Integrand size = 27, antiderivative size = 115 \[ \int \frac {1}{x (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=\frac {4 (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \]
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Time = 0.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1819, 837, 12, 272, 65, 214} \[ \int \frac {1}{x (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4}+\frac {5 d-11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {4 (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {15 d-22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}} \]
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 837
Rule 866
Rule 1819
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d-e x)^3}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx \\ & = \frac {4 (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^3+11 d^2 e x}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2} \\ & = \frac {4 (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-15 d^5 e^2+22 d^4 e^3 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^2} \\ & = \frac {4 (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {15 d^7 e^4}{x \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^4} \\ & = \frac {4 (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^3} \\ & = \frac {4 (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-22 e x}{15 d^4 \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 {4 (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-22 e x}{15 d^4 \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 {4 (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-22 e x}{15 d^4 \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 = 0.40 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (32 d^2+51 d e x+22 e^2 x^2\right )}{(d+e x)^3}-15 \sqrt {d^2} \log (x)+15 \sqrt {d^2} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{15 d^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(331\) vs. \(2(101)=202\).
Time = 0.41 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.89
method | result | size |
default | \(-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{3} \sqrt {d^{2}}}-\frac {-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d e \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d^{2} \left (x +\frac {d}{e}\right )}}{e \,d^{2}}+\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e \,d^{4} \left (x +\frac {d}{e}\right )}-\frac {-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d e \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d^{2} \left (x +\frac {d}{e}\right )}\right )}{5 d}}{e^{2} d}\) | \(332\) |
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Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=\frac {32 \, e^{3} x^{3} + 96 \, d e^{2} x^{2} + 96 \, d^{2} e x + 32 \, d^{3} + 15 \, {\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (22 \, e^{2} x^{2} + 51 \, d e x + 32 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{4} e^{3} x^{3} + 3 \, d^{5} e^{2} x^{2} + 3 \, d^{6} e x + d^{7}\right )}} \]
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\[ \int \frac {1}{x (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {1}{x \sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]
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\[ \int \frac {1}{x (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )}^{3} x} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (102) = 204\).
Time = 0.33 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.84 \[ \int \frac {1}{x (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{d^{4} {\left | e \right |}} - \frac {2 \, {\left (32 \, e + \frac {115 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e x} + \frac {185 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{3} x^{2}} + \frac {135 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{5} x^{3}} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{7} x^{4}}\right )}}{15 \, d^{4} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{5} {\left | e \right |}} \]
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Timed out. \[ \int \frac {1}{x (d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {1}{x\,\sqrt {d^2-e^2\,x^2}\,{\left (d+e\,x\right )}^3} \,d x \]
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