\(\int \frac {1+a x}{\sqrt {a x} \sqrt {1-a x}} \, dx\) [25]

Optimal result

Integrand size = 23, antiderivative size = 37 \[ \int \frac {1+a x}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {a x} \sqrt {1-a x}}{a}-\frac {3 \arcsin (1-2 a x)}{2 a} \]

[Out]

3/2*arcsin(2*a*x-1)/a-(a*x)^(1/2)*(-a*x+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {81, 55, 633, 222} \[ \int \frac {1+a x}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {3 \arcsin (1-2 a x)}{2 a}-\frac {\sqrt {a x} \sqrt {1-a x}}{a} \]

[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 55

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 81

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 222

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]

Rule 633

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]

Rubi steps \begin{align*} \text {integral}& = -\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 \text {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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(37)=74\).

Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.03 \[ \int \frac {1+a x}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {\sqrt {a} x (-1+a x)+6 \sqrt {x} \sqrt {1-a x} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{-1+\sqrt {1-a x}}\right )}{\sqrt {a} \sqrt {-a x (-1+a x)}} \]

[In]

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

[Out]

(Sqrt[a]*x*(-1 + a*x) + 6*Sqrt[x]*Sqrt[1 - a*x]*ArcTan[(Sqrt[a]*Sqrt[x])/(-1 + Sqrt[1 - a*x])])/(Sqrt[a]*Sqrt[
-(a*x*(-1 + a*x))])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.54 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.89

method	result	size
default	\(-\frac {\sqrt {-a x +1}\, x \left (2 \,\operatorname {csgn}\left (a \right ) \sqrt {-x \left (a x -1\right ) a}-3 \arctan \left (\frac {\operatorname {csgn}\left (a \right ) \left (2 a x -1\right )}{2 \sqrt {-x \left (a x -1\right ) a}}\right )\right ) \operatorname {csgn}\left (a \right )}{2 \sqrt {a x}\, \sqrt {-x \left (a x -1\right ) a}}\)	\(70\)
meijerg	\(-\frac {\sqrt {x}\, \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-a \right )^{\frac {3}{2}} \sqrt {-a x +1}}{a}+\frac {\sqrt {\pi }\, \left (-a \right )^{\frac {3}{2}} \arcsin \left (\sqrt {a}\, \sqrt {x}\right )}{a^{\frac {3}{2}}}\right )}{\sqrt {-a}\, \sqrt {a x}\, \sqrt {\pi }}+\frac {2 \sqrt {x}\, \arcsin \left (\sqrt {a}\, \sqrt {x}\right )}{\sqrt {a}\, \sqrt {a x}}\)	\(86\)
risch	\(\frac {x \left (a x -1\right ) \sqrt {a x \left (-a x +1\right )}}{\sqrt {-x \left (a x -1\right ) a}\, \sqrt {a x}\, \sqrt {-a x +1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a^{2} x^{2}+a x}}\right ) \sqrt {a x \left (-a x +1\right )}}{2 \sqrt {a^{2}}\, \sqrt {a x}\, \sqrt {-a x +1}}\)	\(103\)

[In]

int((a*x+1)/(a*x)^(1/2)/(-a*x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(-a*x+1)^(1/2)*x*(2*csgn(a)*(-x*(a*x-1)*a)^(1/2)-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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16 \[ \int \frac {1+a x}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {a x} \sqrt {-a x + 1} + 3 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right )}{a} \]

[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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.69 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.59 \[ \int \frac {1+a x}{\sqrt {a x} \sqrt {1-a x}} \, dx=a \left (\begin {cases} - \frac {i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a^{2}} - \frac {i \sqrt {x} \sqrt {a x - 1}}{a^{\frac {3}{2}}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {\operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a^{2}} + \frac {x^{\frac {3}{2}}}{\sqrt {a} \sqrt {- a x + 1}} - \frac {\sqrt {x}}{a^{\frac {3}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {2 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a} & \text {for}\: \left |{a x}\right | > 1 \\\frac {2 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.11 \[ \int \frac {1+a x}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {3 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{2 \, a} - \frac {\sqrt {-a^{2} x^{2} + a x}}{a} \]

[In]

integrate((a*x+1)/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="maxima")

[Out]

-3/2*arcsin(-(2*a^2*x - a)/a)/a - sqrt(-a^2*x^2 + a*x)/a

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76 \[ \int \frac {1+a x}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {a x} \sqrt {-a x + 1} - 3 \, \arcsin \left (\sqrt {a x}\right )}{a} \]

[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

Mupad [B] (verification not implemented)

Time = 3.75 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.19 \[ \int \frac {1+a x}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {2\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{a}-\frac {4\,\mathrm {atan}\left (\frac {a\,\left (\sqrt {1-a\,x}-1\right )}{\sqrt {a\,x}\,\sqrt {a^2}}\right )}{\sqrt {a^2}}-\frac {\frac {2\,\sqrt {a\,x}}{\sqrt {1-a\,x}-1}-\frac {2\,{\left (a\,x\right )}^{3/2}}{{\left (\sqrt {1-a\,x}-1\right )}^3}}{a\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^2} \]

[In]

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

[Out]

(2*atan((a*x)^(1/2)/((1 - a*x)^(1/2) - 1)))/a - (4*atan((a*((1 - a*x)^(1/2) - 1))/((a*x)^(1/2)*(a^2)^(1/2))))/
(a^2)^(1/2) - ((2*(a*x)^(1/2))/((1 - a*x)^(1/2) - 1) - (2*(a*x)^(3/2))/((1 - a*x)^(1/2) - 1)^3)/(a*((a*x)/((1
- a*x)^(1/2) - 1)^2 + 1)^2)