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 = 0.0125165, 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, Rules used = {80, 53, 619, 216} \[ -\frac{\sqrt{a x} \sqrt{1-a x}}{a}-\frac{3 \sin ^{-1}(1-2 a x)}{2 a} \]

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)

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{1+a x}{\sqrt{a x} \sqrt{1-a x}} \, dx &=-\frac{\sqrt{a x} \sqrt{1-a x}}{a}+\frac{3}{2} \int \frac{1}{\sqrt{a x} \sqrt{1-a x}} \, dx\\ &=-\frac{\sqrt{a x} \sqrt{1-a x}}{a}+\frac{3}{2} \int \frac{1}{\sqrt{a x-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{a x} \sqrt{1-a x}}{a}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{2 a^2}\\ &=-\frac{\sqrt{a x} \sqrt{1-a x}}{a}-\frac{3 \sin ^{-1}(1-2 a x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0242423, 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 = 0.015, size = 70, normalized size = 1.9 \begin{align*} -{\frac{x{\it csgn} \left ( a \right ) }{2}\sqrt{-ax+1} \left ( 2\,\sqrt{-x \left ( ax-1 \right ) a}{\it csgn} \left ( a \right ) -3\,\arctan \left ( 1/2\,{\frac{{\it csgn} \left ( a \right ) \left ( 2\,ax-1 \right ) }{\sqrt{-x \left ( ax-1 \right ) a}}} \right ) \right ){\frac{1}{\sqrt{ax}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) a}}}} \end{align*}

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*(-x*(a*x-1)*a)^(1/2)*csgn(a)-3*arctan(1/2*csgn(a)*(2*a*x-1)/(-x*(a*x-1)*a)^(1/2)))*cs
gn(a)/(a*x)^(1/2)/(-x*(a*x-1)*a)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.30267, size = 100, normalized size = 2.7 \begin{align*} -\frac{\sqrt{a x} \sqrt{-a x + 1} + 3 \, \arctan \left (\frac{\sqrt{a x} \sqrt{-a x + 1}}{a x}\right )}{a} \end{align*}

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, 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 [C]  time = 7.76718, size = 133, normalized size = 3.59 \begin{align*} 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} \end{align*}

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)*s
qrt(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 [A]  time = 2.54551, size = 38, normalized size = 1.03 \begin{align*} -\frac{\sqrt{a x} \sqrt{-a x + 1} - 3 \, \arcsin \left (\sqrt{a x}\right )}{a} \end{align*}

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, algorithm="giac")

[Out]

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