Integrand size = 27, antiderivative size = 171 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4} \]
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Time = 0.14 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1823, 847, 794, 201, 223, 209} \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4} \]
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Rule 201
Rule 209
Rule 223
Rule 794
Rule 847
Rule 866
Rule 1823
Rubi steps \begin{align*} \text {integral}& = \int x^3 (d-e x)^2 \sqrt {d^2-e^2 x^2} \, dx \\ & = -\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x^3 \left (-11 d^2 e^2+14 d e^3 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{7 e^2} \\ & = \frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int x^2 \left (-42 d^3 e^3+66 d^2 e^4 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{42 e^4} \\ & = -\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x \left (-132 d^4 e^4+210 d^3 e^5 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{210 e^6} \\ & = -\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^5 \int \sqrt {d^2-e^2 x^2} \, dx}{4 e^3} \\ & = -\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^7 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^3} \\ & = -\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^7 \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^3} \\ & = -\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.78 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {e \sqrt {d^2-e^2 x^2} \left (-176 d^6+105 d^5 e x-88 d^4 e^2 x^2+70 d^3 e^3 x^3+144 d^2 e^4 x^4-280 d e^5 x^5+120 e^6 x^6\right )-105 d^7 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{840 e^5} \]
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Time = 0.39 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {\left (-120 e^{6} x^{6}+280 d \,e^{5} x^{5}-144 d^{2} e^{4} x^{4}-70 d^{3} x^{3} e^{3}+88 d^{4} e^{2} x^{2}-105 d^{5} e x +176 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{840 e^{4}}-\frac {d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{3} \sqrt {e^{2}}}\) | \(119\) |
default | \(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{4}}-\frac {2 d \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 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}\right )}{e^{3}}-\frac {d^{3} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 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}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{3 d}\right )}{e^{5}}+\frac {3 d^{2} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 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}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{e^{4}}\) | \(565\) |
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Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.68 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {210 \, d^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (120 \, e^{6} x^{6} - 280 \, d e^{5} x^{5} + 144 \, d^{2} e^{4} x^{4} + 70 \, d^{3} e^{3} x^{3} - 88 \, d^{4} e^{2} x^{2} + 105 \, d^{5} e x - 176 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{840 \, e^{4}} \]
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Time = 1.96 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.49 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=d^{2} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} \frac {d^{6} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{4} x}{16 e^{4}} - \frac {d^{2} x^{3}}{24 e^{2}} + \frac {x^{5}}{6}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{5} \sqrt {d^{2}}}{5} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{6}}{105 e^{6}} - \frac {4 d^{4} x^{2}}{105 e^{4}} - \frac {d^{2} x^{4}}{35 e^{2}} + \frac {x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.47 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{4 \, {\left (e^{5} x + d e^{4}\right )}} - \frac {i \, d^{7} \arcsin \left (\frac {e x}{d} + 2\right )}{2 \, e^{4}} - \frac {5 \, d^{7} \arcsin \left (\frac {e x}{d}\right )}{8 \, e^{4}} + \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x}{2 \, e^{3}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5} x}{8 \, e^{3}} + \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{e^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x}{3 \, e^{3}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}}{12 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x}{3 \, e^{3}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{5 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, e^{4}} \]
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Time = 0.33 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.64 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {{\left (13440 \, d^{8} e^{8} \arctan \left (\sqrt {\frac {2 \, d}{e x + d} - 1}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + \frac {{\left (105 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {13}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 3780 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {11}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 189 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 4992 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 1981 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 700 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 105 \, d^{8} e^{8} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} {\left (e x + d\right )}^{7}}{d^{7}}\right )} {\left | e \right |}}{53760 \, d e^{13}} \]
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Timed out. \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {x^3\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \]
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