\(\int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx\) [1757]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [B] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [F(-1)]

Optimal result

Integrand size = 17, antiderivative size = 50 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {x}}-\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{\sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {344, 201, 223, 212} \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=-\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{\sqrt {b}}-\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {x}} \]

[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 201

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 212

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

Rule 223

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 344

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]

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {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 \text {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 \text {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] (verified)

Time = 5.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x}} \left (\sqrt {b+a x}+\frac {a x \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {b+a x}} \]

[In]

Integrate[Sqrt[a + b/x]/x^(3/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.08

method result size
default \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a x +\sqrt {a x +b}\, \sqrt {b}\right )}{\sqrt {x}\, \sqrt {a x +b}\, \sqrt {b}}\) \(54\)
risch \(-\frac {\sqrt {\frac {a x +b}{x}}}{\sqrt {x}}-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{\sqrt {b}\, \sqrt {a x +b}}\) \(57\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.42 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=\left [\frac {a \sqrt {b} x \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, b \sqrt {x} \sqrt {\frac {a x + b}{x}}}{2 \, b x}, \frac {a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) - b \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b x}\right ] \]

[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)]

Sympy [A] (verification not implemented)

Time = 1.43 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=- \frac {\sqrt {a} \sqrt {1 + \frac {b}{a x}}}{\sqrt {x}} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{\sqrt {b}} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (38) = 76\).

Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=\frac {a \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{2 \, \sqrt {b}} - \frac {\sqrt {a + \frac {b}{x}} a \sqrt {x}}{{\left (a + \frac {b}{x}\right )} x - b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=\frac {{\left (\frac {a^{2} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {\sqrt {a x + b} a}{x}\right )} \mathrm {sgn}\left (x\right )}{a} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=\int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \,d x \]

[In]

int((a + b/x)^(1/2)/x^(3/2),x)

[Out]

int((a + b/x)^(1/2)/x^(3/2), x)