\(\int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx\) [220]

Optimal result

Integrand size = 16, antiderivative size = 35 \[ \int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx=\sqrt {x} \sqrt {1-a x}+\frac {\arcsin \left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {52, 56, 222} \[ \int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx=\frac {\arcsin \left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}+\sqrt {x} \sqrt {1-a x} \]

[In]

Int[Sqrt[1 - a*x]/Sqrt[x],x]

[Out]

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 56

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

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx=\sqrt {x} \sqrt {1-a x}+\frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{-1+\sqrt {1-a x}}\right )}{\sqrt {a}} \]

[In]

Integrate[Sqrt[1 - a*x]/Sqrt[x],x]

[Out]

Sqrt[x]*Sqrt[1 - a*x] + (2*ArcTan[(Sqrt[a]*Sqrt[x])/(-1 + Sqrt[1 - a*x])])/Sqrt[a]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(25)=50\).

Time = 0.35 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.63

method	result	size
meijerg	\(\frac {\sqrt {-a}\, \left (-2 \sqrt {\pi }\, \sqrt {x}\, \sqrt {-a}\, \sqrt {-a x +1}-\frac {2 \sqrt {\pi }\, \sqrt {-a}\, \arcsin \left (\sqrt {a}\, \sqrt {x}\right )}{\sqrt {a}}\right )}{2 \sqrt {\pi }\, a}\)	\(57\)
default	\(\sqrt {x}\, \sqrt {-a x +1}+\frac {\sqrt {\left (-a x +1\right ) x}\, \arctan \left (\frac {\sqrt {a}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a \,x^{2}+x}}\right )}{2 \sqrt {-a x +1}\, \sqrt {x}\, \sqrt {a}}\)	\(62\)
risch	\(-\frac {\sqrt {x}\, \left (a x -1\right ) \sqrt {\left (-a x +1\right ) x}}{\sqrt {-x \left (a x -1\right )}\, \sqrt {-a x +1}}+\frac {\sqrt {\left (-a x +1\right ) x}\, \arctan \left (\frac {\sqrt {a}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a \,x^{2}+x}}\right )}{2 \sqrt {-a x +1}\, \sqrt {x}\, \sqrt {a}}\)	\(88\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.63 \[ \int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx=\left [\frac {2 \, \sqrt {-a x + 1} a \sqrt {x} - \sqrt {-a} \log \left (-2 \, a x + 2 \, \sqrt {-a x + 1} \sqrt {-a} \sqrt {x} + 1\right )}{2 \, a}, \frac {\sqrt {-a x + 1} a \sqrt {x} - \sqrt {a} \arctan \left (\frac {\sqrt {-a x + 1}}{\sqrt {a} \sqrt {x}}\right )}{a}\right ] \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.86 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.34 \[ \int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx=\begin {cases} i \sqrt {x} \sqrt {a x - 1} - \frac {i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{\sqrt {a}} & \text {for}\: \left |{a x}\right | > 1 \\- \frac {a x^{\frac {3}{2}}}{\sqrt {- a x + 1}} + \frac {\sqrt {x}}{\sqrt {- a x + 1}} + \frac {\operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{\sqrt {a}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx=-\frac {\arctan \left (\frac {\sqrt {-a x + 1}}{\sqrt {a} \sqrt {x}}\right )}{\sqrt {a}} + \frac {\sqrt {-a x + 1}}{{\left (a - \frac {a x - 1}{x}\right )} \sqrt {x}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (25) = 50\).

Time = 5.54 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx=\frac {a {\left (\frac {\log \left ({\left | -\sqrt {-a x + 1} \sqrt {-a} + \sqrt {{\left (a x - 1\right )} a + a} \right |}\right )}{\sqrt {-a}} + \frac {\sqrt {{\left (a x - 1\right )} a + a} \sqrt {-a x + 1}}{a}\right )}}{{\left | a \right |}} \]

[In]

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

[Out]

a*(log(abs(-sqrt(-a*x + 1)*sqrt(-a) + sqrt((a*x - 1)*a + a)))/sqrt(-a) + sqrt((a*x - 1)*a + a)*sqrt(-a*x + 1)/
a)/abs(a)

Mupad [B] (verification not implemented)

Time = 11.91 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx=\sqrt {x}\,\sqrt {1-a\,x}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {1-a\,x}-1}\right )}{\sqrt {a}} \]

[In]

int((1 - a*x)^(1/2)/x^(1/2),x)

[Out]

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