3.7.49 \(\int \frac {1}{\sqrt {x} \sqrt {1-b x}} \, dx\)

Optimal. Leaf size=19 \[ \frac {2 \sin ^{-1}\left (\sqrt {b} \sqrt {x}\right )}{\sqrt {b}} \]

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {54, 216} \begin {gather*} \frac {2 \sin ^{-1}\left (\sqrt {b} \sqrt {x}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[1 - b*x]),x]

[Out]

(2*ArcSin[Sqrt[b]*Sqrt[x]])/Sqrt[b]

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

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \sin ^{-1}\left (\sqrt {b} \sqrt {x}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcSin[Sqrt[b]*Sqrt[x]])/Sqrt[b]

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IntegrateAlgebraic [A]  time = 0.05, size = 38, normalized size = 2.00 \begin {gather*} \frac {2 \sqrt {-b} \log \left (\sqrt {1-b x}-\sqrt {-b} \sqrt {x}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[x]*Sqrt[1 - b*x]),x]

[Out]

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

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fricas [A]  time = 1.16, size = 57, normalized size = 3.00 \begin {gather*} \left [-\frac {\sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + 1} \sqrt {-b} \sqrt {x} + 1\right )}{b}, -\frac {2 \, \arctan \left (\frac {\sqrt {-b x + 1}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(1/x^(1/2)/(-b*x+1)^(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]2/abs(b)*b^2/b/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+1)+b)-sqrt(-
b)*sqrt(-b*x+1)))

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maple [B]  time = 0.01, size = 48, normalized size = 2.53 \begin {gather*} \frac {\sqrt {\left (-b x +1\right ) x}\, \arctan \left (\frac {\left (x -\frac {1}{2 b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+x}}\right )}{\sqrt {-b x +1}\, \sqrt {b}\, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.89, size = 21, normalized size = 1.11 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {-b x + 1}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(-b*x + 1)/(sqrt(b)*sqrt(x)))/sqrt(b)

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mupad [B]  time = 0.13, size = 23, normalized size = 1.21 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {\sqrt {1-b\,x}-1}{\sqrt {b}\,\sqrt {x}}\right )}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(1 - b*x)^(1/2)),x)

[Out]

-(4*atan(((1 - b*x)^(1/2) - 1)/(b^(1/2)*x^(1/2))))/b^(1/2)

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sympy [A]  time = 1.06, size = 42, normalized size = 2.21 \begin {gather*} \begin {cases} - \frac {2 i \operatorname {acosh}{\left (\sqrt {b} \sqrt {x} \right )}}{\sqrt {b}} & \text {for}\: \left |{b x}\right | > 1 \\\frac {2 \operatorname {asin}{\left (\sqrt {b} \sqrt {x} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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