Integrand size = 24, antiderivative size = 100 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9} \]
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Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9} \]
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Rule 665
Rule 673
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}+\frac {2 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx}{13 d} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}+\frac {2 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9} \, dx}{143 d^2} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.60 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=-\frac {(d-e x)^4 \sqrt {d^2-e^2 x^2} \left (119 d^2+22 d e x+2 e^2 x^2\right )}{1287 d^3 e (d+e x)^7} \]
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Time = 4.36 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (2 x^{2} e^{2}+22 d e x +119 d^{2}\right ) \left (-x^{2} e^{2}+d^{2}\right )^{\frac {7}{2}}}{1287 \left (e x +d \right )^{10} d^{3} e}\) | \(55\) |
trager | \(-\frac {\left (2 e^{6} x^{6}+14 d \,e^{5} x^{5}+43 d^{2} e^{4} x^{4}-352 x^{3} d^{3} e^{3}+628 d^{4} e^{2} x^{2}-454 d^{5} e x +119 d^{6}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{1287 d^{3} \left (e x +d \right )^{7} e}\) | \(93\) |
default | \(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{13 d e \left (x +\frac {d}{e}\right )^{11}}+\frac {2 e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 d e \left (x +\frac {d}{e}\right )^{10}}-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 d^{2} \left (x +\frac {d}{e}\right )^{9}}\right )}{13 d}}{e^{11}}\) | \(145\) |
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (88) = 176\).
Time = 0.39 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.36 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=-\frac {119 \, e^{7} x^{7} + 833 \, d e^{6} x^{6} + 2499 \, d^{2} e^{5} x^{5} + 4165 \, d^{3} e^{4} x^{4} + 4165 \, d^{4} e^{3} x^{3} + 2499 \, d^{5} e^{2} x^{2} + 833 \, d^{6} e x + 119 \, d^{7} + {\left (2 \, e^{6} x^{6} + 14 \, d e^{5} x^{5} + 43 \, d^{2} e^{4} x^{4} - 352 \, d^{3} e^{3} x^{3} + 628 \, d^{4} e^{2} x^{2} - 454 \, d^{5} e x + 119 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1287 \, {\left (d^{3} e^{8} x^{7} + 7 \, d^{4} e^{7} x^{6} + 21 \, d^{5} e^{6} x^{5} + 35 \, d^{6} e^{5} x^{4} + 35 \, d^{7} e^{4} x^{3} + 21 \, d^{8} e^{3} x^{2} + 7 \, d^{9} e^{2} x + d^{10} e\right )}} \]
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Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 790 vs. \(2 (88) = 176\).
Time = 0.22 (sec) , antiderivative size = 790, normalized size of antiderivative = 7.90 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{3 \, {\left (e^{11} x^{10} + 10 \, d e^{10} x^{9} + 45 \, d^{2} e^{9} x^{8} + 120 \, d^{3} e^{8} x^{7} + 210 \, d^{4} e^{7} x^{6} + 252 \, d^{5} e^{6} x^{5} + 210 \, d^{6} e^{5} x^{4} + 120 \, d^{7} e^{4} x^{3} + 45 \, d^{8} e^{3} x^{2} + 10 \, d^{9} e^{2} x + d^{10} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{12 \, {\left (e^{10} x^{9} + 9 \, d e^{9} x^{8} + 36 \, d^{2} e^{8} x^{7} + 84 \, d^{3} e^{7} x^{6} + 126 \, d^{4} e^{6} x^{5} + 126 \, d^{5} e^{5} x^{4} + 84 \, d^{6} e^{4} x^{3} + 36 \, d^{7} e^{3} x^{2} + 9 \, d^{8} e^{2} x + d^{9} e\right )}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{12 \, {\left (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\right )}} + \frac {7 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{26 \, {\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 {7 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{572 \, {\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}{5148 \, {\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}}}{1287 \, {\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}}}{429 \, {\left (d e^{4} x^{3} + 3 \, d^{2} e^{3} x^{2} + 3 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{1287 \, {\left (d^{2} e^{3} x^{2} + 2 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}}}{1287 \, {\left (d^{3} e^{2} x + d^{4} e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (88) = 176\).
Time = 0.31 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.13 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\frac {2 \, {\left (\frac {260 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} + \frac {6708 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {11726 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} + \frac {52481 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + \frac {61776 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{10} x^{5}} + \frac {120120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{12} x^{6}} + \frac {84084 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{14} x^{7}} + \frac {91377 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{16} x^{8}} + \frac {32604 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9}}{e^{18} x^{9}} + \frac {22308 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10}}{e^{20} x^{10}} + \frac {2574 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{11}}{e^{22} x^{11}} + \frac {1287 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{12}}{e^{24} x^{12}} + 119\right )}}{1287 \, d^{3} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{13} {\left | e \right |}} \]
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Time = 11.58 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.99 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\frac {424\,\sqrt {d^2-e^2\,x^2}}{1287\,e\,{\left (d+e\,x\right )}^4}-\frac {1832\,d\,\sqrt {d^2-e^2\,x^2}}{1287\,e\,{\left (d+e\,x\right )}^5}-\frac {\sqrt {d^2-e^2\,x^2}}{429\,d\,e\,{\left (d+e\,x\right )}^3}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{1287\,d^2\,e\,{\left (d+e\,x\right )}^2}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{1287\,d^3\,e\,\left (d+e\,x\right )}+\frac {320\,d^2\,\sqrt {d^2-e^2\,x^2}}{143\,e\,{\left (d+e\,x\right )}^6}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{13\,e\,{\left (d+e\,x\right )}^7} \]
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