Integrand size = 13, antiderivative size = 72 \[ \int \frac {x}{\sqrt {a+\frac {b}{x}}} \, dx=-\frac {3 b \sqrt {a+\frac {b}{x}} x}{4 a^2}+\frac {\sqrt {a+\frac {b}{x}} x^2}{2 a}+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 a^{5/2}} \] Output:
-3/4*b*(a+b/x)^(1/2)*x/a^2+1/2*(a+b/x)^(1/2)*x^2/a+3/4*b^2*arctanh((a+b/x) ^(1/2)/a^(1/2))/a^(5/2)
Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {\sqrt {a} \sqrt {a+\frac {b}{x}} x (-3 b+2 a x)+3 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 a^{5/2}} \] Input:
Integrate[x/Sqrt[a + b/x],x]
Output:
(Sqrt[a]*Sqrt[a + b/x]*x*(-3*b + 2*a*x) + 3*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt [a]])/(4*a^(5/2))
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {798, 52, 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.
\(\displaystyle \int \frac {x}{\sqrt {a+\frac {b}{x}}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {x^3}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {3 b \int \frac {x^2}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{4 a}+\frac {x^2 \sqrt {a+\frac {b}{x}}}{2 a}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {3 b \left (-\frac {b \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{2 a}-\frac {x \sqrt {a+\frac {b}{x}}}{a}\right )}{4 a}+\frac {x^2 \sqrt {a+\frac {b}{x}}}{2 a}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3 b \left (-\frac {\int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{a}-\frac {x \sqrt {a+\frac {b}{x}}}{a}\right )}{4 a}+\frac {x^2 \sqrt {a+\frac {b}{x}}}{2 a}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {x \sqrt {a+\frac {b}{x}}}{a}\right )}{4 a}+\frac {x^2 \sqrt {a+\frac {b}{x}}}{2 a}\) |
Input:
Int[x/Sqrt[a + b/x],x]
Output:
(Sqrt[a + b/x]*x^2)/(2*a) + (3*b*(-((Sqrt[a + b/x]*x)/a) + (b*ArcTanh[Sqrt [a + b/x]/Sqrt[a]])/a^(3/2)))/(4*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] :> 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 = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.19
method | result | size |
risch | \(\frac {\left (2 a x -3 b \right ) \left (a x +b \right )}{4 a^{2} \sqrt {\frac {a x +b}{x}}}+\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {x \left (a x +b \right )}}{8 a^{\frac {5}{2}} x \sqrt {\frac {a x +b}{x}}}\) | \(86\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (4 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} x -8 a^{\frac {3}{2}} \sqrt {x \left (a x +b \right )}\, b +2 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b -b^{2} \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a +4 a \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{2}\right )}{8 \sqrt {x \left (a x +b \right )}\, a^{\frac {7}{2}}}\) | \(143\) |
Input:
int(x/(a+b/x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/4*(2*a*x-3*b)*(a*x+b)/a^2/((a*x+b)/x)^(1/2)+3/8*b^2/a^(5/2)*ln((1/2*b+a* x)/a^(1/2)+(a*x^2+b*x)^(1/2))/x/((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2)
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.88 \[ \int \frac {x}{\sqrt {a+\frac {b}{x}}} \, dx=\left [\frac {3 \, \sqrt {a} b^{2} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (2 \, a^{2} x^{2} - 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{8 \, a^{3}}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (2 \, a^{2} x^{2} - 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{4 \, a^{3}}\right ] \] Input:
integrate(x/(a+b/x)^(1/2),x, algorithm="fricas")
Output:
[1/8*(3*sqrt(a)*b^2*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(2* a^2*x^2 - 3*a*b*x)*sqrt((a*x + b)/x))/a^3, -1/4*(3*sqrt(-a)*b^2*arctan(sqr t(-a)*x*sqrt((a*x + b)/x)/(a*x + b)) - (2*a^2*x^2 - 3*a*b*x)*sqrt((a*x + b )/x))/a^3]
Time = 3.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.39 \[ \int \frac {x}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {x^{\frac {5}{2}}}{2 \sqrt {b} \sqrt {\frac {a x}{b} + 1}} - \frac {\sqrt {b} x^{\frac {3}{2}}}{4 a \sqrt {\frac {a x}{b} + 1}} - \frac {3 b^{\frac {3}{2}} \sqrt {x}}{4 a^{2} \sqrt {\frac {a x}{b} + 1}} + \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{4 a^{\frac {5}{2}}} \] Input:
integrate(x/(a+b/x)**(1/2),x)
Output:
x**(5/2)/(2*sqrt(b)*sqrt(a*x/b + 1)) - sqrt(b)*x**(3/2)/(4*a*sqrt(a*x/b + 1)) - 3*b**(3/2)*sqrt(x)/(4*a**2*sqrt(a*x/b + 1)) + 3*b**2*asinh(sqrt(a)*s qrt(x)/sqrt(b))/(4*a**(5/2))
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.44 \[ \int \frac {x}{\sqrt {a+\frac {b}{x}}} \, dx=-\frac {3 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{8 \, a^{\frac {5}{2}}} - \frac {3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} - 5 \, \sqrt {a + \frac {b}{x}} a b^{2}}{4 \, {\left ({\left (a + \frac {b}{x}\right )}^{2} a^{2} - 2 \, {\left (a + \frac {b}{x}\right )} a^{3} + a^{4}\right )}} \] Input:
integrate(x/(a+b/x)^(1/2),x, algorithm="maxima")
Output:
-3/8*b^2*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(5/2) - 1/4*(3*(a + b/x)^(3/2)*b^2 - 5*sqrt(a + b/x)*a*b^2)/((a + b/x)^2*a^2 - 2 *(a + b/x)*a^3 + a^4)
Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.22 \[ \int \frac {x}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {1}{4} \, \sqrt {a x^{2} + b x} {\left (\frac {2 \, x}{a \mathrm {sgn}\left (x\right )} - \frac {3 \, b}{a^{2} \mathrm {sgn}\left (x\right )}\right )} + \frac {3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, a^{\frac {5}{2}}} - \frac {3 \, b^{2} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{8 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \] Input:
integrate(x/(a+b/x)^(1/2),x, algorithm="giac")
Output:
1/4*sqrt(a*x^2 + b*x)*(2*x/(a*sgn(x)) - 3*b/(a^2*sgn(x))) + 3/8*b^2*log(ab s(b))*sgn(x)/a^(5/2) - 3/8*b^2*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*s qrt(a) + b))/(a^(5/2)*sgn(x))
Time = 0.40 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {x}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4\,a^{5/2}}+\frac {5\,x^2\,\sqrt {a+\frac {b}{x}}}{4\,a}-\frac {3\,x^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}{4\,a^2} \] Input:
int(x/(a + b/x)^(1/2),x)
Output:
(3*b^2*atanh((a + b/x)^(1/2)/a^(1/2)))/(4*a^(5/2)) + (5*x^2*(a + b/x)^(1/2 ))/(4*a) - (3*x^2*(a + b/x)^(3/2))/(4*a^2)
Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {x}{\sqrt {a+\frac {b}{x}}} \, dx=\frac {2 \sqrt {x}\, \sqrt {a x +b}\, a^{2} x -3 \sqrt {x}\, \sqrt {a x +b}\, a b +3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{2}}{4 a^{3}} \] Input:
int(x/(a+b/x)^(1/2),x)
Output:
(2*sqrt(x)*sqrt(a*x + b)*a**2*x - 3*sqrt(x)*sqrt(a*x + b)*a*b + 3*sqrt(a)* log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b))*b**2)/(4*a**3)