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

Optimal. Leaf size=35 \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]

[Out]

Sqrt[x]*Sqrt[1 - a*x] + ArcSin[Sqrt[a]*Sqrt[x]]/Sqrt[a]

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Rubi [A]  time = 0.032716, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - a*x]/Sqrt[x],x]

[Out]

Sqrt[x]*Sqrt[1 - a*x] + ArcSin[Sqrt[a]*Sqrt[x]]/Sqrt[a]

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Rubi in Sympy [A]  time = 5.92644, size = 29, normalized size = 0.83 \[ \sqrt{x} \sqrt{- a x + 1} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)*sqrt(-a*x + 1) + asin(sqrt(a)*sqrt(x))/sqrt(a)

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Mathematica [A]  time = 0.0265896, size = 35, normalized size = 1. \[ \sqrt{x} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[x]*Sqrt[1 - a*x] + ArcSin[Sqrt[a]*Sqrt[x]]/Sqrt[a]

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Maple [B]  time = 0.026, size = 62, normalized size = 1.8 \[ \sqrt{x}\sqrt{-ax+1}+{\frac{1}{2}\sqrt{ \left ( -ax+1 \right ) x}\arctan \left ({1\sqrt{a} \left ( x-{\frac{1}{2\,a}} \right ){\frac{1}{\sqrt{-a{x}^{2}+x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-ax+1}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-a*x+1)^(1/2)/x^(1/2),x)

[Out]

x^(1/2)*(-a*x+1)^(1/2)+1/2*((-a*x+1)*x)^(1/2)/(-a*x+1)^(1/2)/x^(1/2)/a^(1/2)*arc
tan(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 = 0.287702, size = 1, normalized size = 0.03 \[ \left [\frac{2 \, \sqrt{-a x + 1} \sqrt{-a} \sqrt{x} + \log \left (-2 \, \sqrt{-a x + 1} a \sqrt{x} -{\left (2 \, a x - 1\right )} \sqrt{-a}\right )}{2 \, \sqrt{-a}}, \frac{\sqrt{-a x + 1} \sqrt{a} \sqrt{x} - \arctan \left (\frac{\sqrt{-a x + 1}}{\sqrt{a} \sqrt{x}}\right )}{\sqrt{a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*sqrt(-a*x + 1)*sqrt(-a)*sqrt(x) + log(-2*sqrt(-a*x + 1)*a*sqrt(x) - (2*a
*x - 1)*sqrt(-a)))/sqrt(-a), (sqrt(-a*x + 1)*sqrt(a)*sqrt(x) - arctan(sqrt(-a*x
+ 1)/(sqrt(a)*sqrt(x))))/sqrt(a)]

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Sympy [A]  time = 5.87043, size = 83, normalized size = 2.37 \[ \begin{cases} \frac{i a x^{\frac{3}{2}}}{\sqrt{a x - 1}} - \frac{i \sqrt{x}}{\sqrt{a x - 1}} - \frac{i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} & \text{for}\: \left |{a x}\right | > 1 \\\sqrt{x} \sqrt{- a x + 1} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*a*x**(3/2)/sqrt(a*x - 1) - I*sqrt(x)/sqrt(a*x - 1) - I*acosh(sqrt(a
)*sqrt(x))/sqrt(a), Abs(a*x) > 1), (sqrt(x)*sqrt(-a*x + 1) + asin(sqrt(a)*sqrt(x
))/sqrt(a), 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