Integrand size = 27, antiderivative size = 145 \[ \int \frac {(d+e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {2 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6} \]
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Time = 0.18 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1819, 821, 272, 65, 214} \[ \int \frac {(d+e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {2 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}+\frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1819
Rubi steps \begin{align*} \text {integral}& = \frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^2-10 d e x-8 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2} \\ & = \frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^2+30 d e x+26 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4} \\ & = \frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^2-30 d e x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6} \\ & = \frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {(2 e) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^5} \\ & = \frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {e \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{d^5} \\ & = \frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {2 \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^5 e} \\ & = \frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (15 d^4-76 d^3 e x+32 d^2 e^2 x^2+82 d e^3 x^3-56 e^4 x^4\right )}{x (-d+e x)^3 (d+e x)}-30 \sqrt {d^2} e \log (x)+30 \sqrt {d^2} e \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{15 d^7} \]
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Time = 0.38 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.72
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{6} x}-\frac {2 e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{5} \sqrt {d^{2}}}+\frac {29 \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{60 d^{5} e \left (x -\frac {d}{e}\right )^{2}}-\frac {313 \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{120 d^{6} \left (x -\frac {d}{e}\right )}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{8 d^{6} \left (x +\frac {d}{e}\right )}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{10 d^{4} e^{2} \left (x -\frac {d}{e}\right )^{3}}\) | \(250\) |
default | \(e^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+d^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{d^{2}}\right )+2 d e \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )\) | \(288\) |
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Time = 0.37 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.34 \[ \int \frac {(d+e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {46 \, e^{5} x^{5} - 92 \, d e^{4} x^{4} + 92 \, d^{3} e^{2} x^{2} - 46 \, d^{4} e x + 30 \, {\left (e^{5} x^{5} - 2 \, d e^{4} x^{4} + 2 \, d^{3} e^{2} x^{2} - d^{4} e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (56 \, e^{4} x^{4} - 82 \, d e^{3} x^{3} - 32 \, d^{2} e^{2} x^{2} + 76 \, d^{3} e x - 15 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{6} e^{4} x^{5} - 2 \, d^{7} e^{3} x^{4} + 2 \, d^{9} e x^{2} - d^{10} x\right )}} \]
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\[ \int \frac {(d+e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {7 \, e^{2} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}} + \frac {2 \, e}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d} + \frac {28 \, e^{2} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}} + \frac {2 \, e}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} - \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x} + \frac {56 \, e^{2} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}} - \frac {2 \, e \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{6}} + \frac {2 \, e}{\sqrt {-e^{2} x^{2} + d^{2}} d^{5}} \]
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\[ \int \frac {(d+e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{x^2\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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