3.1726 \(\int \frac {1}{\sqrt {a+\frac {b}{x}} x} \, dx\)

Optimal. Leaf size=25 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[Out]

2*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 63, 208} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x]*x),x]

[Out]

(2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x]*x),x]

[Out]

(2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a]

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fricas [A]  time = 1.11, size = 60, normalized size = 2.40 \[ \left [\frac {\log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right )}{\sqrt {a}}, -\frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right )}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 25, normalized size = 1.00 \[ -\frac {2 \, \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 121, normalized size = 4.84 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (-b \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-b \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}-2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}\right ) x}{2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a+b/x)^(1/2),x)

[Out]

-1/2*((a*x+b)/x)^(1/2)*x*(2*((a*x+b)*x)^(1/2)*a^(1/2)-b*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))-
2*(a*x^2+b*x)^(1/2)*a^(1/2)-b*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^(1/2)))/((a*x+b)*x)^(1/2)/b/a^(1/
2)

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maxima [A]  time = 2.34, size = 37, normalized size = 1.48 \[ -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/sqrt(a)

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mupad [B]  time = 1.41, size = 19, normalized size = 0.76 \[ \frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b/x)^(1/2)),x)

[Out]

(2*atanh((a + b/x)^(1/2)/a^(1/2)))/a^(1/2)

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sympy [A]  time = 1.20, size = 22, normalized size = 0.88 \[ \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*asinh(sqrt(a)*sqrt(x)/sqrt(b))/sqrt(a)

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