3.7.32 \(\int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx\) [632]

Optimal. Leaf size=24 \[ \frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 24, 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 \text {ArcSin}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[2 - b*x]),x]

[Out]

(2*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/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 {2-b x}} \, dx &=2 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{\sqrt {b}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 35, normalized size = 1.46 \begin {gather*} -\frac {2 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{\sqrt {-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*Sqrt[2 - b*x]),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(17)=34\).
time = 0.12, size = 50, normalized size = 2.08

method	result	size
meijerg	\(\frac {2 \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {b}}\)	\(18\)
default	\(\frac {\sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{\sqrt {-b x +2}\, \sqrt {x}\, \sqrt {b}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 21, normalized size = 0.88 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\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+2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.05, size = 56, normalized size = 2.33 \begin {gather*} \left [-\frac {\sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{b}, -\frac {2 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\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+2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.52, size = 56, normalized size = 2.33 \begin {gather*} \begin {cases} - \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{\sqrt {b}} & \text {for}\: \left |{b x}\right | > 2 \\\frac {2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \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+2)**(1/2),x)

[Out]

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

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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+2)^(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]%%%}+%%%{-8,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,[2,1]%%%}+%%%{6,[2,0]%%%}+%%%
{-4,[1,2]%%%}+%%%{-28

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Mupad [B]
time = 0.03, size = 27, normalized size = 1.12 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {\sqrt {2}-\sqrt {2-b\,x}}{\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)*(2 - b*x)^(1/2)),x)

[Out]

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

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