Integrand size = 27, antiderivative size = 87 \[ \int \frac {x^2 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {d (d+e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7 (d+e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x}{15 d^2 e^2 \sqrt {d^2-e^2 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1649, 792, 197} \[ \int \frac {x^2 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x}{15 d^2 e^2 \sqrt {d^2-e^2 x^2}}+\frac {d (d+e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7 (d+e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rule 197
Rule 792
Rule 1649
Rubi steps \begin{align*} \text {integral}& = \frac {d (d+e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {\left (\frac {2 d^2}{e^2}+\frac {5 d x}{e}\right ) (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d} \\ & = \frac {d (d+e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7 (d+e x)}{15 e^3 \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}{15 e^2} \\ & = \frac {d (d+e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7 (d+e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x}{15 d^2 e^2 \sqrt {d^2-e^2 x^2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \frac {x^2 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-4 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )}{15 d^2 e^3 (d-e x)^3 (d+e x)} \]
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Time = 0.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{3} \left (-e^{3} x^{3}+2 d \,e^{2} x^{2}-8 d^{2} e x +4 d^{3}\right )}{15 d^{2} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(66\) |
trager | \(-\frac {\left (-e^{3} x^{3}+2 d \,e^{2} x^{2}-8 d^{2} e x +4 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{2} e^{3} \left (-e x +d \right )^{3} \left (e x +d \right )}\) | \(68\) |
default | \(e^{2} \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+2 d e \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )\) | \(282\) |
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Time = 0.34 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.34 \[ \int \frac {x^2 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {4 \, e^{4} x^{4} - 8 \, d e^{3} x^{3} + 8 \, d^{3} e x - 4 \, d^{4} + {\left (e^{3} x^{3} - 2 \, d e^{2} x^{2} + 8 \, d^{2} e x - 4 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{7} x^{4} - 2 \, d^{3} e^{6} x^{3} + 2 \, d^{5} e^{4} x - d^{6} e^{3}\right )}} \]
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\[ \int \frac {x^2 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x^{2} \left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.51 \[ \int \frac {x^2 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} - \frac {d^{2} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {4 \, d^{3}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} + \frac {x}{30 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} + \frac {x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2}} \]
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\[ \int \frac {x^2 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
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Time = 11.40 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {\sqrt {d^2-e^2\,x^2}\,\left (4\,d^3-8\,d^2\,e\,x+2\,d\,e^2\,x^2-e^3\,x^3\right )}{15\,d^2\,e^3\,\left (d+e\,x\right )\,{\left (d-e\,x\right )}^3} \]
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