Integrand size = 25, antiderivative size = 148 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 x}{315 d^7 e^2 \sqrt {d^2-e^2 x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {810, 792, 198, 197} \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {16 x}{315 d^7 e^2 \sqrt {d^2-e^2 x^2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rule 197
Rule 198
Rule 792
Rule 810
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {\int \frac {x \left (2 d^2 e-6 d e^2 x\right )}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx}{9 d^2 e^2} \\ & = \frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{21 d e^2} \\ & = \frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{105 d^3 e^2} \\ & = \frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{315 d^5 e^2} \\ & = \frac {x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac {2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 x}{315 d^7 e^2 \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-10 d^8+10 d^7 e x+35 d^6 e^2 x^2+70 d^5 e^3 x^3-70 d^4 e^4 x^4-56 d^3 e^5 x^5+56 d^2 e^6 x^6+16 d e^7 x^7-16 e^8 x^8\right )}{315 d^7 e^3 (d-e x)^5 (d+e x)^4} \]
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Time = 0.36 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (16 e^{8} x^{8}-16 d \,e^{7} x^{7}-56 d^{2} e^{6} x^{6}+56 d^{3} e^{5} x^{5}+70 d^{4} x^{4} e^{4}-70 d^{5} e^{3} x^{3}-35 d^{6} e^{2} x^{2}-10 d^{7} e x +10 d^{8}\right )}{315 d^{7} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\) | \(121\) |
trager | \(-\frac {\left (16 e^{8} x^{8}-16 d \,e^{7} x^{7}-56 d^{2} e^{6} x^{6}+56 d^{3} e^{5} x^{5}+70 d^{4} x^{4} e^{4}-70 d^{5} e^{3} x^{3}-35 d^{6} e^{2} x^{2}-10 d^{7} e x +10 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{315 d^{7} \left (-e x +d \right )^{5} \left (e x +d \right )^{4} e^{3}}\) | \(123\) |
default | \(e \left (\frac {x^{2}}{7 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {2 d^{2}}{63 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}\right )+d \left (\frac {x}{8 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}-\frac {d^{2} \left (\frac {x}{9 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{7 d^{2}}\right )}{9 d^{2}}}{d^{2}}\right )}{8 e^{2}}\right )\) | \(199\) |
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Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (130) = 260\).
Time = 0.50 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.06 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=-\frac {10 \, e^{9} x^{9} - 10 \, d e^{8} x^{8} - 40 \, d^{2} e^{7} x^{7} + 40 \, d^{3} e^{6} x^{6} + 60 \, d^{4} e^{5} x^{5} - 60 \, d^{5} e^{4} x^{4} - 40 \, d^{6} e^{3} x^{3} + 40 \, d^{7} e^{2} x^{2} + 10 \, d^{8} e x - 10 \, d^{9} - {\left (16 \, e^{8} x^{8} - 16 \, d e^{7} x^{7} - 56 \, d^{2} e^{6} x^{6} + 56 \, d^{3} e^{5} x^{5} + 70 \, d^{4} e^{4} x^{4} - 70 \, d^{5} e^{3} x^{3} - 35 \, d^{6} e^{2} x^{2} - 10 \, d^{7} e x + 10 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{7} e^{12} x^{9} - d^{8} e^{11} x^{8} - 4 \, d^{9} e^{10} x^{7} + 4 \, d^{10} e^{9} x^{6} + 6 \, d^{11} e^{8} x^{5} - 6 \, d^{12} e^{7} x^{4} - 4 \, d^{13} e^{6} x^{3} + 4 \, d^{14} e^{5} x^{2} + d^{15} e^{4} x - d^{16} e^{3}\right )}} \]
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Result contains complex when optimal does not.
Time = 16.45 (sec) , antiderivative size = 1401, normalized size of antiderivative = 9.47 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\text {Too large to display} \]
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none
Time = 0.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.07 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {x^{2}}{7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e} + \frac {d x}{9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e^{2}} - \frac {2 \, d^{2}}{63 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e^{3}} - \frac {x}{63 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d e^{2}} - \frac {2 \, x}{105 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{2}} - \frac {8 \, x}{315 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} e^{2}} - \frac {16 \, x}{315 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} e^{2}} \]
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\[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int { \frac {{\left (e x + d\right )} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {11}{2}}} \,d x } \]
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Time = 11.62 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.36 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}}{144\,d^3\,e^3\,{\left (d-e\,x\right )}^5}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1}{252\,e^3}-\frac {17\,x}{252\,d\,e^2}\right )}{{\left (d+e\,x\right )}^4\,{\left (d-e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {5}{144\,d^2\,e^3}+\frac {131\,x}{5040\,d^3\,e^2}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{315\,d^5\,e^2\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{315\,d^7\,e^2\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
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