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

Optimal. Leaf size=39 \[ x \sqrt{a+\frac{b}{x}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.018864, antiderivative size = 39, normalized size of antiderivative = 1., 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, 47, 63, 208} \[ x \sqrt{a+\frac{b}{x}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x],x]

[Out]

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

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]

Rule 47

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

Rubi steps

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

Mathematica [A]  time = 0.0163468, size = 39, normalized size = 1. \[ x \sqrt{a+\frac{b}{x}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x],x]

[Out]

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

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Maple [B]  time = 0.003, size = 74, normalized size = 1.9 \begin{align*}{\frac{x}{2}\sqrt{{\frac{ax+b}{x}}} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.32817, size = 234, normalized size = 6. \begin{align*} \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}, \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}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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, (a*x*sqrt((a*x +
b)/x) - sqrt(-a)*b*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a))/a]

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Sympy [A]  time = 1.82385, size = 42, normalized size = 1.08 \begin{align*} \sqrt{b} \sqrt{x} \sqrt{\frac{a x}{b} + 1} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*sqrt(x)*sqrt(a*x/b + 1) + b*asinh(sqrt(a)*sqrt(x)/sqrt(b))/sqrt(a)

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Giac [B]  time = 1.14471, size = 86, normalized size = 2.21 \begin{align*} -\frac{b \log \left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ) \mathrm{sgn}\left (x\right )}{2 \, \sqrt{a}} + \frac{b \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{2 \, \sqrt{a}} + \sqrt{a x^{2} + b x} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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