Integrand size = 27, antiderivative size = 88 \[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \]
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Time = 0.05 (sec) , antiderivative size = 88, 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 = {871, 837, 12, 272, 65, 214} \[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}} \]
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 837
Rule 871
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-3 d e^2+2 e^3 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2} \\ & = \frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {3 d^3 e^4}{x \sqrt {d^2-e^2 x^2}} \, dx}{3 d^6 e^4} \\ & = \frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^3} \\ & = \frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^3} \\ & = \frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\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 )}{d^3 e^2} \\ & = \frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\frac {d \left (4 d^2+d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{(d-e x) (d+e x)^2}-3 \sqrt {d^2} \log (x)+3 \sqrt {d^2} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{3 d^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(170\) vs. \(2(78)=156\).
Time = 0.37 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.94
method | result | size |
default | \(\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d}-\frac {-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d}\) | \(171\) |
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Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.76 \[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {4 \, e^{3} x^{3} + 4 \, d e^{2} x^{2} - 4 \, d^{2} e x - 4 \, d^{3} + 3 \, {\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (2 \, e^{2} x^{2} - d e x - 4 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{4} e^{3} x^{3} + d^{5} e^{2} x^{2} - d^{6} e x - d^{7}\right )}} \]
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\[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
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\[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x} \,d x } \]
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\[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x} \,d x } \]
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Timed out. \[ \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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