3.1757 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^{3/2}} \, dx\)

Optimal. Leaf size=50 \[ -\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{x}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0260107, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {337, 195, 217, 206} \[ -\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{x}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]/x^(3/2),x]

[Out]

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

Rule 337

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, -Dist[k/c, Subst[
Int[(a + b/(c^n*x^(k*n)))^p/x^(k*(m + 1) + 1), x], x, 1/(c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && ILtQ[n,
0] && FractionQ[m]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.123413, size = 74, normalized size = 1.48 \[ \frac{\sqrt{a+\frac{b}{x}} \left (-\frac{a^{3/2} x^{3/2} \sqrt{\frac{b}{a x}+1} \sinh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}}\right )}{\sqrt{b} (a x+b)}-1\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]/x^(3/2),x]

[Out]

(Sqrt[a + b/x]*(-1 - (a^(3/2)*Sqrt[1 + b/(a*x)]*x^(3/2)*ArcSinh[Sqrt[b]/(Sqrt[a]*Sqrt[x])])/(Sqrt[b]*(b + a*x)
)))/Sqrt[x]

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Maple [A]  time = 0.013, size = 54, normalized size = 1.1 \begin{align*} -{\sqrt{{\frac{ax+b}{x}}} \left ({\it Artanh} \left ({\sqrt{ax+b}{\frac{1}{\sqrt{b}}}} \right ) ax+\sqrt{ax+b}\sqrt{b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{ax+b}}}{\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(1/2)/x^(3/2),x)

[Out]

-((a*x+b)/x)^(1/2)*(arctanh((a*x+b)^(1/2)/b^(1/2))*a*x+(a*x+b)^(1/2)*b^(1/2))/x^(1/2)/(a*x+b)^(1/2)/b^(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^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51862, size = 290, normalized size = 5.8 \begin{align*} \left [\frac{a \sqrt{b} x \log \left (\frac{a x - 2 \, \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}}}{2 \, b x}, \frac{a \sqrt{-b} x \arctan \left (\frac{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b}\right ) - b \sqrt{x} \sqrt{\frac{a x + b}{x}}}{b x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)/x^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.32732, size = 44, normalized size = 0.88 \begin{align*} - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x}}}{\sqrt{x}} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} \sqrt{x}} \right )}}{\sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(1/2)/x**(3/2),x)

[Out]

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

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Giac [A]  time = 1.30258, size = 54, normalized size = 1.08 \begin{align*} a{\left (\frac{\arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{a x + b}}{a x}\right )} \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^(3/2),x, algorithm="giac")

[Out]

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