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

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

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {50, 54, 216} \begin {gather*} \sqrt {x} \sqrt {1-a x}+\frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \end {gather*}

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]

Rule 50

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 54

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 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 {\sqrt {1-a x}}{\sqrt {x}} \, dx &=\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}+\operatorname {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*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 1.00 \begin {gather*} \sqrt {x} \sqrt {1-a x}+\frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.08, size = 54, normalized size = 1.54 \begin {gather*} \sqrt {x} \sqrt {1-a x}+\frac {\sqrt {-a} \log \left (\sqrt {1-a x}-\sqrt {-a} \sqrt {x}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - a*x]/Sqrt[x],x]

[Out]

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

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fricas [A]  time = 0.40, size = 92, normalized size = 2.63 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{4,[1,
1]%%%}+%%%{4,[1,0]%%%}+%%%{-4,[0,1]%%%}+%%%{-4,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,[2,1]%%%}+%%%{6,[2,0]%%%}+%%%
{-4,[1,2]%%%}+%%%{-16,[1,1]%%%}+%%%{-4,[1,0]%%%}+%%%{6,[0,2]%%%}+%%%{4,[0,1]%%%}+%%%{6,[0,0]%%%},0,%%%{4,[3,3]
%%%}+%%%{-4,[3,2]%%%}+%%%{-4,[3,1]%%%}+%%%{4,[3,0]%%%}+%%%{4,[2,3]%%%}+%%%{-52,[2,2]%%%}+%%%{12,[2,1]%%%}+%%%{
4,[2,0]%%%}+%%%{-4,[1,3]%%%}+%%%{-12,[1,2]%%%}+%%%{52,[1,1]%%%}+%%%{-4,[1,0]%%%}+%%%{-4,[0,3]%%%}+%%%{4,[0,2]%
%%}+%%%{4,[0,1]%%%}+%%%{-4,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{-4,[4,3]%%%}+%%%{6,[4,2]%%%}+%%%{-4,[4,1]%%%}+%%%{1
,[4,0]%%%}+%%%{4,[3,4]%%%}+%%%{-8,[3,3]%%%}+%%%{8,[3,2]%%%}+%%%{-8,[3,1]%%%}+%%%{4,[3,0]%%%}+%%%{6,[2,4]%%%}+%
%%{-8,[2,3]%%%}+%%%{20,[2,2]%%%}+%%%{-8,[2,1]%%%}+%%%{6,[2,0]%%%}+%%%{4,[1,4]%%%}+%%%{-8,[1,3]%%%}+%%%{8,[1,2]
%%%}+%%%{-8,[1,1]%%%}+%%%{4,[1,0]%%%}+%%%{1,[0,4]%%%}+%%%{-4,[0,3]%%%}+%%%{6,[0,2]%%%}+%%%{-4,[0,1]%%%}+%%%{1,
[0,0]%%%}] at parameters values [-15.6438432182,61.7937478349]Warning, choosing root of [1,0,%%%{4,[1,1]%%%}+%
%%{4,[1,0]%%%}+%%%{-4,[0,1]%%%}+%%%{-4,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,[2,1]%%%}+%%%{6,[2,0]%%%}+%%%{-4,[1,2
]%%%}+%%%{-16,[1,1]%%%}+%%%{-4,[1,0]%%%}+%%%{6,[0,2]%%%}+%%%{4,[0,1]%%%}+%%%{6,[0,0]%%%},0,%%%{4,[3,3]%%%}+%%%
{-4,[3,2]%%%}+%%%{-4,[3,1]%%%}+%%%{4,[3,0]%%%}+%%%{4,[2,3]%%%}+%%%{-52,[2,2]%%%}+%%%{12,[2,1]%%%}+%%%{4,[2,0]%
%%}+%%%{-4,[1,3]%%%}+%%%{-12,[1,2]%%%}+%%%{52,[1,1]%%%}+%%%{-4,[1,0]%%%}+%%%{-4,[0,3]%%%}+%%%{4,[0,2]%%%}+%%%{
4,[0,1]%%%}+%%%{-4,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{-4,[4,3]%%%}+%%%{6,[4,2]%%%}+%%%{-4,[4,1]%%%}+%%%{1,[4,0]%%
%}+%%%{4,[3,4]%%%}+%%%{-8,[3,3]%%%}+%%%{8,[3,2]%%%}+%%%{-8,[3,1]%%%}+%%%{4,[3,0]%%%}+%%%{6,[2,4]%%%}+%%%{-8,[2
,3]%%%}+%%%{20,[2,2]%%%}+%%%{-8,[2,1]%%%}+%%%{6,[2,0]%%%}+%%%{4,[1,4]%%%}+%%%{-8,[1,3]%%%}+%%%{8,[1,2]%%%}+%%%
{-8,[1,1]%%%}+%%%{4,[1,0]%%%}+%%%{1,[0,4]%%%}+%%%{-4,[0,3]%%%}+%%%{6,[0,2]%%%}+%%%{-4,[0,1]%%%}+%%%{1,[0,0]%%%
}] at parameters values [-29.292030761,78.6493344628]1/abs(a)*a^2/a*(1/a*sqrt(-a*x+1)*sqrt(-a*(-a*x+1)+a)+1/sq
rt(-a)*ln(abs(sqrt(-a*(-a*x+1)+a)-sqrt(-a)*sqrt(-a*x+1))))

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maple [B]  time = 0.01, size = 62, normalized size = 1.77 \begin {gather*} \sqrt {-a x +1}\, \sqrt {x}+\frac {\sqrt {\left (-a x +1\right ) x}\, \arctan \left (\frac {\left (x -\frac {1}{2 a}\right ) \sqrt {a}}{\sqrt {-a \,x^{2}+x}}\right )}{2 \sqrt {-a x +1}\, \sqrt {a}\, \sqrt {x}} \end {gather*}

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

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maxima [A]  time = 0.96, size = 48, normalized size = 1.37 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 2.99, size = 38, normalized size = 1.09 \begin {gather*} \sqrt {x}\,\sqrt {1-a\,x}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {1-a\,x}-1}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 1.90, size = 83, normalized size = 2.37 \begin {gather*} \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} \end {gather*}

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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