Integrand size = 13, antiderivative size = 69 \[ \int \sqrt {a+\frac {b}{x}} x \, dx=\frac {b \sqrt {a+\frac {b}{x}} x}{4 a}+\frac {1}{2} \sqrt {a+\frac {b}{x}} x^2-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 a^{3/2}} \] Output:
1/4*b*(a+b/x)^(1/2)*x/a+1/2*(a+b/x)^(1/2)*x^2-1/4*b^2*arctanh((a+b/x)^(1/2 )/a^(1/2))/a^(3/2)
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \sqrt {a+\frac {b}{x}} x \, dx=\frac {\sqrt {a} \sqrt {a+\frac {b}{x}} x (b+2 a x)-b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 a^{3/2}} \] Input:
Integrate[Sqrt[a + b/x]*x,x]
Output:
(Sqrt[a]*Sqrt[a + b/x]*x*(b + 2*a*x) - b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]) /(4*a^(3/2))
Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {798, 51, 52, 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.
\(\displaystyle \int x \sqrt {a+\frac {b}{x}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \sqrt {a+\frac {b}{x}} x^3d\frac {1}{x}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} x^2 \sqrt {a+\frac {b}{x}}-\frac {1}{4} b \int \frac {x^2}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} x^2 \sqrt {a+\frac {b}{x}}-\frac {1}{4} b \left (-\frac {b \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{2 a}-\frac {x \sqrt {a+\frac {b}{x}}}{a}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} x^2 \sqrt {a+\frac {b}{x}}-\frac {1}{4} b \left (-\frac {\int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{a}-\frac {x \sqrt {a+\frac {b}{x}}}{a}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} x^2 \sqrt {a+\frac {b}{x}}-\frac {1}{4} b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {x \sqrt {a+\frac {b}{x}}}{a}\right )\) |
Input:
Int[Sqrt[a + b/x]*x,x]
Output:
(Sqrt[a + b/x]*x^2)/2 - (b*(-((Sqrt[a + b/x]*x)/a) + (b*ArcTanh[Sqrt[a + b /x]/Sqrt[a]])/a^(3/2)))/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22
method | result | size |
risch | \(\frac {\left (2 a x +b \right ) x \sqrt {\frac {a x +b}{x}}}{4 a}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{8 a^{\frac {3}{2}} \left (a x +b \right )}\) | \(84\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (4 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} x +2 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b -b^{2} \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \right )}{8 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}}}\) | \(96\) |
Input:
int((a+b/x)^(1/2)*x,x,method=_RETURNVERBOSE)
Output:
1/4*(2*a*x+b)*x/a*((a*x+b)/x)^(1/2)-1/8*b^2/a^(3/2)*ln((1/2*b+a*x)/a^(1/2) +(a*x^2+b*x)^(1/2))*((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2)/(a*x+b)
Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.88 \[ \int \sqrt {a+\frac {b}{x}} x \, dx=\left [\frac {\sqrt {a} b^{2} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (2 \, a^{2} x^{2} + a b x\right )} \sqrt {\frac {a x + b}{x}}}{8 \, a^{2}}, \frac {\sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (2 \, a^{2} x^{2} + a b x\right )} \sqrt {\frac {a x + b}{x}}}{4 \, a^{2}}\right ] \] Input:
integrate((a+b/x)^(1/2)*x,x, algorithm="fricas")
Output:
[1/8*(sqrt(a)*b^2*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(2*a^ 2*x^2 + a*b*x)*sqrt((a*x + b)/x))/a^2, 1/4*(sqrt(-a)*b^2*arctan(sqrt(-a)*x *sqrt((a*x + b)/x)/(a*x + b)) + (2*a^2*x^2 + a*b*x)*sqrt((a*x + b)/x))/a^2 ]
Time = 2.43 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.41 \[ \int \sqrt {a+\frac {b}{x}} x \, dx=\frac {a x^{\frac {5}{2}}}{2 \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {3 \sqrt {b} x^{\frac {3}{2}}}{4 \sqrt {\frac {a x}{b} + 1}} + \frac {b^{\frac {3}{2}} \sqrt {x}}{4 a \sqrt {\frac {a x}{b} + 1}} - \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{4 a^{\frac {3}{2}}} \] Input:
integrate((a+b/x)**(1/2)*x,x)
Output:
a*x**(5/2)/(2*sqrt(b)*sqrt(a*x/b + 1)) + 3*sqrt(b)*x**(3/2)/(4*sqrt(a*x/b + 1)) + b**(3/2)*sqrt(x)/(4*a*sqrt(a*x/b + 1)) - b**2*asinh(sqrt(a)*sqrt(x )/sqrt(b))/(4*a**(3/2))
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.45 \[ \int \sqrt {a+\frac {b}{x}} x \, dx=\frac {b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{8 \, a^{\frac {3}{2}}} + \frac {{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} + \sqrt {a + \frac {b}{x}} a b^{2}}{4 \, {\left ({\left (a + \frac {b}{x}\right )}^{2} a - 2 \, {\left (a + \frac {b}{x}\right )} a^{2} + a^{3}\right )}} \] Input:
integrate((a+b/x)^(1/2)*x,x, algorithm="maxima")
Output:
1/8*b^2*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(3/2) + 1/4*((a + b/x)^(3/2)*b^2 + sqrt(a + b/x)*a*b^2)/((a + b/x)^2*a - 2*(a + b /x)*a^2 + a^3)
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.10 \[ \int \sqrt {a+\frac {b}{x}} x \, dx=-\frac {b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, a^{\frac {3}{2}}} + \frac {1}{8} \, {\left (2 \, \sqrt {a x^{2} + b x} {\left (2 \, x + \frac {b}{a}\right )} + \frac {b^{2} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{a^{\frac {3}{2}}}\right )} \mathrm {sgn}\left (x\right ) \] Input:
integrate((a+b/x)^(1/2)*x,x, algorithm="giac")
Output:
-1/8*b^2*log(abs(b))*sgn(x)/a^(3/2) + 1/8*(2*sqrt(a*x^2 + b*x)*(2*x + b/a) + b^2*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/a^(3/2))*sg n(x)
Time = 0.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78 \[ \int \sqrt {a+\frac {b}{x}} x \, dx=\frac {x^2\,\sqrt {a+\frac {b}{x}}}{4}-\frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4\,a^{3/2}}+\frac {x^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}{4\,a} \] Input:
int(x*(a + b/x)^(1/2),x)
Output:
(x^2*(a + b/x)^(1/2))/4 - (b^2*atanh((a + b/x)^(1/2)/a^(1/2)))/(4*a^(3/2)) + (x^2*(a + b/x)^(3/2))/(4*a)
Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.81 \[ \int \sqrt {a+\frac {b}{x}} x \, dx=\frac {2 \sqrt {x}\, \sqrt {a x +b}\, a^{2} x +\sqrt {x}\, \sqrt {a x +b}\, a b -\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{2}}{4 a^{2}} \] Input:
int((a+b/x)^(1/2)*x,x)
Output:
(2*sqrt(x)*sqrt(a*x + b)*a**2*x + sqrt(x)*sqrt(a*x + b)*a*b - sqrt(a)*log( (sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*b**2)/(4*a**2)