Integrand size = 27, antiderivative size = 114 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3} \]
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Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {864, 849, 821, 272, 65, 214} \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\frac {e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x} \]
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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^4 \sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {\int \frac {3 d^2 e-2 d e^2 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}+\frac {\int \frac {4 d^3 e^2-3 d^2 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{6 d^4} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {e^3 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {e^3 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e \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^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\frac {\left (-2 d^2+3 d e x-4 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{6 d^3 x^3}+\frac {\sqrt {d^2} e^3 \log (x)}{2 d^4}-\frac {\sqrt {d^2} e^3 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{2 d^4} \]
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Time = 0.43 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (4 e^{2} x^{2}-3 d e x +2 d^{2}\right )}{6 d^{3} x^{3}}+\frac {e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\) | \(88\) |
default | \(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d^{3} x^{3}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d^{2} x}-\frac {2 e^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{d^{2}}\right )}{d^{3}}-\frac {e^{3} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{d^{4}}-\frac {e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}\right )}{d^{2}}+\frac {e^{3} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d^{4}}\) | \(351\) |
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Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=-\frac {3 \, e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (4 \, e^{2} x^{2} - 3 \, d e x + 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, d^{3} x^{3}} \]
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\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{4} \left (d + e x\right )}\, dx \]
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\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\int { \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (98) = 196\).
Time = 0.29 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.24 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\frac {{\left (e^{4} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{2}}{x} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{x^{2}}\right )} e^{6} x^{3}}{24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{3} {\left | e \right |}} + \frac {e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, d^{3} {\left | e \right |}} - \frac {\frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{6} e^{4}}{x} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{6} e^{2}}{x^{2}} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{6}}{x^{3}}}{24 \, d^{9} e^{2} {\left | e \right |}} \]
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Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\int \frac {\sqrt {d^2-e^2\,x^2}}{x^4\,\left (d+e\,x\right )} \,d x \]
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