a}} \right ){\frac{1}{\sqrt{-a{x}^{2}+x}}}} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-ax+1}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(-a*x+1)^(1/2),x)

[Out]

1/2*x^(3/2)*(-a*x+1)^(1/2)-1/4*x^(1/2)*(-a*x+1)^(1/2)/a+1/8/a^(3/2)*((-a*x+1)*x)
^(1/2)/(-a*x+1)^(1/2)/x^(1/2)*arctan(a^(1/2)*(x-1/2/a)/(-a*x^2+x)^(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 + 1)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287385, size = 1, normalized size = 0.02 \[ \left [\frac{2 \,{\left (2 \, a x - 1\right )} \sqrt{-a x + 1} \sqrt{-a} \sqrt{x} + \log \left (-2 \, \sqrt{-a x + 1} a \sqrt{x} -{\left (2 \, a x - 1\right )} \sqrt{-a}\right )}{8 \, \sqrt{-a} a}, \frac{{\left (2 \, a x - 1\right )} \sqrt{-a x + 1} \sqrt{a} \sqrt{x} - \arctan \left (\frac{\sqrt{-a x + 1}}{\sqrt{a} \sqrt{x}}\right )}{4 \, a^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-a*x + 1)*sqrt(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 10.2354, size = 148, normalized size = 2.35 \[ \begin{cases} \frac{i a x^{\frac{5}{2}}}{2 \sqrt{a x - 1}} - \frac{3 i x^{\frac{3}{2}}}{4 \sqrt{a x - 1}} + \frac{i \sqrt{x}}{4 a \sqrt{a x - 1}} - \frac{i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{4 a^{\frac{3}{2}}} & \text{for}\: \left |{a x}\right | > 1 \\- \frac{a x^{\frac{5}{2}}}{2 \sqrt{- a x + 1}} + \frac{3 x^{\frac{3}{2}}}{4 \sqrt{- a x + 1}} - \frac{\sqrt{x}}{4 a \sqrt{- a x + 1}} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(-a*x + 1)*sqrt(x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError