Integrand size = 15, antiderivative size = 38 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^7} \, dx=\frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^2}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, 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} \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^7} \, dx=\frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^2}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \text {Subst}\left (\int x \sqrt {a+b x} \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = -\left (\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = \frac {2 a \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b^2}-\frac {2 \left (a+\frac {b}{x^3}\right )^{5/2}}{15 b^2} \\ \end{align*}
Time = 1.60 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^7} \, dx=-\frac {2 \sqrt {a+\frac {b}{x^3}} \left (3 b^2+a b x^3-2 a^2 x^6\right )}{45 b^2 x^6} \]
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Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03
method | result | size |
gosper | \(\frac {2 \left (a \,x^{3}+b \right ) \left (2 a \,x^{3}-3 b \right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{45 b^{2} x^{6}}\) | \(39\) |
risch | \(\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \left (2 a^{2} x^{6}-a b \,x^{3}-3 b^{2}\right )}{45 x^{6} b^{2}}\) | \(43\) |
trager | \(\frac {2 \left (2 a^{2} x^{6}-a b \,x^{3}-3 b^{2}\right ) \sqrt {-\frac {-a \,x^{3}-b}{x^{3}}}}{45 x^{6} b^{2}}\) | \(47\) |
default | \(\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \sqrt {a \,x^{4}+b x}\, \left (2 a^{2} x^{6}-a b \,x^{3}-3 b^{2}\right )}{45 x^{6} \sqrt {x \left (a \,x^{3}+b \right )}\, b^{2}}\) | \(65\) |
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^7} \, dx=\frac {2 \, {\left (2 \, a^{2} x^{6} - a b x^{3} - 3 \, b^{2}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{45 \, b^{2} x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (34) = 68\).
Time = 0.87 (sec) , antiderivative size = 313, normalized size of antiderivative = 8.24 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^7} \, dx=\frac {4 a^{\frac {11}{2}} b^{\frac {3}{2}} x^{9} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} + \frac {2 a^{\frac {9}{2}} b^{\frac {5}{2}} x^{6} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} - \frac {8 a^{\frac {7}{2}} b^{\frac {7}{2}} x^{3} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} - \frac {6 a^{\frac {5}{2}} b^{\frac {9}{2}} \sqrt {\frac {a x^{3}}{b} + 1}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} - \frac {4 a^{6} b x^{\frac {21}{2}}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} - \frac {4 a^{5} b^{2} x^{\frac {15}{2}}}{45 a^{\frac {7}{2}} b^{3} x^{\frac {21}{2}} + 45 a^{\frac {5}{2}} b^{4} x^{\frac {15}{2}}} \]
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Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^7} \, dx=-\frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}}}{15 \, b^{2}} + \frac {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a}{9 \, b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^7} \, dx=-\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} - 5 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a\right )}}{45 \, b^{2}} \]
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Time = 6.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^7} \, dx=-\frac {2\,\sqrt {a+\frac {b}{x^3}}\,\left (-2\,a^2\,x^6+a\,b\,x^3+3\,b^2\right )}{45\,b^2\,x^6} \]
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