\(\int \frac {1}{(a+\frac {b}{x})^{3/2} x} \, dx\) [1735]

Optimal result

Integrand size = 15, antiderivative size = 42 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x} \, dx=-\frac {2}{a \sqrt {a+\frac {b}{x}}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 53, 65, 214} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {a+\frac {b}{x}}} \]

[In]

Int[1/((a + b/x)^(3/2)*x),x]

[Out]

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

Rule 53

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 65

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 214

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

Rule 272

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

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x} \, dx=-\frac {2 \sqrt {a+\frac {b}{x}} x}{a (b+a x)}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \]

[In]

Integrate[1/((a + b/x)^(3/2)*x),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(197\) vs. \(2(34)=68\).

Time = 0.05 (sec) , antiderivative size = 198, normalized size of antiderivative = 4.71

method	result	size
default	\(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (\ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b \,x^{2}-2 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, x^{2}+2 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{2} x +2 a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}}-4 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b x +\ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{3}-2 \sqrt {a}\, \sqrt {x \left (a x +b \right )}\, b^{2}\right )}{a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b \left (a x +b \right )^{2}}\)	\(198\)

[In]

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

[Out]

((a*x+b)/x)^(1/2)*x/a^(3/2)*(ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*b*x^2-2*a^(5/2)*(x*(a*x
+b))^(1/2)*x^2+2*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*b^2*x+2*a^(3/2)*(x*(a*x+b))^(3/2)-4*a
^(3/2)*(x*(a*x+b))^(1/2)*b*x+ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^3-2*a^(1/2)*(x*(a*x+b))^(
1/2)*b^2)/(x*(a*x+b))^(1/2)/b/(a*x+b)^2

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.05 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x} \, dx=\left [-\frac {2 \, a x \sqrt {\frac {a x + b}{x}} - {\left (a x + b\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right )}{a^{3} x + a^{2} b}, -\frac {2 \, {\left (a x \sqrt {\frac {a x + b}{x}} + {\left (a x + b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right )\right )}}{a^{3} x + a^{2} b}\right ] \]

[In]

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

[Out]

[-(2*a*x*sqrt((a*x + b)/x) - (a*x + b)*sqrt(a)*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b))/(a^3*x + a^2*b)
, -2*(a*x*sqrt((a*x + b)/x) + (a*x + b)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a))/(a^3*x + a^2*b)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (32) = 64\).

Time = 0.94 (sec) , antiderivative size = 148, normalized size of antiderivative = 3.52 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x} \, dx=- \frac {2 a^{3} x \sqrt {1 + \frac {b}{a x}}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} - \frac {a^{3} x \log {\left (\frac {b}{a x} \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} + \frac {2 a^{3} x \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} - \frac {a^{2} b \log {\left (\frac {b}{a x} \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} + \frac {2 a^{2} b \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{a^{\frac {9}{2}} x + a^{\frac {7}{2}} b} \]

[In]

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

[Out]

-2*a**3*x*sqrt(1 + b/(a*x))/(a**(9/2)*x + a**(7/2)*b) - a**3*x*log(b/(a*x))/(a**(9/2)*x + a**(7/2)*b) + 2*a**3
*x*log(sqrt(1 + b/(a*x)) + 1)/(a**(9/2)*x + a**(7/2)*b) - a**2*b*log(b/(a*x))/(a**(9/2)*x + a**(7/2)*b) + 2*a*
*2*b*log(sqrt(1 + b/(a*x)) + 1)/(a**(9/2)*x + a**(7/2)*b)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x} \, dx=-\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {2}{\sqrt {a + \frac {b}{x}} a} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (34) = 68\).

Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x} \, dx=\frac {{\left (\log \left ({\left | b \right |}\right ) + 2\right )} \mathrm {sgn}\left (x\right )}{a^{\frac {3}{2}}} - \frac {\log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} - \frac {2 \, b}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )} a^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.86 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a\,\sqrt {a+\frac {b}{x}}} \]

[In]

int(1/(x*(a + b/x)^(3/2)),x)

[Out]

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