Integrand size = 13, antiderivative size = 67 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\frac {5}{4} b \sqrt {a+\frac {b}{x}} x+\frac {1}{2} a \sqrt {a+\frac {b}{x}} x^2+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 \sqrt {a}} \] Output:
5/4*b*(a+b/x)^(1/2)*x+1/2*a*(a+b/x)^(1/2)*x^2+3/4*b^2*arctanh((a+b/x)^(1/2 )/a^(1/2))/a^(1/2)
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\frac {1}{4} \left (\sqrt {a+\frac {b}{x}} x (5 b+2 a x)+\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}\right ) \] Input:
Integrate[(a + b/x)^(3/2)*x,x]
Output:
(Sqrt[a + b/x]*x*(5*b + 2*a*x) + (3*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sq rt[a])/4
Time = 0.28 (sec) , antiderivative size = 65, 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, 51, 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.
\(\displaystyle \int x \left (a+\frac {b}{x}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \left (a+\frac {b}{x}\right )^{3/2} x^3d\frac {1}{x}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} x^2 \left (a+\frac {b}{x}\right )^{3/2}-\frac {3}{4} b \int \sqrt {a+\frac {b}{x}} x^2d\frac {1}{x}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} x^2 \left (a+\frac {b}{x}\right )^{3/2}-\frac {3}{4} b \left (\frac {1}{2} b \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}-x \sqrt {a+\frac {b}{x}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} x^2 \left (a+\frac {b}{x}\right )^{3/2}-\frac {3}{4} b \left (\int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}-x \sqrt {a+\frac {b}{x}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} x^2 \left (a+\frac {b}{x}\right )^{3/2}-\frac {3}{4} b \left (x \left (-\sqrt {a+\frac {b}{x}}\right )-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}\right )\) |
Input:
Int[(a + b/x)^(3/2)*x,x]
Output:
((a + b/x)^(3/2)*x^2)/2 - (3*b*(-(Sqrt[a + b/x]*x) - (b*ArcTanh[Sqrt[a + b /x]/Sqrt[a]])/Sqrt[a]))/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] :> 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 = 0.15 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {\left (2 a x +5 b \right ) x \sqrt {\frac {a x +b}{x}}}{4}+\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x \left (a x +b \right )}}{8 \sqrt {a}\, \left (a x +b \right )}\) | \(83\) |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (4 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} x +10 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b +3 b^{2} \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \right )}{8 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}}}\) | \(96\) |
Input:
int((a+b/x)^(3/2)*x,x,method=_RETURNVERBOSE)
Output:
1/4*(2*a*x+5*b)*x*((a*x+b)/x)^(1/2)+3/8*b^2*ln((1/2*b+a*x)/a^(1/2)+(a*x^2+ b*x)^(1/2))/a^(1/2)*((a*x+b)/x)^(1/2)*(x*(a*x+b))^(1/2)/(a*x+b)
Time = 0.09 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.01 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\left [\frac {3 \, \sqrt {a} b^{2} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (2 \, a^{2} x^{2} + 5 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{8 \, a}, -\frac {3 \, \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) - {\left (2 \, a^{2} x^{2} + 5 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{4 \, a}\right ] \] Input:
integrate((a+b/x)^(3/2)*x,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 + 5*a*b*x)*sqrt((a*x + b)/x))/a, -1/4*(3*sqrt(-a)*b^2*arctan(sqrt( -a)*x*sqrt((a*x + b)/x)/(a*x + b)) - (2*a^2*x^2 + 5*a*b*x)*sqrt((a*x + b)/ x))/a]
Time = 2.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\frac {a \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}}{2} + \frac {5 b^{\frac {3}{2}} \sqrt {x} \sqrt {\frac {a x}{b} + 1}}{4} + \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{4 \sqrt {a}} \] Input:
integrate((a+b/x)**(3/2)*x,x)
Output:
a*sqrt(b)*x**(3/2)*sqrt(a*x/b + 1)/2 + 5*b**(3/2)*sqrt(x)*sqrt(a*x/b + 1)/ 4 + 3*b**2*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(4*sqrt(a))
Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.46 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=-\frac {3 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{8 \, \sqrt {a}} + \frac {5 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {a + \frac {b}{x}} a b^{2}}{4 \, {\left ({\left (a + \frac {b}{x}\right )}^{2} - 2 \, {\left (a + \frac {b}{x}\right )} a + a^{2}\right )}} \] Input:
integrate((a+b/x)^(3/2)*x,x, algorithm="maxima")
Output:
-3/8*b^2*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/sqrt(a) + 1/4*(5*(a + b/x)^(3/2)*b^2 - 3*sqrt(a + b/x)*a*b^2)/((a + b/x)^2 - 2*(a + b/x)*a + a^2)
Time = 0.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.15 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=-\frac {3 \, b^{2} \log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, \sqrt {a}} + \frac {3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, \sqrt {a}} + \frac {1}{4} \, \sqrt {a x^{2} + b x} {\left (2 \, a x \mathrm {sgn}\left (x\right ) + 5 \, b \mathrm {sgn}\left (x\right )\right )} \] Input:
integrate((a+b/x)^(3/2)*x,x, algorithm="giac")
Output:
-3/8*b^2*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))*sgn(x)/sq rt(a) + 3/8*b^2*log(abs(b))*sgn(x)/sqrt(a) + 1/4*sqrt(a*x^2 + b*x)*(2*a*x* sgn(x) + 5*b*sgn(x))
Time = 0.37 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\frac {5\,x^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}{4}+\frac {3\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4\,\sqrt {a}}-\frac {3\,a\,x^2\,\sqrt {a+\frac {b}{x}}}{4} \] Input:
int(x*(a + b/x)^(3/2),x)
Output:
(5*x^2*(a + b/x)^(3/2))/4 + (3*b^2*atanh((a + b/x)^(1/2)/a^(1/2)))/(4*a^(1 /2)) - (3*a*x^2*(a + b/x)^(1/2))/4
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \left (a+\frac {b}{x}\right )^{3/2} x \, dx=\frac {2 \sqrt {x}\, \sqrt {a x +b}\, a^{2} x +5 \sqrt {x}\, \sqrt {a x +b}\, a b +3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b^{2}}{4 a} \] Input:
int((a+b/x)^(3/2)*x,x)
Output:
(2*sqrt(x)*sqrt(a*x + b)*a**2*x + 5*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)