3.7.49 \(\int \frac {1}{\sqrt {x} \sqrt {1-b x}} \, dx\) [649]

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

[Out]

2*arcsin(b^(1/2)*x^(1/2))/b^(1/2)

________________________________________________________________________________________

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 = {56, 222} \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 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*} \int \frac {1}{\sqrt {x} \sqrt {1-b x}} \, dx &=2 \text {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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 35, normalized size = 1.84 \begin {gather*} -\frac {2 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {1-b x}\right )}{\sqrt {-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.10, size = 35, normalized size = 1.84 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {-2 I \text {ArcCosh}\left [\sqrt {b} \sqrt {x}\right ]}{\sqrt {b}},\text {Abs}\left [b x\right ]>1\right \}\right \},\frac {2 \text {ArcSin}\left [\sqrt {b} \sqrt {x}\right ]}{\sqrt {b}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{-2 I ArcCosh[Sqrt[b] Sqrt[x]] / Sqrt[b], Abs[b x] > 1}}, 2 ArcSin[Sqrt[b] Sqrt[x]] / Sqrt[b]]

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(13)=26\).
time = 0.13, size = 48, normalized size = 2.53

method	result	size
meijerg	\(\frac {2 \arcsin \left (\sqrt {b}\, \sqrt {x}\right )}{\sqrt {b}}\)	\(14\)
default	\(\frac {\sqrt {x \left (-b x +1\right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{2 b}\right )}{\sqrt {-x^{2} b +x}}\right )}{\sqrt {x}\, \sqrt {-b x +1}\, \sqrt {b}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

________________________________________________________________________________________

Maxima [A]
time = 0.35, 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)

________________________________________________________________________________________

Fricas [A]
time = 0.32, 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)]

________________________________________________________________________________________

Sympy [A]
time = 0.51, 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))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (13) = 26\).
time = 0.00, size = 35, normalized size = 1.84 \begin {gather*} -\frac {2 \ln \left (\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)

[Out]

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

________________________________________________________________________________________

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)

________________________________________________________________________________________