Integrand size = 26, antiderivative size = 79 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )} \]
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Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 14} \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx=-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )} \]
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Rule 14
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^{10}} \, dx}{a b+b^2 x^3} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a b}{x^{10}}+\frac {b^2}{x^7}\right ) \, dx}{a b+b^2 x^3} \\ & = -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{9 x^9 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{6 x^6 \left (a+b x^3\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx=-\frac {\sqrt {\left (a+b x^3\right )^2} \left (2 a+3 b x^3\right )}{18 x^9 \left (a+b x^3\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.30
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {csgn}\left (b \,x^{3}+a \right ) \left (3 b \,x^{3}+2 a \right )}{18 x^{9}}\) | \(24\) |
risch | \(\frac {\left (-\frac {b \,x^{3}}{6}-\frac {a}{9}\right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{x^{9} \left (b \,x^{3}+a \right )}\) | \(35\) |
gosper | \(-\frac {\left (3 b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{18 x^{9} \left (b \,x^{3}+a \right )}\) | \(36\) |
default | \(-\frac {\left (3 b \,x^{3}+2 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{18 x^{9} \left (b \,x^{3}+a \right )}\) | \(36\) |
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Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx=-\frac {3 \, b x^{3} + 2 \, a}{18 \, x^{9}} \]
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Timed out. \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (53) = 106\).
Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx=-\frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{3}}{6 \, a^{3}} - \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} b^{2}}{6 \, a^{2} x^{3}} + \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}} b}{6 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac {3}{2}}}{9 \, a^{2} x^{9}} \]
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Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx=-\frac {3 \, b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 2 \, a \mathrm {sgn}\left (b x^{3} + a\right )}{18 \, x^{9}} \]
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Time = 8.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^{10}} \, dx=-\frac {\left (3\,b\,x^3+2\,a\right )\,\sqrt {{\left (b\,x^3+a\right )}^2}}{18\,x^9\,\left (b\,x^3+a\right )} \]
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