Integrand size = 25, antiderivative size = 159 \[ \int x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3} \]
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Time = 0.06 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {811, 655, 201, 223, 209} \[ \int x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2} \]
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Rule 201
Rule 209
Rule 223
Rule 655
Rule 811
Rubi steps \begin{align*} \text {integral}& = -\frac {\int (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^2}+\frac {d^2 \int (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2} \\ & = -\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {d \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^2}+\frac {d^3 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2} \\ & = \frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {\left (5 d^3\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{6 e^2}+\frac {\left (3 d^5\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{4 e^2} \\ & = \frac {3 d^5 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {\left (5 d^5\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{8 e^2}+\frac {\left (3 d^7\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^2} \\ & = \frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac {\left (5 d^7\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{16 e^2}+\frac {\left (3 d^7\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^2} \\ & = \frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {3 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3}-\frac {\left (5 d^7\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^2} \\ & = \frac {d^5 x \sqrt {d^2-e^2 x^2}}{16 e^2}+\frac {d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac {d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac {d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^3} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.79 \[ \int x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (96 d^6+105 d^5 e x+48 d^4 e^2 x^2-490 d^3 e^3 x^3-384 d^2 e^4 x^4+280 d e^5 x^5+240 e^6 x^6\right )+210 d^7 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{1680 e^3} \]
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Time = 0.36 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {\left (240 e^{6} x^{6}+280 d \,e^{5} x^{5}-384 d^{2} e^{4} x^{4}-490 d^{3} x^{3} e^{3}+48 d^{4} e^{2} x^{2}+105 d^{5} e x +96 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{1680 e^{3}}+\frac {d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e^{2} \sqrt {e^{2}}}\) | \(119\) |
default | \(e \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 e^{4}}\right )+d \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6 e^{2}}\right )\) | \(153\) |
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Time = 0.34 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.73 \[ \int x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=-\frac {210 \, d^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (240 \, e^{6} x^{6} + 280 \, d e^{5} x^{5} - 384 \, d^{2} e^{4} x^{4} - 490 \, d^{3} e^{3} x^{3} + 48 \, d^{4} e^{2} x^{2} + 105 \, d^{5} e x + 96 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1680 \, e^{3}} \]
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Time = 0.53 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.09 \[ \int x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\begin {cases} \frac {d^{7} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{2}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{6}}{35 e^{3}} - \frac {d^{5} x}{16 e^{2}} - \frac {d^{4} x^{2}}{35 e} + \frac {7 d^{3} x^{3}}{24} + \frac {8 d^{2} e x^{4}}{35} - \frac {d e^{2} x^{5}}{6} - \frac {e^{3} x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\left (\frac {d x^{3}}{3} + \frac {e x^{4}}{4}\right ) \left (d^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.87 \[ \int x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {d^{7} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{16 \, \sqrt {e^{2}} e^{2}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{5} x}{16 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x}{24 \, e^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x^{2}}{7 \, e} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x}{6 \, e^{2}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{35 \, e^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.67 \[ \int x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {d^{7} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{16 \, e^{2} {\left | e \right |}} - \frac {1}{1680} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {96 \, d^{6}}{e^{3}} + {\left (\frac {105 \, d^{5}}{e^{2}} + 2 \, {\left (\frac {24 \, d^{4}}{e} - {\left (245 \, d^{3} + 4 \, {\left (48 \, d^{2} e - 5 \, {\left (6 \, e^{3} x + 7 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
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Timed out. \[ \int x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\int x^2\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right ) \,d x \]
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