3.1725 \(\int \frac {1}{\sqrt {a+\frac {b}{x}}} \, dx\)

Optimal. Leaf size=43 \[ \frac {x \sqrt {a+\frac {b}{x}}}{a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \]

[Out]

-b*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(3/2)+x*(a+b/x)^(1/2)/a

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {242, 51, 63, 208} \[ \frac {x \sqrt {a+\frac {b}{x}}}{a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b/x],x]

[Out]

(Sqrt[a + b/x]*x)/a - (b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 242

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x}}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{a}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{a}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{a}\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 1.00 \[ \frac {x \sqrt {a+\frac {b}{x}}}{a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b/x],x]

[Out]

(Sqrt[a + b/x]*x)/a - (b*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)

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fricas [A]  time = 0.98, size = 98, normalized size = 2.28 \[ \left [\frac {2 \, a x \sqrt {\frac {a x + b}{x}} + \sqrt {a} b \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right )}{2 \, a^{2}}, \frac {a x \sqrt {\frac {a x + b}{x}} + \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right )}{a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.23, size = 71, normalized size = 1.65 \[ -\frac {b \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{2 \, a^{\frac {3}{2}}} + \frac {b \log \left ({\left | -2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} - b \right |}\right )}{2 \, a^{\frac {3}{2}} \mathrm {sgn}\relax (x)} + \frac {\sqrt {a x^{2} + b x}}{a \mathrm {sgn}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*log(abs(b))*sgn(x)/a^(3/2) + 1/2*b*log(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))/(a^(3/2)*sg
n(x)) + sqrt(a*x^2 + b*x)/(a*sgn(x))

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maple [A]  time = 0.00, size = 70, normalized size = 1.63 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (b \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\right ) x}{2 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((a*x+b)/x)^(1/2)*x*(b*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))-2*((a*x+b)*x)^(1/2)*a^(1/2))
/((a*x+b)*x)^(1/2)/a^(3/2)

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maxima [A]  time = 2.23, size = 67, normalized size = 1.56 \[ \frac {\sqrt {a + \frac {b}{x}} b}{{\left (a + \frac {b}{x}\right )} a - a^{2}} + \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{2 \, a^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a + b/x)*b/((a + b/x)*a - a^2) + 1/2*b*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(3/2)

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mupad [B]  time = 1.20, size = 66, normalized size = 1.53 \[ \frac {2\,x\,\left (\frac {3\,\sqrt {b}\,\sqrt {b+a\,x}}{2\,a\,x}+\frac {b^{3/2}\,\mathrm {asin}\left (\frac {\sqrt {a}\,\sqrt {x}\,1{}\mathrm {i}}{\sqrt {b}}\right )\,3{}\mathrm {i}}{2\,a^{3/2}\,x^{3/2}}\right )\,\sqrt {\frac {a\,x}{b}+1}}{3\,\sqrt {a+\frac {b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x)^(1/2),x)

[Out]

(2*x*((3*b^(1/2)*(b + a*x)^(1/2))/(2*a*x) + (b^(3/2)*asin((a^(1/2)*x^(1/2)*1i)/b^(1/2))*3i)/(2*a^(3/2)*x^(3/2)
))*((a*x)/b + 1)^(1/2))/(3*(a + b/x)^(1/2))

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sympy [A]  time = 2.63, size = 44, normalized size = 1.02 \[ \frac {\sqrt {b} \sqrt {x} \sqrt {\frac {a x}{b} + 1}}{a} - \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{a^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*sqrt(x)*sqrt(a*x/b + 1)/a - b*asinh(sqrt(a)*sqrt(x)/sqrt(b))/a**(3/2)

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