Integrand size = 27, antiderivative size = 152 \[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {5 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6} \]
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Time = 0.08 (sec) , antiderivative size = 152, 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 = {871, 837, 849, 821, 272, 65, 214} \[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {5 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 837
Rule 849
Rule 871
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-5 d e^2+4 e^3 x}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2} \\ & = \frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^3 e^4+8 d^2 e^5 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^6 e^4} \\ & = \frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {\int \frac {-16 d^4 e^5+15 d^3 e^6 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{6 d^8 e^4} \\ & = \frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {\left (5 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^5} \\ & = \frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {\left (5 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^5} \\ & = \frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {5 \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^5} \\ & = \frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {5 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (3 d^4-3 d^3 e x-23 d^2 e^2 x^2+d e^3 x^3+16 e^4 x^4\right )}{x^2 (-d+e x) (d+e x)^2}-15 \sqrt {d^2} e^2 \log (x)+15 \sqrt {d^2} e^2 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{6 d^7} \]
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Time = 0.40 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.36
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-2 e x +d \right )}{2 d^{6} x^{2}}-\frac {5 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{5} \sqrt {d^{2}}}+\frac {23 e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{12 d^{6} \left (x +\frac {d}{e}\right )}-\frac {e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{4 d^{6} \left (x -\frac {d}{e}\right )}+\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{6 d^{5} \left (x +\frac {d}{e}\right )^{2}}\) | \(206\) |
default | \(\frac {-\frac {1}{2 d^{2} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {3 e^{2} \left (\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}\right )}{2 d^{2}}}{d}+\frac {e^{2} \left (\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}\right )}{d^{3}}-\frac {e \left (-\frac {1}{d^{2} x \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {2 e^{2} x}{d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}-\frac {e^{2} \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{d^{3}}\) | \(325\) |
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Time = 0.29 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {14 \, e^{5} x^{5} + 14 \, d e^{4} x^{4} - 14 \, d^{2} e^{3} x^{3} - 14 \, d^{3} e^{2} x^{2} + 15 \, {\left (e^{5} x^{5} + d e^{4} x^{4} - d^{2} e^{3} x^{3} - d^{3} e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (16 \, e^{4} x^{4} + d e^{3} x^{3} - 23 \, d^{2} e^{2} x^{2} - 3 \, d^{3} e x + 3 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, {\left (d^{6} e^{3} x^{5} + d^{7} e^{2} x^{4} - d^{8} e x^{3} - d^{9} x^{2}\right )}} \]
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\[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
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\[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{3}} \,d x } \]
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\[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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