Integrand size = 11, antiderivative size = 60 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {3 b}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {x}{a \sqrt {a+\frac {b}{x}}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \] Output:
3*b/a^2/(a+b/x)^(1/2)+x/a/(a+b/x)^(1/2)-3*b*arctanh((a+b/x)^(1/2)/a^(1/2)) /a^(5/2)
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} x (3 b+a x)}{a^2 (b+a x)}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \] Input:
Integrate[(a + b/x)^(-3/2),x]
Output:
(Sqrt[a + b/x]*x*(3*b + a*x))/(a^2*(b + a*x)) - (3*b*ArcTanh[Sqrt[a + b/x] /Sqrt[a]])/a^(5/2)
Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {773, 52, 61, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 773 |
\(\displaystyle -\int \frac {x^2}{\left (a+\frac {b}{x}\right )^{3/2}}d\frac {1}{x}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {3 b \int \frac {x}{\left (a+\frac {b}{x}\right )^{3/2}}d\frac {1}{x}}{2 a}+\frac {x}{a \sqrt {a+\frac {b}{x}}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {3 b \left (\frac {\int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{a}+\frac {2}{a \sqrt {a+\frac {b}{x}}}\right )}{2 a}+\frac {x}{a \sqrt {a+\frac {b}{x}}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3 b \left (\frac {2 \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{a b}+\frac {2}{a \sqrt {a+\frac {b}{x}}}\right )}{2 a}+\frac {x}{a \sqrt {a+\frac {b}{x}}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {3 b \left (\frac {2}{a \sqrt {a+\frac {b}{x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{2 a}+\frac {x}{a \sqrt {a+\frac {b}{x}}}\) |
Input:
Int[(a + b/x)^(-3/2),x]
Output:
x/(a*Sqrt[a + b/x]) + (3*b*(2/(a*Sqrt[a + b/x]) - (2*ArcTanh[Sqrt[a + b/x] /Sqrt[a]])/a^(3/2)))/(2*a)
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] - Simp[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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] - Simp[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] && 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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
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] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(115\) vs. \(2(50)=100\).
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.93
method | result | size |
risch | \(\frac {a x +b}{a^{2} \sqrt {\frac {a x +b}{x}}}+\frac {\left (-\frac {3 b \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right )}{2 a^{\frac {5}{2}}}+\frac {2 b \sqrt {a \left (x +\frac {b}{a}\right )^{2}-b \left (x +\frac {b}{a}\right )}}{a^{3} \left (x +\frac {b}{a}\right )}\right ) \sqrt {x \left (a x +b \right )}}{x \sqrt {\frac {a x +b}{x}}}\) | \(116\) |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, x \left (3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b \,x^{2}-6 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} x^{2}+6 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{2} x +4 a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}}-12 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b x +3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{3}-6 \sqrt {x \left (a x +b \right )}\, \sqrt {a}\, b^{2}\right )}{2 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, \left (a x +b \right )^{2}}\) | \(198\) |
Input:
int(1/(a+b/x)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/a^2*(a*x+b)/((a*x+b)/x)^(1/2)+(-3/2*b/a^(5/2)*ln((1/2*b+a*x)/a^(1/2)+(a* x^2+b*x)^(1/2))+2*b/a^3/(x+b/a)*(a*(x+b/a)^2-b*(x+b/a))^(1/2))/x/((a*x+b)/ x)^(1/2)*(x*(a*x+b))^(1/2)
Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.68 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a b x + b^{2}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (a^{2} x^{2} + 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{2 \, {\left (a^{4} x + a^{3} b\right )}}, \frac {3 \, {\left (a b x + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (a^{2} x^{2} + 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{a^{4} x + a^{3} b}\right ] \] Input:
integrate(1/(a+b/x)^(3/2),x, algorithm="fricas")
Output:
[1/2*(3*(a*b*x + b^2)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(a^2*x^2 + 3*a*b*x)*sqrt((a*x + b)/x))/(a^4*x + a^3*b), (3*(a*b*x + b^2)*sqrt(-a)*arctan(sqrt(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) + (a^2*x^2 + 3*a*b*x)*sqrt((a*x + b)/x))/(a^4*x + a^3*b)]
Time = 2.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {x^{\frac {3}{2}}}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {3 \sqrt {b} \sqrt {x}}{a^{2} \sqrt {\frac {a x}{b} + 1}} - \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{a^{\frac {5}{2}}} \] Input:
integrate(1/(a+b/x)**(3/2),x)
Output:
x**(3/2)/(a*sqrt(b)*sqrt(a*x/b + 1)) + 3*sqrt(b)*sqrt(x)/(a**2*sqrt(a*x/b + 1)) - 3*b*asinh(sqrt(a)*sqrt(x)/sqrt(b))/a**(5/2)
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {3 \, {\left (a + \frac {b}{x}\right )} b - 2 \, a b}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} - \sqrt {a + \frac {b}{x}} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{2 \, a^{\frac {5}{2}}} \] Input:
integrate(1/(a+b/x)^(3/2),x, algorithm="maxima")
Output:
(3*(a + b/x)*b - 2*a*b)/((a + b/x)^(3/2)*a^2 - sqrt(a + b/x)*a^3) + 3/2*b* log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (50) = 100\).
Time = 0.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=-\frac {{\left (3 \, b \log \left ({\left | b \right |}\right ) + 4 \, b\right )} \mathrm {sgn}\left (x\right )}{2 \, a^{\frac {5}{2}}} + \frac {3 \, b \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{2 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} + \frac {\sqrt {a x^{2} + b x}}{a^{2} \mathrm {sgn}\left (x\right )} + \frac {2 \, b^{2}}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )} a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \] Input:
integrate(1/(a+b/x)^(3/2),x, algorithm="giac")
Output:
-1/2*(3*b*log(abs(b)) + 4*b)*sgn(x)/a^(5/2) + 3/2*b*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(a^(5/2)*sgn(x)) + sqrt(a*x^2 + b*x)/(a^ 2*sgn(x)) + 2*b^2/(((sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b)*a^(5/2)*s gn(x))
Time = 0.46 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {2\,x\,{\left (\frac {a\,x}{b}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {5}{2};\ \frac {7}{2};\ -\frac {a\,x}{b}\right )}{5\,{\left (a+\frac {b}{x}\right )}^{3/2}} \] Input:
int(1/(a + b/x)^(3/2),x)
Output:
(2*x*((a*x)/b + 1)^(3/2)*hypergeom([3/2, 5/2], 7/2, -(a*x)/b))/(5*(a + b/x )^(3/2))
Time = 0.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx=\frac {-12 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b +9 \sqrt {a}\, \sqrt {a x +b}\, b +4 \sqrt {x}\, a^{2} x +12 \sqrt {x}\, a b}{4 \sqrt {a x +b}\, a^{3}} \] Input:
int(1/(a+b/x)^(3/2),x)
Output:
( - 12*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b) )*b + 9*sqrt(a)*sqrt(a*x + b)*b + 4*sqrt(x)*a**2*x + 12*sqrt(x)*a*b)/(4*sq rt(a*x + b)*a**3)