\(\int \frac {1}{\sqrt {a+\frac {b}{x}} x^{3/2}} \, dx\) [1782]

Optimal result

Integrand size = 17, antiderivative size = 30 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^{3/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{\sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {344, 223, 212} \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^{3/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{\sqrt {b}} \]

[In]

Int[1/(Sqrt[a + b/x]*x^(3/2)),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 2.73 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^{3/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {x} \sqrt {\frac {b+a x}{x}}}{\sqrt {b}}\right )}{\sqrt {b}} \]

[In]

Integrate[1/(Sqrt[a + b/x]*x^(3/2)),x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30

method	result	size
default	\(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )}{\sqrt {a x +b}\, \sqrt {b}}\)	\(39\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^{3/2}} \, dx=\left [\frac {\log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right )}{\sqrt {b}}, \frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right )}{b}\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^{3/2}} \, dx=- \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{\sqrt {b}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^{3/2}} \, dx=\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{\sqrt {b}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^{3/2}} \, dx=\frac {2 \, {\left (\frac {\arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {\arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right )}{\sqrt {-b}}\right )}}{\mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} x^{3/2}} \, dx=\int \frac {1}{x^{3/2}\,\sqrt {a+\frac {b}{x}}} \,d x \]

[In]

int(1/(x^(3/2)*(a + b/x)^(1/2)),x)

[Out]

int(1/(x^(3/2)*(a + b/x)^(1/2)), x)