Integrand size = 17, antiderivative size = 99 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{9/2}} \, dx=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{5/2}}+\frac {10}{3 b^2 \sqrt {a+\frac {b}{x}} x^{3/2}}-\frac {5 \sqrt {a+\frac {b}{x}}}{b^3 \sqrt {x}}+\frac {5 a \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{7/2}} \]
2/3/b/(a+b/x)^(3/2)/x^(5/2)+5*a*arctanh(b^(1/2)/(a+b/x)^(1/2)/x^(1/2))/b^( 7/2)+10/3/b^2/x^(3/2)/(a+b/x)^(1/2)-5*(a+b/x)^(1/2)/b^3/x^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{9/2}} \, dx=-\frac {2 \sqrt {1+\frac {b}{a x}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {7}{2},\frac {9}{2},-\frac {b}{a x}\right )}{7 a^2 \sqrt {a+\frac {b}{x}} x^{7/2}} \]
(-2*Sqrt[1 + b/(a*x)]*Hypergeometric2F1[5/2, 7/2, 9/2, -(b/(a*x))])/(7*a^2 *Sqrt[a + b/x]*x^(7/2))
Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {860, 252, 252, 262, 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.
| \(\displaystyle \int \frac {1}{x^{9/2} \left (a+\frac {b}{x}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 860 |
| \(\displaystyle -2 \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^3}d\frac {1}{\sqrt {x}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle -2 \left (\frac {5 \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^2}d\frac {1}{\sqrt {x}}}{3 b}-\frac {1}{3 b x^{5/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle -2 \left (\frac {5 \left (\frac {3 \int \frac {1}{\sqrt {a+\frac {b}{x}} x}d\frac {1}{\sqrt {x}}}{b}-\frac {1}{b x^{3/2} \sqrt {a+\frac {b}{x}}}\right )}{3 b}-\frac {1}{3 b x^{5/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -2 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {a+\frac {b}{x}}}{2 b \sqrt {x}}-\frac {a \int \frac {1}{\sqrt {a+\frac {b}{x}}}d\frac {1}{\sqrt {x}}}{2 b}\right )}{b}-\frac {1}{b x^{3/2} \sqrt {a+\frac {b}{x}}}\right )}{3 b}-\frac {1}{3 b x^{5/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -2 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {a+\frac {b}{x}}}{2 b \sqrt {x}}-\frac {a \int \frac {1}{1-\frac {b}{x}}d\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}}{2 b}\right )}{b}-\frac {1}{b x^{3/2} \sqrt {a+\frac {b}{x}}}\right )}{3 b}-\frac {1}{3 b x^{5/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -2 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {a+\frac {b}{x}}}{2 b \sqrt {x}}-\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{2 b^{3/2}}\right )}{b}-\frac {1}{b x^{3/2} \sqrt {a+\frac {b}{x}}}\right )}{3 b}-\frac {1}{3 b x^{5/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\) |
-2*(-1/3*1/(b*(a + b/x)^(3/2)*x^(5/2)) + (5*(-(1/(b*Sqrt[a + b/x]*x^(3/2)) ) + (3*(Sqrt[a + b/x]/(2*b*Sqrt[x]) - (a*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sq rt[x])])/(2*b^(3/2))))/b))/(3*b))
3.19.2.3.1 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_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[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 = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(-\frac {a x +b}{b^{3} x^{\frac {3}{2}} \sqrt {\frac {a x +b}{x}}}-\frac {a \left (-\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {8}{\sqrt {a x +b}}+\frac {4 b}{3 \left (a x +b \right )^{\frac {3}{2}}}\right ) \sqrt {a x +b}}{2 b^{3} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\) | \(90\) |
| default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-15 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {a x +b}\, a^{2} x^{2}+3 b^{\frac {5}{2}}+20 b^{\frac {3}{2}} a x +15 a^{2} x^{2} \sqrt {b}-15 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a b x \sqrt {a x +b}\right )}{3 \sqrt {x}\, \left (a x +b \right )^{2} b^{\frac {7}{2}}}\) | \(102\) |
-1/b^3*(a*x+b)/x^(3/2)/((a*x+b)/x)^(1/2)-1/2/b^3*a*(-10/b^(1/2)*arctanh((a *x+b)^(1/2)/b^(1/2))+8/(a*x+b)^(1/2)+4/3*b/(a*x+b)^(3/2))/x^(1/2)/((a*x+b) /x)^(1/2)*(a*x+b)^(1/2)
Time = 0.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.52 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{9/2}} \, dx=\left [\frac {15 \, {\left (a^{3} x^{3} + 2 \, a^{2} b x^{2} + a b^{2} x\right )} \sqrt {b} \log \left (\frac {a x + 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (15 \, a^{2} b x^{2} + 20 \, a b^{2} x + 3 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{6 \, {\left (a^{2} b^{4} x^{3} + 2 \, a b^{5} x^{2} + b^{6} x\right )}}, -\frac {15 \, {\left (a^{3} x^{3} + 2 \, a^{2} b x^{2} + a b^{2} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (15 \, a^{2} b x^{2} + 20 \, a b^{2} x + 3 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{2} b^{4} x^{3} + 2 \, a b^{5} x^{2} + b^{6} x\right )}}\right ] \]
[1/6*(15*(a^3*x^3 + 2*a^2*b*x^2 + a*b^2*x)*sqrt(b)*log((a*x + 2*sqrt(b)*sq rt(x)*sqrt((a*x + b)/x) + 2*b)/x) - 2*(15*a^2*b*x^2 + 20*a*b^2*x + 3*b^3)* sqrt(x)*sqrt((a*x + b)/x))/(a^2*b^4*x^3 + 2*a*b^5*x^2 + b^6*x), -1/3*(15*( a^3*x^3 + 2*a^2*b*x^2 + a*b^2*x)*sqrt(-b)*arctan(sqrt(-b)*sqrt(x)*sqrt((a* x + b)/x)/b) + (15*a^2*b*x^2 + 20*a*b^2*x + 3*b^3)*sqrt(x)*sqrt((a*x + b)/ x))/(a^2*b^4*x^3 + 2*a*b^5*x^2 + b^6*x)]
Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (85) = 170\).
Time = 81.46 (sec) , antiderivative size = 818, normalized size of antiderivative = 8.26 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{9/2}} \, dx=- \frac {15 a^{4} b^{13} x^{4} \log {\left (\frac {a x}{b} \right )}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} + \frac {30 a^{4} b^{13} x^{4} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} - \frac {30 a^{3} b^{14} x^{3} \sqrt {\frac {a x}{b} + 1}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} - \frac {45 a^{3} b^{14} x^{3} \log {\left (\frac {a x}{b} \right )}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} + \frac {90 a^{3} b^{14} x^{3} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} - \frac {70 a^{2} b^{15} x^{2} \sqrt {\frac {a x}{b} + 1}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} - \frac {45 a^{2} b^{15} x^{2} \log {\left (\frac {a x}{b} \right )}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} + \frac {90 a^{2} b^{15} x^{2} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} - \frac {46 a b^{16} x \sqrt {\frac {a x}{b} + 1}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} - \frac {15 a b^{16} x \log {\left (\frac {a x}{b} \right )}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} + \frac {30 a b^{16} x \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} - \frac {6 b^{17} \sqrt {\frac {a x}{b} + 1}}{6 a^{3} b^{\frac {33}{2}} x^{4} + 18 a^{2} b^{\frac {35}{2}} x^{3} + 18 a b^{\frac {37}{2}} x^{2} + 6 b^{\frac {39}{2}} x} \]
-15*a**4*b**13*x**4*log(a*x/b)/(6*a**3*b**(33/2)*x**4 + 18*a**2*b**(35/2)* x**3 + 18*a*b**(37/2)*x**2 + 6*b**(39/2)*x) + 30*a**4*b**13*x**4*log(sqrt( a*x/b + 1) + 1)/(6*a**3*b**(33/2)*x**4 + 18*a**2*b**(35/2)*x**3 + 18*a*b** (37/2)*x**2 + 6*b**(39/2)*x) - 30*a**3*b**14*x**3*sqrt(a*x/b + 1)/(6*a**3* b**(33/2)*x**4 + 18*a**2*b**(35/2)*x**3 + 18*a*b**(37/2)*x**2 + 6*b**(39/2 )*x) - 45*a**3*b**14*x**3*log(a*x/b)/(6*a**3*b**(33/2)*x**4 + 18*a**2*b**( 35/2)*x**3 + 18*a*b**(37/2)*x**2 + 6*b**(39/2)*x) + 90*a**3*b**14*x**3*log (sqrt(a*x/b + 1) + 1)/(6*a**3*b**(33/2)*x**4 + 18*a**2*b**(35/2)*x**3 + 18 *a*b**(37/2)*x**2 + 6*b**(39/2)*x) - 70*a**2*b**15*x**2*sqrt(a*x/b + 1)/(6 *a**3*b**(33/2)*x**4 + 18*a**2*b**(35/2)*x**3 + 18*a*b**(37/2)*x**2 + 6*b* *(39/2)*x) - 45*a**2*b**15*x**2*log(a*x/b)/(6*a**3*b**(33/2)*x**4 + 18*a** 2*b**(35/2)*x**3 + 18*a*b**(37/2)*x**2 + 6*b**(39/2)*x) + 90*a**2*b**15*x* *2*log(sqrt(a*x/b + 1) + 1)/(6*a**3*b**(33/2)*x**4 + 18*a**2*b**(35/2)*x** 3 + 18*a*b**(37/2)*x**2 + 6*b**(39/2)*x) - 46*a*b**16*x*sqrt(a*x/b + 1)/(6 *a**3*b**(33/2)*x**4 + 18*a**2*b**(35/2)*x**3 + 18*a*b**(37/2)*x**2 + 6*b* *(39/2)*x) - 15*a*b**16*x*log(a*x/b)/(6*a**3*b**(33/2)*x**4 + 18*a**2*b**( 35/2)*x**3 + 18*a*b**(37/2)*x**2 + 6*b**(39/2)*x) + 30*a*b**16*x*log(sqrt( a*x/b + 1) + 1)/(6*a**3*b**(33/2)*x**4 + 18*a**2*b**(35/2)*x**3 + 18*a*b** (37/2)*x**2 + 6*b**(39/2)*x) - 6*b**17*sqrt(a*x/b + 1)/(6*a**3*b**(33/2)*x **4 + 18*a**2*b**(35/2)*x**3 + 18*a*b**(37/2)*x**2 + 6*b**(39/2)*x)
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{9/2}} \, dx=-\frac {15 \, {\left (a + \frac {b}{x}\right )}^{2} a x^{2} - 10 \, {\left (a + \frac {b}{x}\right )} a b x - 2 \, a b^{2}}{3 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{3} x^{\frac {5}{2}} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{4} x^{\frac {3}{2}}\right )}} - \frac {5 \, a \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{2 \, b^{\frac {7}{2}}} \]
-1/3*(15*(a + b/x)^2*a*x^2 - 10*(a + b/x)*a*b*x - 2*a*b^2)/((a + b/x)^(5/2 )*b^3*x^(5/2) - (a + b/x)^(3/2)*b^4*x^(3/2)) - 5/2*a*log((sqrt(a + b/x)*sq rt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b)))/b^(7/2)
Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{9/2}} \, dx=-\frac {5 \, a \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{3}} - \frac {2 \, {\left (6 \, {\left (a x + b\right )} a + a b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{3}} - \frac {\sqrt {a x + b}}{b^{3} x} \]
-5*a*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^3) - 2/3*(6*(a*x + b)*a + a*b)/((a*x + b)^(3/2)*b^3) - sqrt(a*x + b)/(b^3*x)
Timed out. \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{9/2}} \, dx=\int \frac {1}{x^{9/2}\,{\left (a+\frac {b}{x}\right )}^{5/2}} \,d x \]