Integrand size = 27, antiderivative size = 91 \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 d-3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e^3 \sqrt {d^2-e^2 x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {864, 833, 792, 197} \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 d-3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e^3 \sqrt {d^2-e^2 x^2}} \]
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Rule 197
Rule 792
Rule 833
Rule 864
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 (d-e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \\ & = \frac {x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x \left (2 d^3-3 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2} \\ & = \frac {x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 d-3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 e^3} \\ & = \frac {x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 d-3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e^3 \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^4-2 d^3 e x+3 d^2 e^2 x^2+3 d e^3 x^3+3 e^4 x^4\right )}{15 d^2 e^4 (d-e x)^2 (d+e x)^3} \]
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Time = 0.38 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (-3 e^{4} x^{4}-3 d \,e^{3} x^{3}-3 d^{2} e^{2} x^{2}+2 d^{3} e x +2 d^{4}\right )}{15 d^{2} e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(70\) |
trager | \(-\frac {\left (-3 e^{4} x^{4}-3 d \,e^{3} x^{3}-3 d^{2} e^{2} x^{2}+2 d^{3} e x +2 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{2} e^{4} \left (e x +d \right )^{3} \left (-e x +d \right )^{2}}\) | \(79\) |
default | \(\frac {\frac {x}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2}}}{e}+\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{3}}-\frac {d}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{3} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{e^{4}}\) | \(311\) |
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Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (80) = 160\).
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.88 \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, e^{5} x^{5} + 2 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} - 4 \, d^{3} e^{2} x^{2} + 2 \, d^{4} e x + 2 \, d^{5} - {\left (3 \, e^{4} x^{4} + 3 \, d e^{3} x^{3} + 3 \, d^{2} e^{2} x^{2} - 2 \, d^{3} e x - 2 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{9} x^{5} + d^{3} e^{8} x^{4} - 2 \, d^{4} e^{7} x^{3} - 2 \, d^{5} e^{6} x^{2} + d^{6} e^{5} x + d^{7} e^{4}\right )}} \]
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\[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {x^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {d^{2}}{5 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4}\right )}} + \frac {2 \, x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} - \frac {d}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} - \frac {x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{3}} \]
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\[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}} \,d x } \]
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Time = 12.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.86 \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (-2\,d^4-2\,d^3\,e\,x+3\,d^2\,e^2\,x^2+3\,d\,e^3\,x^3+3\,e^4\,x^4\right )}{15\,d^2\,e^4\,{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^2} \]
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