Integrand size = 27, antiderivative size = 140 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^2} \, dx=\frac {e^3 \sqrt {d^2-e^2 x^2}}{4 d x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}-\frac {e^5 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^2} \]
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Time = 0.11 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {866, 1821, 849, 821, 272, 43, 65, 214} \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^2} \, dx=-\frac {e^5 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}+\frac {e^3 \sqrt {d^2-e^2 x^2}}{4 d x^2} \]
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 866
Rule 1821
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^6} \, dx \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}-\frac {\int \frac {\left (10 d^3 e-7 d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^5} \, dx}{5 d^2} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}+\frac {\int \frac {\left (28 d^4 e^2-10 d^3 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^4} \, dx}{20 d^4} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}-\frac {e^3 \int \frac {\sqrt {d^2-e^2 x^2}}{x^3} \, dx}{2 d} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}-\frac {e^3 \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{4 d} \\ & = \frac {e^3 \sqrt {d^2-e^2 x^2}}{4 d x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}+\frac {e^5 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{8 d} \\ & = \frac {e^3 \sqrt {d^2-e^2 x^2}}{4 d x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}-\frac {e^3 \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 )}{4 d} \\ & = \frac {e^3 \sqrt {d^2-e^2 x^2}}{4 d x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{2 d x^4}-\frac {7 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 x^3}-\frac {e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^2} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-12 d^4+30 d^3 e x-16 d^2 e^2 x^2-15 d e^3 x^3+28 e^4 x^4\right )}{60 d^2 x^5}-\frac {\sqrt {d^2} e^5 \log (x)}{4 d^3}+\frac {\sqrt {d^2} e^5 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{4 d^3} \]
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Time = 0.54 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-28 e^{4} x^{4}+15 d \,e^{3} x^{3}+16 d^{2} e^{2} x^{2}-30 d^{3} e x +12 d^{4}\right )}{60 x^{5} d^{2}}-\frac {e^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{4 d \sqrt {d^{2}}}\) | \(110\) |
default | \(\text {Expression too large to display}\) | \(1349\) |
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Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^2} \, dx=\frac {15 \, e^{5} x^{5} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (28 \, e^{4} x^{4} - 15 \, d e^{3} x^{3} - 16 \, d^{2} e^{2} x^{2} + 30 \, d^{3} e x - 12 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{60 \, d^{2} x^{5}} \]
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Result contains complex when optimal does not.
Time = 4.88 (sec) , antiderivative size = 660, normalized size of antiderivative = 4.71 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^2} \, dx=d^{2} \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.11 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^2} \, dx=-\frac {e^{5} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{4 \, d^{2}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{5}}{4 \, d^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{4 \, d^{3} x^{2}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{15 \, d^{2} x^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{2 \, d x^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{5 \, x^{5}} \]
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.96 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^2} \, dx=-\frac {1}{960} \, {\left (\frac {240 \, e^{4} \log \left (\sqrt {\frac {2 \, d}{e x + d} - 1} + 1\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{2}} - \frac {240 \, e^{4} \log \left ({\left | \sqrt {\frac {2 \, d}{e x + d} - 1} - 1 \right |}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{2}} + \frac {8 \, {\left (15 \, e^{4} \log \left (2\right ) - 30 \, e^{4} \log \left (i + 1\right ) + 56 i \, e^{4}\right )} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{2}} - \frac {15 \, e^{4} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 250 \, e^{4} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 128 \, e^{4} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 70 \, e^{4} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 15 \, e^{4} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{2} {\left (\frac {d}{e x + d} - 1\right )}^{5}}\right )} {\left | e \right |} \]
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Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^2} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^6\,{\left (d+e\,x\right )}^2} \,d x \]
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