Integrand size = 17, antiderivative size = 75 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}+\frac {2}{b^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{5/2}} \]
2/3/b/(a+b/x)^(3/2)/x^(3/2)-2*arctanh(b^(1/2)/(a+b/x)^(1/2)/x^(1/2))/b^(5/ 2)+2/b^2/(a+b/x)^(1/2)/x^(1/2)
Time = 10.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\frac {2 \left (\sqrt {b} \sqrt {x} (4 b+3 a x)-3 \sqrt {a} \sqrt {1+\frac {b}{a x}} x (b+a x) \text {arcsinh}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )\right )}{3 b^{5/2} \sqrt {a+\frac {b}{x}} x (b+a x)} \]
(2*(Sqrt[b]*Sqrt[x]*(4*b + 3*a*x) - 3*Sqrt[a]*Sqrt[1 + b/(a*x)]*x*(b + a*x )*ArcSinh[Sqrt[b]/(Sqrt[a]*Sqrt[x])]))/(3*b^(5/2)*Sqrt[a + b/x]*x*(b + a*x ))
Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {860, 252, 252, 224, 219}
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.
\(\displaystyle \int \frac {1}{x^{7/2} \left (a+\frac {b}{x}\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 860 |
\(\displaystyle -2 \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^2}d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle -2 \left (\frac {\int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x}d\frac {1}{\sqrt {x}}}{b}-\frac {1}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 252 |
\(\displaystyle -2 \left (\frac {\frac {\int \frac {1}{\sqrt {a+\frac {b}{x}}}d\frac {1}{\sqrt {x}}}{b}-\frac {1}{b \sqrt {x} \sqrt {a+\frac {b}{x}}}}{b}-\frac {1}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -2 \left (\frac {\frac {\int \frac {1}{1-\frac {b}{x}}d\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}}{b}-\frac {1}{b \sqrt {x} \sqrt {a+\frac {b}{x}}}}{b}-\frac {1}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{3/2}}-\frac {1}{b \sqrt {x} \sqrt {a+\frac {b}{x}}}}{b}-\frac {1}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\) |
-2*(-1/3*1/(b*(a + b/x)^(3/2)*x^(3/2)) + (-(1/(b*Sqrt[a + b/x]*Sqrt[x])) + ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])]/b^(3/2))/b)
3.19.1.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[-k/c Subst[Int[(a + b/(c^n*x^(k*n)))^p/x^(k*(m + 1 ) + 1), x], x, 1/(c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && ILtQ[n, 0] && FractionQ[m]
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (-3 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {a x +b}\, a x +4 b^{\frac {3}{2}}+3 x a \sqrt {b}-3 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) b \sqrt {a x +b}\right )}{3 b^{\frac {5}{2}} \left (a x +b \right )^{2}}\) | \(85\) |
2/3*((a*x+b)/x)^(1/2)*x^(1/2)*(-3*arctanh((a*x+b)^(1/2)/b^(1/2))*(a*x+b)^( 1/2)*a*x+4*b^(3/2)+3*x*a*b^(1/2)-3*arctanh((a*x+b)^(1/2)/b^(1/2))*b*(a*x+b )^(1/2))/b^(5/2)/(a*x+b)^2
Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.73 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {b} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (3 \, a b x + 4 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}}, \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (3 \, a b x + 4 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}\right )}}{3 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}}\right ] \]
[1/3*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(b)*log((a*x - 2*sqrt(b)*sqrt(x)*sqr t((a*x + b)/x) + 2*b)/x) + 2*(3*a*b*x + 4*b^2)*sqrt(x)*sqrt((a*x + b)/x))/ (a^2*b^3*x^2 + 2*a*b^4*x + b^5), 2/3*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(-b) *arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/b) + (3*a*b*x + 4*b^2)*sqrt(x)* sqrt((a*x + b)/x))/(a^2*b^3*x^2 + 2*a*b^4*x + b^5)]
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (63) = 126\).
Time = 27.06 (sec) , antiderivative size = 697, normalized size of antiderivative = 9.29 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\frac {3 a^{3} b^{4} x^{3} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {6 a^{3} b^{4} x^{3} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {6 a^{2} b^{5} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {9 a^{2} b^{5} x^{2} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {18 a^{2} b^{5} x^{2} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {14 a b^{6} x \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {9 a b^{6} x \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {18 a b^{6} x \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {8 b^{7} \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {3 b^{7} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {6 b^{7} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} \]
3*a**3*b**4*x**3*log(a*x/b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) - 6*a**3*b**4*x**3*log(sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b **(19/2)) + 6*a**2*b**5*x**2*sqrt(a*x/b + 1)/(3*a**3*b**(13/2)*x**3 + 9*a* *2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) + 9*a**2*b**5*x**2*log( a*x/b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) - 18*a**2*b**5*x**2*log(sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2 )*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) + 14*a*b** 6*x*sqrt(a*x/b + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b **(17/2)*x + 3*b**(19/2)) + 9*a*b**6*x*log(a*x/b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) - 18*a*b**6*x*log( sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a* b**(17/2)*x + 3*b**(19/2)) + 8*b**7*sqrt(a*x/b + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) + 3*b**7*log(a*x /b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b **(19/2)) - 6*b**7*log(sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2)*x**3 + 9*a** 2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2))
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{b^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} x + b\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} x^{\frac {3}{2}}} \]
log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b)))/b ^(5/2) + 2/3*(3*(a + b/x)*x + b)/((a + b/x)^(3/2)*b^2*x^(3/2))
Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {2 \, {\left (3 \, a x + 4 \, b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}} \]
Timed out. \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\int \frac {1}{x^{7/2}\,{\left (a+\frac {b}{x}\right )}^{5/2}} \,d x \]