Integrand size = 24, antiderivative size = 33 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {665} \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \]
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Rule 665
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 d e (d+e x)^9} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=-\frac {(d-e x)^4 \sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5} \]
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Time = 3.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}{9 \left (e x +d \right )^{8} d e}\) | \(36\) |
default | \(-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{9 e^{10} d \left (x +\frac {d}{e}\right )^{9}}\) | \(46\) |
trager | \(-\frac {\left (e^{4} x^{4}-4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}-4 d^{3} e x +d^{4}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{9 d \left (e x +d \right )^{5} e}\) | \(68\) |
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Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (29) = 58\).
Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 4.91 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=-\frac {e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5} + {\left (e^{4} x^{4} - 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} - 4 \, d^{3} e x + d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{9 \, {\left (d e^{6} x^{5} + 5 \, d^{2} e^{5} x^{4} + 10 \, d^{3} e^{4} x^{3} + 10 \, d^{4} e^{3} x^{2} + 5 \, d^{5} e^{2} x + d^{6} e\right )}} \]
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Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (29) = 58\).
Time = 0.20 (sec) , antiderivative size = 539, normalized size of antiderivative = 16.33 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{e^{9} x^{8} + 8 \, d e^{8} x^{7} + 28 \, d^{2} e^{7} x^{6} + 56 \, d^{3} e^{6} x^{5} + 70 \, d^{4} e^{5} x^{4} + 56 \, d^{5} e^{4} x^{3} + 28 \, d^{6} e^{3} x^{2} + 8 \, d^{7} e^{2} x + d^{8} e} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{2 \, {\left (e^{8} x^{7} + 7 \, d e^{7} x^{6} + 21 \, d^{2} e^{6} x^{5} + 35 \, d^{3} e^{5} x^{4} + 35 \, d^{4} e^{4} x^{3} + 21 \, d^{5} e^{3} x^{2} + 7 \, d^{6} e^{2} x + d^{7} e\right )}} - \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{6 \, {\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} + \frac {35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{9 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{18 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d}{6 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{9 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{9 \, {\left (d e^{2} x + d^{2} e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (29) = 58\).
Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 5.06 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=\frac {2 \, {\left (\frac {36 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {126 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + \frac {84 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{12} x^{6}} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{16} x^{8}} + 1\right )}}{9 \, d {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{9} {\left | e \right |}} \]
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Time = 10.62 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.27 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx=\frac {8\,\sqrt {d^2-e^2\,x^2}}{9\,e\,{\left (d+e\,x\right )}^2}-\frac {8\,d\,\sqrt {d^2-e^2\,x^2}}{3\,e\,{\left (d+e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{9\,d\,e\,\left (d+e\,x\right )}+\frac {32\,d^2\,\sqrt {d^2-e^2\,x^2}}{9\,e\,{\left (d+e\,x\right )}^4}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{9\,e\,{\left (d+e\,x\right )}^5} \]
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