Integrand size = 27, antiderivative size = 209 \[ \int \frac {(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}-\frac {7 e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^8} \]
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Time = 0.31 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1819, 1821, 821, 272, 65, 214} \[ \int \frac {(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {7 e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^8}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1819
Rule 1821
Rubi steps \begin{align*} \text {integral}& = \frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^2-10 d e x-10 e^2 x^2-\frac {10 e^3 x^3}{d}-\frac {8 e^4 x^4}{d^2}}{x^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2} \\ & = \frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^2+30 d e x+45 e^2 x^2+\frac {60 e^3 x^3}{d}+\frac {46 e^4 x^4}{d^2}}{x^4 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4} \\ & = \frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^2-30 d e x-60 e^2 x^2-\frac {90 e^3 x^3}{d}}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6} \\ & = \frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}+\frac {\int \frac {90 d^3 e+210 d^2 e^2 x+270 d e^3 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{45 d^8} \\ & = \frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {\int \frac {-420 d^4 e^2-630 d^3 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{90 d^{10}} \\ & = \frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}+\frac {\left (7 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^7} \\ & = \frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}+\frac {\left (7 e^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^7} \\ & = \frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}-\frac {(7 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 )}{d^7} \\ & = \frac {2 e^3 (d+e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^3 (20 d+23 e x)}{15 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^3 (45 d+53 e x)}{15 d^8 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^6 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d^7 x^2}-\frac {14 e^2 \sqrt {d^2-e^2 x^2}}{3 d^8 x}-\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^8} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (5 d^6+5 d^5 e x+40 d^4 e^2 x^2-246 d^3 e^3 x^3+122 d^2 e^4 x^4+247 d e^5 x^5-176 e^6 x^6\right )}{x^3 (-d+e x)^3 (d+e x)}-105 \sqrt {d^2} e^3 \log (x)+105 \sqrt {d^2} e^3 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{15 d^9} \]
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Time = 0.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.29
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (14 e^{2} x^{2}+3 d e x +d^{2}\right )}{3 d^{8} x^{3}}-\frac {7 e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{7} \sqrt {d^{2}}}-\frac {e^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{8 d^{8} \left (x +\frac {d}{e}\right )}-\frac {833 e^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{120 d^{8} \left (x -\frac {d}{e}\right )}+\frac {49 e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{60 d^{7} \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{10 d^{6} \left (x -\frac {d}{e}\right )^{3}}\) | \(270\) |
default | \(e^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{d^{2}}\right )+d^{2} \left (-\frac {1}{3 d^{2} x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {8 e^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{d^{2}}\right )}{3 d^{2}}\right )+2 d e \left (-\frac {1}{2 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 e^{2} \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )}{2 d^{2}}\right )\) | \(381\) |
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Time = 0.33 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.09 \[ \int \frac {(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {116 \, e^{7} x^{7} - 232 \, d e^{6} x^{6} + 232 \, d^{3} e^{4} x^{4} - 116 \, d^{4} e^{3} x^{3} + 105 \, {\left (e^{7} x^{7} - 2 \, d e^{6} x^{6} + 2 \, d^{3} e^{4} x^{4} - d^{4} e^{3} x^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (176 \, e^{6} x^{6} - 247 \, d e^{5} x^{5} - 122 \, d^{2} e^{4} x^{4} + 246 \, d^{3} e^{3} x^{3} - 40 \, d^{4} e^{2} x^{2} - 5 \, d^{5} e x - 5 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{8} e^{4} x^{7} - 2 \, d^{9} e^{3} x^{6} + 2 \, d^{11} e x^{4} - d^{12} x^{3}\right )}} \]
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\[ \int \frac {(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {22 \, e^{4} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}} + \frac {7 \, e^{3}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}} + \frac {88 \, e^{4} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}} + \frac {7 \, e^{3}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} - \frac {11 \, e^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x} + \frac {176 \, e^{4} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8}} - \frac {7 \, e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{8}} + \frac {7 \, e^{3}}{\sqrt {-e^{2} x^{2} + d^{2}} d^{7}} - \frac {e}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{2}} - \frac {1}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x^{3}} \]
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\[ \int \frac {(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^2}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{x^4\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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