Integrand size = 15, antiderivative size = 59 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^3} \]
Time = 1.85 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 \sqrt {a+\frac {b}{x^3}} \left (15 b^3+3 a b^2 x^3-4 a^2 b x^6+8 a^3 x^9\right )}{315 b^3 x^9} \]
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 53, 2009}
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.
\(\displaystyle \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -\frac {1}{3} \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^6}d\frac {1}{x^3}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle -\frac {1}{3} \int \left (\frac {\left (a+\frac {b}{x^3}\right )^{5/2}}{b^2}-\frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{b^2}+\frac {a^2 \sqrt {a+\frac {b}{x^3}}}{b^2}\right )d\frac {1}{x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{3/2}}{3 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{7 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{5/2}}{5 b^3}\right )\) |
((-2*a^2*(a + b/x^3)^(3/2))/(3*b^3) + (4*a*(a + b/x^3)^(5/2))/(5*b^3) - (2 *(a + b/x^3)^(7/2))/(7*b^3))/3
3.20.96.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
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 = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(-\frac {2 \left (a \,x^{3}+b \right ) \left (8 a^{2} x^{6}-12 a b \,x^{3}+15 b^{2}\right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{315 b^{3} x^{9}}\) | \(50\) |
risch | \(-\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \left (8 a^{3} x^{9}-4 a^{2} b \,x^{6}+3 a \,b^{2} x^{3}+15 b^{3}\right )}{315 x^{9} b^{3}}\) | \(54\) |
trager | \(-\frac {2 \left (8 a^{3} x^{9}-4 a^{2} b \,x^{6}+3 a \,b^{2} x^{3}+15 b^{3}\right ) \sqrt {-\frac {-a \,x^{3}-b}{x^{3}}}}{315 x^{9} b^{3}}\) | \(58\) |
default | \(-\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \sqrt {a \,x^{4}+b x}\, \left (8 a^{3} x^{9}-4 a^{2} b \,x^{6}+3 a \,b^{2} x^{3}+15 b^{3}\right )}{315 x^{9} \sqrt {x \left (a \,x^{3}+b \right )}\, b^{3}}\) | \(76\) |
Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 \, {\left (8 \, a^{3} x^{9} - 4 \, a^{2} b x^{6} + 3 \, a b^{2} x^{3} + 15 \, b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{315 \, b^{3} x^{9}} \]
Leaf count of result is larger than twice the leaf count of optimal. 913 vs. \(2 (54) = 108\).
Time = 1.33 (sec) , antiderivative size = 913, normalized size of antiderivative = 15.47 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=- \frac {16 a^{\frac {19}{2}} b^{\frac {9}{2}} x^{18} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {40 a^{\frac {17}{2}} b^{\frac {11}{2}} x^{15} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {30 a^{\frac {15}{2}} b^{\frac {13}{2}} x^{12} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {40 a^{\frac {13}{2}} b^{\frac {15}{2}} x^{9} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {100 a^{\frac {11}{2}} b^{\frac {17}{2}} x^{6} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {96 a^{\frac {9}{2}} b^{\frac {19}{2}} x^{3} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} - \frac {30 a^{\frac {7}{2}} b^{\frac {21}{2}} \sqrt {\frac {a x^{3}}{b} + 1}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} + \frac {16 a^{10} b^{4} x^{\frac {39}{2}}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} + \frac {48 a^{9} b^{5} x^{\frac {33}{2}}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} + \frac {48 a^{8} b^{6} x^{\frac {27}{2}}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} + \frac {16 a^{7} b^{7} x^{\frac {21}{2}}}{315 a^{\frac {13}{2}} b^{7} x^{\frac {39}{2}} + 945 a^{\frac {11}{2}} b^{8} x^{\frac {33}{2}} + 945 a^{\frac {9}{2}} b^{9} x^{\frac {27}{2}} + 315 a^{\frac {7}{2}} b^{10} x^{\frac {21}{2}}} \]
-16*a**(19/2)*b**(9/2)*x**18*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39 /2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a** (7/2)*b**10*x**(21/2)) - 40*a**(17/2)*b**(11/2)*x**15*sqrt(a*x**3/b + 1)/( 315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2) *b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 30*a**(15/2)*b**(13/2)*x **12*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8 *x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 40*a**(13/2)*b**(15/2)*x**9*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39 /2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a** (7/2)*b**10*x**(21/2)) - 100*a**(11/2)*b**(17/2)*x**6*sqrt(a*x**3/b + 1)/( 315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2) *b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 96*a**(9/2)*b**(19/2)*x* *3*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 945*a**(11/2)*b**8*x **(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b**10*x**(21/2)) - 3 0*a**(7/2)*b**(21/2)*sqrt(a*x**3/b + 1)/(315*a**(13/2)*b**7*x**(39/2) + 94 5*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b* *10*x**(21/2)) + 16*a**10*b**4*x**(39/2)/(315*a**(13/2)*b**7*x**(39/2) + 9 45*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2)*b **10*x**(21/2)) + 48*a**9*b**5*x**(33/2)/(315*a**(13/2)*b**7*x**(39/2) + 9 45*a**(11/2)*b**8*x**(33/2) + 945*a**(9/2)*b**9*x**(27/2) + 315*a**(7/2...
Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}}}{21 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a}{15 \, b^{3}} - \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a^{2}}{9 \, b^{3}} \]
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 \, {\left (15 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} - 42 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a + 35 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a^{2}\right )}}{315 \, b^{3}} \]
Time = 6.61 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=\frac {8\,a^2\,\sqrt {a+\frac {b}{x^3}}}{315\,b^2\,x^3}-\frac {16\,a^3\,\sqrt {a+\frac {b}{x^3}}}{315\,b^3}-\frac {2\,a\,\sqrt {a+\frac {b}{x^3}}}{105\,b\,x^6}-\frac {2\,\sqrt {a+\frac {b}{x^3}}}{21\,x^9} \]