Integrand size = 15, antiderivative size = 60 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x} \, dx=-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2}{a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \] Output:
-2/3/a/(a+b/x)^(3/2)-2/a^2/(a+b/x)^(1/2)+2*arctanh((a+b/x)^(1/2)/a^(1/2))/ a^(5/2)
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x} \, dx=-\frac {2 \sqrt {a+\frac {b}{x}} x (3 b+4 a x)}{3 a^2 (b+a x)^2}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}} \] Input:
Integrate[1/((a + b/x)^(5/2)*x),x]
Output:
(-2*Sqrt[a + b/x]*x*(3*b + 4*a*x))/(3*a^2*(b + a*x)^2) + (2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(5/2)
Time = 0.31 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 61, 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.
\(\displaystyle \int \frac {1}{x \left (a+\frac {b}{x}\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\int \frac {x}{\left (a+\frac {b}{x}\right )^{5/2}}d\frac {1}{x}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {\int \frac {x}{\left (a+\frac {b}{x}\right )^{3/2}}d\frac {1}{x}}{a}-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {\frac {\int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{a}+\frac {2}{a \sqrt {a+\frac {b}{x}}}}{a}-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {\frac {2 \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{a b}+\frac {2}{a \sqrt {a+\frac {b}{x}}}}{a}-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\frac {2}{a \sqrt {a+\frac {b}{x}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}}{a}-\frac {2}{3 a \left (a+\frac {b}{x}\right )^{3/2}}\) |
Input:
Int[1/((a + b/x)^(5/2)*x),x]
Output:
-2/(3*a*(a + b/x)^(3/2)) - (2/(a*Sqrt[a + b/x]) - (2*ArcTanh[Sqrt[a + b/x] /Sqrt[a]])/a^(3/2))/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] && 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[(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]]
Leaf count of result is larger than twice the leaf count of optimal. \(273\) vs. \(2(48)=96\).
Time = 0.24 (sec) , antiderivative size = 274, normalized size of antiderivative = 4.57
method | result | size |
default | \(\frac {\sqrt {\frac {a x +b}{x}}\, x \left (-6 \sqrt {x \left (a x +b \right )}\, a^{\frac {7}{2}} x^{3}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b \,x^{3}+6 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} x -18 \sqrt {x \left (a x +b \right )}\, a^{\frac {5}{2}} b \,x^{2}+9 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{2} b^{2} x^{2}+4 b \,a^{\frac {3}{2}} \left (x \left (a x +b \right )\right )^{\frac {3}{2}}-18 \sqrt {x \left (a x +b \right )}\, a^{\frac {3}{2}} b^{2} x +9 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a \,b^{3} x -6 \sqrt {x \left (a x +b \right )}\, \sqrt {a}\, b^{3}+3 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) b^{4}\right )}{3 a^{\frac {5}{2}} \sqrt {x \left (a x +b \right )}\, b \left (a x +b \right )^{3}}\) | \(274\) |
Input:
int(1/(a+b/x)^(5/2)/x,x,method=_RETURNVERBOSE)
Output:
1/3*((a*x+b)/x)^(1/2)*x/a^(5/2)*(-6*(x*(a*x+b))^(1/2)*a^(7/2)*x^3+3*ln(1/2 *(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*b*x^3+6*(x*(a*x+b))^(3 /2)*a^(5/2)*x-18*(x*(a*x+b))^(1/2)*a^(5/2)*b*x^2+9*ln(1/2*(2*(x*(a*x+b))^( 1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*b^2*x^2+4*b*a^(3/2)*(x*(a*x+b))^(3/2)-1 8*(x*(a*x+b))^(1/2)*a^(3/2)*b^2*x+9*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2* a*x+b)/a^(1/2))*a*b^3*x-6*(x*(a*x+b))^(1/2)*a^(1/2)*b^3+3*ln(1/2*(2*(x*(a* x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^4)/(x*(a*x+b))^(1/2)/b/(a*x+b)^3
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (48) = 96\).
Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 3.37 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) - 2 \, {\left (4 \, a^{2} x^{2} + 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (4 \, a^{2} x^{2} + 3 \, a b x\right )} \sqrt {\frac {a x + b}{x}}\right )}}{3 \, {\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}}\right ] \] Input:
integrate(1/(a+b/x)^(5/2)/x,x, algorithm="fricas")
Output:
[1/3*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(a)*log(2*a*x + 2*sqrt(a)*x*sqrt((a* x + b)/x) + b) - 2*(4*a^2*x^2 + 3*a*b*x)*sqrt((a*x + b)/x))/(a^5*x^2 + 2*a ^4*b*x + a^3*b^2), -2/3*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(-a)*arctan(sqrt( -a)*x*sqrt((a*x + b)/x)/(a*x + b)) + (4*a^2*x^2 + 3*a*b*x)*sqrt((a*x + b)/ x))/(a^5*x^2 + 2*a^4*b*x + a^3*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (48) = 96\).
Time = 1.93 (sec) , antiderivative size = 700, normalized size of antiderivative = 11.67 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x} \, dx=- \frac {8 a^{7} x^{3} \sqrt {1 + \frac {b}{a x}}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {3 a^{7} x^{3} \log {\left (\frac {b}{a x} \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} + \frac {6 a^{7} x^{3} \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {14 a^{6} b x^{2} \sqrt {1 + \frac {b}{a x}}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {9 a^{6} b x^{2} \log {\left (\frac {b}{a x} \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} + \frac {18 a^{6} b x^{2} \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {6 a^{5} b^{2} x \sqrt {1 + \frac {b}{a x}}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {9 a^{5} b^{2} x \log {\left (\frac {b}{a x} \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} + \frac {18 a^{5} b^{2} x \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} - \frac {3 a^{4} b^{3} \log {\left (\frac {b}{a x} \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} + \frac {6 a^{4} b^{3} \log {\left (\sqrt {1 + \frac {b}{a x}} + 1 \right )}}{3 a^{\frac {19}{2}} x^{3} + 9 a^{\frac {17}{2}} b x^{2} + 9 a^{\frac {15}{2}} b^{2} x + 3 a^{\frac {13}{2}} b^{3}} \] Input:
integrate(1/(a+b/x)**(5/2)/x,x)
Output:
-8*a**7*x**3*sqrt(1 + b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9* a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 3*a**7*x**3*log(b/(a*x))/(3*a**(19/ 2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) + 6* a**7*x**3*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x** 2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 14*a**6*b*x**2*sqrt(1 + b/(a* x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13 /2)*b**3) - 9*a**6*b*x**2*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x **2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) + 18*a**6*b*x**2*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 6*a**5*b**2*x*sqrt(1 + b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 9*a**5*b**2 *x*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2* x + 3*a**(13/2)*b**3) + 18*a**5*b**2*x*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(1 9/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 3*a**4*b**3*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15 /2)*b**2*x + 3*a**(13/2)*b**3) + 6*a**4*b**3*log(sqrt(1 + b/(a*x)) + 1)/(3 *a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b* *3)
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x} \, dx=-\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {2 \, {\left (4 \, a + \frac {3 \, b}{x}\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}} \] Input:
integrate(1/(a+b/x)^(5/2)/x,x, algorithm="maxima")
Output:
-log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(5/2) - 2/3*(4 *a + 3*b/x)/((a + b/x)^(3/2)*a^2)
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (48) = 96\).
Time = 0.14 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.43 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x} \, dx=\frac {{\left (3 \, \log \left ({\left | b \right |}\right ) + 8\right )} \mathrm {sgn}\left (x\right )}{3 \, a^{\frac {5}{2}}} - \frac {\log \left ({\left | 2 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b \right |}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (6 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b + 9 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{2} + 4 \, b^{3}\right )}}{3 \, {\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} + b\right )}^{3} a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \] Input:
integrate(1/(a+b/x)^(5/2)/x,x, algorithm="giac")
Output:
1/3*(3*log(abs(b)) + 8)*sgn(x)/a^(5/2) - log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(a^(5/2)*sgn(x)) - 2/3*(6*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b + 9*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^2 + 4*b^3)/(((s qrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b)^3*a^(5/2)*sgn(x))
Time = 0.80 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2\,\left (a+\frac {b}{x}\right )}{a^2}+\frac {2}{3\,a}}{{\left (a+\frac {b}{x}\right )}^{3/2}} \] Input:
int(1/(x*(a + b/x)^(5/2)),x)
Output:
(2*atanh((a + b/x)^(1/2)/a^(1/2)))/a^(5/2) - ((2*(a + b/x))/a^2 + 2/(3*a)) /(a + b/x)^(3/2)
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x} \, dx=\frac {2 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) a x +2 \sqrt {a}\, \sqrt {a x +b}\, \mathrm {log}\left (\frac {\sqrt {a x +b}+\sqrt {x}\, \sqrt {a}}{\sqrt {b}}\right ) b -\frac {8 \sqrt {x}\, a^{2} x}{3}-2 \sqrt {x}\, a b}{\sqrt {a x +b}\, a^{3} \left (a x +b \right )} \] Input:
int(1/(a+b/x)^(5/2)/x,x)
Output:
(2*(3*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt(b)) *a*x + 3*sqrt(a)*sqrt(a*x + b)*log((sqrt(a*x + b) + sqrt(x)*sqrt(a))/sqrt( b))*b - 4*sqrt(x)*a**2*x - 3*sqrt(x)*a*b))/(3*sqrt(a*x + b)*a**3*(a*x + b) )