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

Optimal. Leaf size=37 \[ -\frac{\sqrt{a x} \sqrt{1-a x}}{a}-\frac{3 \sin ^{-1}(1-2 a x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0646522, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{\sqrt{a x} \sqrt{1-a x}}{a}-\frac{3 \sin ^{-1}(1-2 a x)}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[(1 + a*x)/(Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.75011, size = 29, normalized size = 0.78 \[ - \frac{\sqrt{a x} \sqrt{- a x + 1}}{a} + \frac{3 \operatorname{asin}{\left (2 a x - 1 \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0530202, size = 61, normalized size = 1.65 \[ \frac{\sqrt{a} x (a x-1)+3 \sqrt{x} \sqrt{1-a x} \sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a} \sqrt{-a x (a x-1)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.024, size = 70, normalized size = 1.9 \[ -{\frac{x{\it csgn} \left ( a \right ) }{2}\sqrt{-ax+1} \left ( 2\,{\it csgn} \left ( a \right ) \sqrt{-x \left ( ax-1 \right ) a}-3\,\arctan \left ( 1/2\,{\frac{ \left ( 2\,ax-1 \right ){\it csgn} \left ( a \right ) }{\sqrt{-x \left ( ax-1 \right ) a}}} \right ) \right ){\frac{1}{\sqrt{ax}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-a*x+1)^(1/2)*x*(2*csgn(a)*(-x*(a*x-1)*a)^(1/2)-3*arctan(1/2*(2*a*x-1)*csg
n(a)/(-x*(a*x-1)*a)^(1/2)))*csgn(a)/(a*x)^(1/2)/(-x*(a*x-1)*a)^(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((a*x + 1)/(sqrt(a*x)*sqrt(-a*x + 1)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239394, size = 58, normalized size = 1.57 \[ -\frac{\sqrt{a x} \sqrt{-a x + 1} + 3 \, \arctan \left (\frac{\sqrt{a x} \sqrt{-a x + 1}}{a x}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(sqrt(a*x)*sqrt(-a*x + 1) + 3*arctan(sqrt(a*x)*sqrt(-a*x + 1)/(a*x)))/a

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Sympy [A]  time = 18.9627, size = 133, normalized size = 3.59 \[ a \left (\begin{cases} - \frac{i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{a^{2}} - \frac{i \sqrt{x} \sqrt{a x - 1}}{a^{\frac{3}{2}}} & \text{for}\: \left |{a x}\right | > 1 \\\frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{a^{2}} + \frac{x^{\frac{3}{2}}}{\sqrt{a} \sqrt{- a x + 1}} - \frac{\sqrt{x}}{a^{\frac{3}{2}} \sqrt{- a x + 1}} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{2 i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{a} & \text{for}\: \left |{a x}\right | > 1 \\\frac{2 \operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241976, size = 38, normalized size = 1.03 \[ -\frac{\sqrt{a x} \sqrt{-a x + 1} - 3 \, \arcsin \left (\sqrt{a x}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(sqrt(a*x)*sqrt(-a*x + 1) - 3*arcsin(sqrt(a*x)))/a