Integrand size = 25, antiderivative size = 121 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx=\frac {x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {8 x}{105 d^5 e^2 \sqrt {d^2-e^2 x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {810, 792, 198, 197} \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx=\frac {x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 x}{105 d^5 e^2 \sqrt {d^2-e^2 x^2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 792
Rule 810
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {\int \frac {x \left (2 d^2 e-4 d e^2 x\right )}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d^2 e^2} \\ & = \frac {x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 d e^2} \\ & = \frac {x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{105 d^3 e^2} \\ & = \frac {x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {8 x}{105 d^5 e^2 \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-6 d^6+6 d^5 e x+15 d^4 e^2 x^2+20 d^3 e^3 x^3-20 d^2 e^4 x^4-8 d e^5 x^5+8 e^6 x^6\right )}{105 d^5 e^3 (d-e x)^4 (d+e x)^3} \]
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Time = 0.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (-8 e^{6} x^{6}+8 d \,e^{5} x^{5}+20 d^{2} e^{4} x^{4}-20 d^{3} x^{3} e^{3}-15 d^{4} e^{2} x^{2}-6 d^{5} e x +6 d^{6}\right )}{105 d^{5} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}\) | \(99\) |
trager | \(-\frac {\left (-8 e^{6} x^{6}+8 d \,e^{5} x^{5}+20 d^{2} e^{4} x^{4}-20 d^{3} x^{3} e^{3}-15 d^{4} e^{2} x^{2}-6 d^{5} e x +6 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{105 d^{5} \left (-e x +d \right )^{4} \left (e x +d \right )^{3} e^{3}}\) | \(101\) |
default | \(e \left (\frac {x^{2}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}-\frac {2 d^{2}}{35 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\right )+d \left (\frac {x}{6 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}-\frac {d^{2} \left (\frac {x}{7 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{7 d^{2}}}{d^{2}}\right )}{6 e^{2}}\right )\) | \(173\) |
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Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (107) = 214\).
Time = 0.41 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.98 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx=-\frac {6 \, e^{7} x^{7} - 6 \, d e^{6} x^{6} - 18 \, d^{2} e^{5} x^{5} + 18 \, d^{3} e^{4} x^{4} + 18 \, d^{4} e^{3} x^{3} - 18 \, d^{5} e^{2} x^{2} - 6 \, d^{6} e x + 6 \, d^{7} - {\left (8 \, e^{6} x^{6} - 8 \, d e^{5} x^{5} - 20 \, d^{2} e^{4} x^{4} + 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 6 \, d^{5} e x - 6 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{105 \, {\left (d^{5} e^{10} x^{7} - d^{6} e^{9} x^{6} - 3 \, d^{7} e^{8} x^{5} + 3 \, d^{8} e^{7} x^{4} + 3 \, d^{9} e^{6} x^{3} - 3 \, d^{10} e^{5} x^{2} - d^{11} e^{4} x + d^{12} e^{3}\right )}} \]
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Result contains complex when optimal does not.
Time = 8.46 (sec) , antiderivative size = 903, normalized size of antiderivative = 7.46 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx=d \left (\begin {cases} \frac {35 i d^{4} x^{3}}{- 105 d^{13} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {28 i d^{2} e^{2} x^{5}}{- 105 d^{13} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {8 i e^{4} x^{7}}{- 105 d^{13} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {35 d^{4} x^{3}}{- 105 d^{13} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {28 d^{2} e^{2} x^{5}}{- 105 d^{13} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {8 e^{4} x^{7}}{- 105 d^{13} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 315 d^{11} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} - 315 d^{9} e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 105 d^{7} e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \frac {2 d^{2}}{- 35 d^{6} e^{4} \sqrt {d^{2} - e^{2} x^{2}} + 105 d^{4} e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}} - 105 d^{2} e^{8} x^{4} \sqrt {d^{2} - e^{2} x^{2}} + 35 e^{10} x^{6} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {7 e^{2} x^{2}}{- 35 d^{6} e^{4} \sqrt {d^{2} - e^{2} x^{2}} + 105 d^{4} e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}} - 105 d^{2} e^{8} x^{4} \sqrt {d^{2} - e^{2} x^{2}} + 35 e^{10} x^{6} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 \left (d^{2}\right )^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx=\frac {x^{2}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e} + \frac {d x}{7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}} - \frac {2 \, d^{2}}{35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}} - \frac {x}{35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{2}} - \frac {4 \, x}{105 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e^{2}} - \frac {8 \, x}{105 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5} e^{2}} \]
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\[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx=\int { \frac {{\left (e x + d\right )} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}}} \,d x } \]
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Time = 11.54 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.36 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}}{56\,d^2\,e^3\,{\left (d-e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {2}{35\,e^3}-\frac {3\,x}{70\,d\,e^2}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1}{56\,d^2\,e^3}+\frac {4\,x}{105\,d^3\,e^2}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{105\,d^5\,e^2\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
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