Integrand size = 27, antiderivative size = 105 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=-\frac {1}{2} e (4 d+e x) \sqrt {d^2-e^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}-\frac {1}{2} d^2 e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+2 d^2 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {866, 1821, 829, 858, 223, 209, 272, 65, 214} \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=-\frac {1}{2} d^2 e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+2 d^2 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}-\frac {1}{2} e (4 d+e x) \sqrt {d^2-e^2 x^2} \]
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 829
Rule 858
Rule 866
Rule 1821
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^2} \, dx \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}-\frac {\int \frac {\left (2 d^3 e+d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{d^2} \\ & = -\frac {1}{2} e (4 d+e x) \sqrt {d^2-e^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}+\frac {\int \frac {-4 d^5 e^3-d^4 e^4 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^2 e^2} \\ & = -\frac {1}{2} e (4 d+e x) \sqrt {d^2-e^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}-\left (2 d^3 e\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{2} \left (d^2 e^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {1}{2} e (4 d+e x) \sqrt {d^2-e^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}-\left (d^3 e\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{2} \left (d^2 e^2\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = -\frac {1}{2} e (4 d+e x) \sqrt {d^2-e^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}-\frac {1}{2} d^2 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e} \\ & = -\frac {1}{2} e (4 d+e x) \sqrt {d^2-e^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{x}-\frac {1}{2} d^2 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+2 d^2 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^2-4 d e x+e^2 x^2\right )}{2 x}+d^2 e \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+2 d \sqrt {d^2} e \log (x)-2 d \sqrt {d^2} e \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]
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Time = 0.43 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.33
method | result | size |
risch | \(-\frac {d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{x}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x}{2}-\frac {e^{2} d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}+\frac {2 e \,d^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}-2 e d \sqrt {-e^{2} x^{2}+d^{2}}\) | \(140\) |
default | \(\frac {-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \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 )}{d^{2}}}{d^{2}}-\frac {2 e \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{d^{3}}+\frac {\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}}{d^{2}}+\frac {2 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 )}{d^{3}}\) | \(674\) |
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Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.06 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=\frac {2 \, d^{2} e x \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 4 \, d^{2} e x \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - 4 \, d^{2} e x + {\left (e^{2} x^{2} - 4 \, d e x - 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2 \, x} \]
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Result contains complex when optimal does not.
Time = 3.48 (sec) , antiderivative size = 330, normalized size of antiderivative = 3.14 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=d^{2} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \frac {d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {d^{2} - e^{2} x^{2}}}{2} & \text {for}\: e^{2} \neq 0 \\x \sqrt {d^{2}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.20 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=-\frac {d^{2} e^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} + 2 \, d^{2} e \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{2} x - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} d e - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{x} \]
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Exception generated. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^2} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^2\,{\left (d+e\,x\right )}^2} \,d x \]
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