Integrand size = 25, antiderivative size = 58 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {810, 12, 267} \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}} \]
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Rule 12
Rule 267
Rule 810
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {2 d^2 e x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2} \\ & = \frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 \int \frac {x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 e} \\ & = \frac {x^2 (d+e x)}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{3 e^3 \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^2+2 d e x+e^2 x^2\right )}{3 d e^3 (d-e x)^2 (d+e x)} \]
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Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{2} \left (-e^{2} x^{2}-2 d e x +2 d^{2}\right )}{3 d \,e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(55\) |
trager | \(-\frac {\left (-e^{2} x^{2}-2 d e x +2 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3 d \,e^{3} \left (-e x +d \right )^{2} \left (e x +d \right )}\) | \(57\) |
default | \(e \left (\frac {x^{2}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\right )+d \left (\frac {x}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2}}\right )\) | \(120\) |
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (50) = 100\).
Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.79 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, e^{3} x^{3} - 2 \, d e^{2} x^{2} - 2 \, d^{2} e x + 2 \, d^{3} - {\left (e^{2} x^{2} + 2 \, d e x - 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d e^{6} x^{3} - d^{2} e^{5} x^{2} - d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
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Result contains complex when optimal does not.
Time = 3.44 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.98 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=d \left (\begin {cases} \frac {i x^{3}}{- 3 d^{5} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{3} e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {x^{3}}{- 3 d^{5} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}} + 3 d^{3} e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \frac {2 d^{2}}{- 3 d^{2} e^{4} \sqrt {d^{2} - e^{2} x^{2}} + 3 e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}} - \frac {3 e^{2} x^{2}}{- 3 d^{2} e^{4} \sqrt {d^{2} - e^{2} x^{2}} + 3 e^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{4}}{4 \left (d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \]
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none
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.52 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} + \frac {d x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} - \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2}} \]
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\[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 11.49 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (-2\,d^2+2\,d\,e\,x+e^2\,x^2\right )}{3\,d\,e^3\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^2} \]
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