Integrand size = 27, antiderivative size = 110 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\frac {8 e^2 (d-e x)}{d \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {15 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d} \]
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Time = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1819, 1821, 821, 272, 65, 214} \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=-\frac {15 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d}+\frac {8 e^2 (d-e x)}{d \sqrt {d^2-e^2 x^2}}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2} \]
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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^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \\ & = \frac {8 e^2 (d-e x)}{d \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-d^4+4 d^3 e x-7 d^2 e^2 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{d^2} \\ & = \frac {8 e^2 (d-e x)}{d \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}+\frac {\int \frac {-8 d^5 e+15 d^4 e^2 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{2 d^4} \\ & = \frac {8 e^2 (d-e x)}{d \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d x}+\frac {1}{2} \left (15 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {8 e^2 (d-e x)}{d \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d x}+\frac {1}{4} \left (15 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right ) \\ & = \frac {8 e^2 (d-e x)}{d \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {15}{2} \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 ) \\ & = \frac {8 e^2 (d-e x)}{d \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {15 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\frac {1}{2} \left (\frac {\sqrt {d^2-e^2 x^2} \left (-d^2+7 d e x+24 e^2 x^2\right )}{d x^2 (d+e x)}-\frac {15 e^2 \log (x)}{\sqrt {d^2}}+\frac {15 e^2 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{\sqrt {d^2}}\right ) \]
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Time = 0.60 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-8 e x +d \right )}{2 d \,x^{2}}-\frac {15 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}+\frac {8 e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d \left (x +\frac {d}{e}\right )}\) | \(115\) |
default | \(\text {Expression too large to display}\) | \(1461\) |
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Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\frac {16 \, e^{3} x^{3} + 16 \, d e^{2} x^{2} + 15 \, {\left (e^{3} x^{3} + d e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (24 \, e^{2} x^{2} + 7 \, d e x - d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2 \, {\left (d e x^{3} + d^{2} x^{2}\right )}} \]
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\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{x^{3} \left (d + e x\right )^{4}}\, dx \]
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\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (99) = 198\).
Time = 0.32 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.29 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\frac {{\left (e^{3} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e}{x} - \frac {144 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e x^{2}}\right )} e^{4} x^{2}}{8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )} {\left | e \right |}} - \frac {15 \, e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, d {\left | e \right |}} + \frac {\frac {16 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e {\left | e \right |}}{x} - \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d {\left | e \right |}}{e x^{2}}}{8 \, d^{2} e^{2}} \]
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Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^3\,{\left (d+e\,x\right )}^4} \,d x \]
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