Integrand size = 27, antiderivative size = 215 \[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}+\frac {7 e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^9} \]
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Time = 0.15 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {871, 837, 849, 821, 272, 65, 214} \[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {7 e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^9}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}+\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
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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}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-8 d e^2+7 e^3 x}{x^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2} \\ & = \frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-48 d^3 e^4+35 d^2 e^5 x}{x^4 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^4} \\ & = \frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-192 d^5 e^6+105 d^4 e^7 x}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^6} \\ & = \frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {\int \frac {-315 d^6 e^7+384 d^5 e^8 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{45 d^{12} e^6} \\ & = \frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {\int \frac {-768 d^7 e^8+315 d^6 e^9 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{90 d^{14} e^6} \\ & = \frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}-\frac {\left (7 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^8} \\ & = \frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 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 )}{4 d^8} \\ & = \frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 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 )}{2 d^8} \\ & = \frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}+\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^9} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (10 d^7-5 d^6 e x+75 d^5 e^2 x^2+236 d^4 e^3 x^3-244 d^3 e^4 x^4-489 d^2 e^5 x^5+151 d e^6 x^6+256 e^7 x^7\right )}{x^3 (d-e x)^2 (d+e x)^3}-105 \sqrt {d^2} e^3 \log (x)+105 \sqrt {d^2} e^3 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{30 d^{10}} \]
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Time = 0.44 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.45
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (22 e^{2} x^{2}-3 d e x +2 d^{2}\right )}{6 d^{9} x^{3}}+\frac {7 e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{8} \sqrt {d^{2}}}-\frac {331 e^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{80 d^{9} \left (x +\frac {d}{e}\right )}-\frac {35 e^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{48 d^{9} \left (x -\frac {d}{e}\right )}-\frac {9 e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 d^{8} \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 d^{7} \left (x +\frac {d}{e}\right )^{3}}+\frac {e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{24 d^{8} \left (x -\frac {d}{e}\right )^{2}}\) | \(312\) |
default | \(\frac {-\frac {1}{3 d^{2} x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 e^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 e^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}\right )}{d^{2}}}{d}+\frac {e^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 e^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}\right )}{d^{3}}-\frac {e^{3} \left (\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}\right )}{d^{4}}-\frac {e \left (-\frac {1}{2 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {5 e^{2} \left (\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}\right )}{2 d^{2}}\right )}{d^{2}}+\frac {e^{3} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{d^{4}}\) | \(569\) |
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Time = 0.36 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {116 \, e^{8} x^{8} + 116 \, d e^{7} x^{7} - 232 \, d^{2} e^{6} x^{6} - 232 \, d^{3} e^{5} x^{5} + 116 \, d^{4} e^{4} x^{4} + 116 \, d^{5} e^{3} x^{3} + 105 \, {\left (e^{8} x^{8} + d e^{7} x^{7} - 2 \, d^{2} e^{6} x^{6} - 2 \, d^{3} e^{5} x^{5} + d^{4} e^{4} x^{4} + d^{5} e^{3} x^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (256 \, e^{7} x^{7} + 151 \, d e^{6} x^{6} - 489 \, d^{2} e^{5} x^{5} - 244 \, d^{3} e^{4} x^{4} + 236 \, d^{4} e^{3} x^{3} + 75 \, d^{5} e^{2} x^{2} - 5 \, d^{6} e x + 10 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (d^{9} e^{5} x^{8} + d^{10} e^{4} x^{7} - 2 \, d^{11} e^{3} x^{6} - 2 \, d^{12} e^{2} x^{5} + d^{13} e x^{4} + d^{14} x^{3}\right )}} \]
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\[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
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\[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )} x^{4}} \,d x } \]
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\[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{x^4\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \]
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