Integrand size = 25, antiderivative size = 116 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {d^4 x \sqrt {d^2-e^2 x^2}}{16 e}-\frac {d^2 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e}-\frac {(6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^2}-\frac {d^6 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^2} \]
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Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {799, 794, 201, 223, 209} \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {d^6 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^2}-\frac {d^2 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e}-\frac {(6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^2}-\frac {d^4 x \sqrt {d^2-e^2 x^2}}{16 e} \]
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Rule 201
Rule 209
Rule 223
Rule 794
Rule 799
Rubi steps \begin{align*} \text {integral}& = \frac {\int x \left (d^2 e-d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{d e} \\ & = -\frac {(6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^2}-\frac {d^2 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{6 e} \\ & = -\frac {d^2 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e}-\frac {(6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^2}-\frac {d^4 \int \sqrt {d^2-e^2 x^2} \, dx}{8 e} \\ & = -\frac {d^4 x \sqrt {d^2-e^2 x^2}}{16 e}-\frac {d^2 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e}-\frac {(6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^2}-\frac {d^6 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{16 e} \\ & = -\frac {d^4 x \sqrt {d^2-e^2 x^2}}{16 e}-\frac {d^2 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e}-\frac {(6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^2}-\frac {d^6 \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} \\ & = -\frac {d^4 x \sqrt {d^2-e^2 x^2}}{16 e}-\frac {d^2 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e}-\frac {(6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 e^2}-\frac {d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e^2} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.98 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-48 d^5+15 d^4 e x+96 d^3 e^2 x^2-70 d^2 e^3 x^3-48 d e^4 x^4+40 e^5 x^5\right )+30 d^6 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{240 e^2} \]
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Time = 0.39 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {\left (-40 e^{5} x^{5}+48 d \,e^{4} x^{4}+70 d^{2} e^{3} x^{3}-96 d^{3} e^{2} x^{2}-15 d^{4} e x +48 d^{5}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{240 e^{2}}-\frac {d^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e \sqrt {e^{2}}}\) | \(108\) |
default | \(\frac {\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}}{e}-\frac {d \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{e^{2}}\) | \(295\) |
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Time = 0.32 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {30 \, d^{6} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (40 \, e^{5} x^{5} - 48 \, d e^{4} x^{4} - 70 \, d^{2} e^{3} x^{3} + 96 \, d^{3} e^{2} x^{2} + 15 \, d^{4} e x - 48 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, e^{2}} \]
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Time = 1.42 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.96 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=d^{3} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2}}{3 e^{2}} + \frac {x^{2}}{3}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{2} \sqrt {d^{2}}}{2} & \text {otherwise} \end {cases}\right ) - d^{2} e \left (\begin {cases} \frac {d^{4} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2} x}{8 e^{2}} + \frac {x^{3}}{4}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{3} \sqrt {d^{2}}}{3} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} \frac {d^{6} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{4} x}{16 e^{4}} - \frac {d^{2} x^{3}}{24 e^{2}} + \frac {x^{5}}{6}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{5} \sqrt {d^{2}}}{5} & \text {otherwise} \end {cases}\right ) \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.52 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {3 i \, d^{6} \arcsin \left (\frac {e x}{d} + 2\right )}{8 \, e^{2}} + \frac {5 \, d^{6} \arcsin \left (\frac {e x}{d}\right )}{16 \, e^{2}} - \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4} x}{8 \, e} + \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} x}{16 \, e} - \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5}}{4 \, e^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x}{24 \, e} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x}{6 \, e} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{5 \, e^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {d^{6} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{16 \, e {\left | e \right |}} - \frac {1}{240} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {48 \, d^{5}}{e^{2}} - {\left (\frac {15 \, d^{4}}{e} + 2 \, {\left (48 \, d^{3} - {\left (35 \, d^{2} e - 4 \, {\left (5 \, e^{3} x - 6 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} \]
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Timed out. \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {x\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{d+e\,x} \,d x \]
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