Integrand size = 27, antiderivative size = 196 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {27 e^5 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4} \]
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Time = 0.34 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1819, 1821, 821, 272, 65, 214} \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\frac {27 e^5 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 866
Rule 1819
Rule 1821
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d-e x)^4}{x^6 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \\ & = -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-d^4+4 d^3 e x-7 d^2 e^2 x^2+8 d e^3 x^3-8 e^4 x^4+\frac {8 e^5 x^5}{d}}{x^6 \sqrt {d^2-e^2 x^2}} \, dx}{d^2} \\ & = -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {\int \frac {-20 d^5 e+39 d^4 e^2 x-40 d^3 e^3 x^2+40 d^2 e^4 x^3-40 d e^5 x^4}{x^5 \sqrt {d^2-e^2 x^2}} \, dx}{5 d^4} \\ & = -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {\int \frac {-156 d^6 e^2+220 d^5 e^3 x-160 d^4 e^4 x^2+160 d^3 e^5 x^3}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{20 d^6} \\ & = -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {\int \frac {-660 d^7 e^3+792 d^6 e^4 x-480 d^5 e^5 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{60 d^8} \\ & = -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {\int \frac {-1584 d^8 e^4+1620 d^7 e^5 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{120 d^{10}} \\ & = -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {\left (27 e^5\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^3} \\ & = -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {\left (27 e^5\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^3} \\ & = -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {\left (27 e^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 )}{2 d^3} \\ & = -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {27 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=-\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (2 d^5-8 d^4 e x+16 d^3 e^2 x^2-29 d^2 e^3 x^3+77 d e^4 x^4+212 e^5 x^5\right )}{x^5 (d+e x)}-135 \sqrt {d^2} e^5 \log (x)+135 \sqrt {d^2} e^5 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{10 d^5} \]
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Time = 0.87 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (132 e^{4} x^{4}-55 d \,e^{3} x^{3}+26 d^{2} e^{2} x^{2}-10 d^{3} e x +2 d^{4}\right )}{10 x^{5} d^{4}}+\frac {27 e^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{3} \sqrt {d^{2}}}-\frac {8 e^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d^{4} \left (x +\frac {d}{e}\right )}\) | \(155\) |
default | \(\text {Expression too large to display}\) | \(1997\) |
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Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.75 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=-\frac {80 \, e^{6} x^{6} + 80 \, d e^{5} x^{5} + 135 \, {\left (e^{6} x^{6} + d e^{5} x^{5}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (212 \, e^{5} x^{5} + 77 \, d e^{4} x^{4} - 29 \, d^{2} e^{3} x^{3} + 16 \, d^{3} e^{2} x^{2} - 8 \, d^{4} e x + 2 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{10 \, {\left (d^{4} e x^{6} + d^{5} x^{5}\right )}} \]
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\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{x^{6} \left (d + e x\right )^{4}}\, dx \]
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\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{6}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (173) = 346\).
Time = 0.31 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.29 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\frac {{\left (e^{6} - \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{4}}{x} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{2}}{x^{2}} - \frac {185 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{x^{3}} + \frac {870 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{2} x^{4}} + \frac {3670 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{4} x^{5}}\right )} e^{10} x^{5}}{160 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{4} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )} {\left | e \right |}} + \frac {27 \, e^{6} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, d^{4} {\left | e \right |}} - \frac {\frac {1110 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{16} e^{8}}{x} - \frac {240 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{16} e^{6}}{x^{2}} + \frac {55 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{16} e^{4}}{x^{3}} - \frac {10 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{16} e^{2}}{x^{4}} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{16}}{x^{5}}}{160 \, d^{20} e^{4} {\left | e \right |}} \]
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Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^6\,{\left (d+e\,x\right )}^4} \,d x \]
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