Integrand size = 27, antiderivative size = 82 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}+\frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}-\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^2} \]
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Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {864, 849, 821, 272, 65, 214} \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)} \, dx=-\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^2}+\frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}-\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 864
Rubi steps \begin{align*} \text {integral}& = \int \frac {d-e x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}-\frac {\int \frac {2 d^2 e-d e^2 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{2 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}+\frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}+\frac {e^2 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}+\frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}+\frac {e^2 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}+\frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}-\frac {\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} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{2 d x^2}+\frac {e \sqrt {d^2-e^2 x^2}}{d^2 x}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^2} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)} \, dx=-\frac {d (d-2 e x) \sqrt {d^2-e^2 x^2}+\sqrt {d^2} e^2 x^2 \log (x)-\sqrt {d^2} e^2 x^2 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{2 d^3 x^2} \]
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Time = 0.40 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-2 e x +d \right )}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d \sqrt {d^{2}}}\) | \(75\) |
default | \(\frac {-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}}{d}+\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{d^{3}}-\frac {e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d^{2} x}-\frac {2 e^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{d^{2}}\right )}{d^{2}}-\frac {e^{2} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d^{3}}\) | \(326\) |
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Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)} \, dx=\frac {e^{2} x^{2} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}} {\left (2 \, e x - d\right )}}{2 \, d^{2} x^{2}} \]
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\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{3} \left (d + e x\right )}\, dx \]
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\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)} \, dx=\int { \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (72) = 144\).
Time = 0.28 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)} \, dx=\frac {{\left (e^{3} - \frac {4 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e}{x}\right )} e^{4} x^{2}}{8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} {\left | e \right |}} - \frac {e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, d^{2} {\left | e \right |}} + \frac {\frac {4 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} e {\left | e \right |}}{x} - \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} {\left | e \right |}}{e x^{2}}}{8 \, d^{4} e^{2}} \]
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Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)} \, dx=\int \frac {\sqrt {d^2-e^2\,x^2}}{x^3\,\left (d+e\,x\right )} \,d x \]
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