Integrand size = 15, antiderivative size = 59 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^3} \]
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Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 a^2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^3}-\frac {2 \left (a+\frac {b}{x^3}\right )^{7/2}}{21 b^3}+\frac {4 a \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^3} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \text {Subst}\left (\int x^2 \sqrt {a+b x} \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = -\left (\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = -\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} \\ \end{align*}
Time = 1.85 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 \sqrt {a+\frac {b}{x^3}} \left (15 b^3+3 a b^2 x^3-4 a^2 b x^6+8 a^3 x^9\right )}{315 b^3 x^9} \]
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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 | \(-\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \sqrt {a \,x^{4}+b x}\, \left (8 a^{3} x^{9}-4 a^{2} b \,x^{6}+3 a \,b^{2} x^{3}+15 b^{3}\right )}{315 x^{9} \sqrt {x \left (a \,x^{3}+b \right )}\, b^{3}}\) | \(76\) |
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Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 \, {\left (8 \, a^{3} x^{9} - 4 \, a^{2} b x^{6} + 3 \, a b^{2} x^{3} + 15 \, b^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{315 \, b^{3} x^{9}} \]
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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}}} \]
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Time = 0.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}}}{21 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a}{15 \, b^{3}} - \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a^{2}}{9 \, b^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=-\frac {2 \, {\left (15 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {7}{2}} - 42 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a + 35 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a^{2}\right )}}{315 \, b^{3}} \]
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Time = 6.61 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^{10}} \, dx=\frac {8\,a^2\,\sqrt {a+\frac {b}{x^3}}}{315\,b^2\,x^3}-\frac {16\,a^3\,\sqrt {a+\frac {b}{x^3}}}{315\,b^3}-\frac {2\,a\,\sqrt {a+\frac {b}{x^3}}}{105\,b\,x^6}-\frac {2\,\sqrt {a+\frac {b}{x^3}}}{21\,x^9} \]
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