Integrand size = 27, antiderivative size = 216 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} d^3 e^5 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1821, 827, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx=\frac {13}{2} d^3 e^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} d^3 e^5 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}+\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2} \]
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 827
Rule 858
Rule 1821
Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-15 d^4 e-13 d^3 e^2 x-5 d^2 e^3 x^2\right )}{x^5} \, dx}{5 d^2} \\ & = -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {\int \frac {\left (52 d^5 e^2-25 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx}{20 d^4} \\ & = -\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {\left (150 d^6 e^3+312 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx}{72 d^4} \\ & = \frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {\int \frac {\left (-1248 d^7 e^4+600 d^6 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx}{192 d^4} \\ & = \frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {-1200 d^8 e^5-2496 d^7 e^6 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{384 d^4} \\ & = \frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {1}{8} \left (25 d^4 e^5\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{2} \left (13 d^3 e^6\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {1}{16} \left (25 d^4 e^5\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{2} \left (13 d^3 e^6\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = \frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{8} \left (25 d^4 e^3\right ) \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 ) \\ & = \frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} d^3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-24 d^7-90 d^6 e x-32 d^5 e^2 x^2+345 d^4 e^3 x^3+656 d^3 e^4 x^4+80 d^2 e^5 x^5+180 d e^6 x^6+40 e^7 x^7\right )}{120 x^5}-13 d^3 e^5 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} \left (d^2\right )^{3/2} e^5 \log (x)+\frac {25}{8} \left (d^2\right )^{3/2} e^5 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]
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Time = 0.46 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {d^{3} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (-656 e^{4} x^{4}-345 d \,e^{3} x^{3}+32 d^{2} e^{2} x^{2}+90 d^{3} e x +24 d^{4}\right )}{120 x^{5}}+\frac {e^{7} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3}+\frac {2 e^{5} d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3}+\frac {13 e^{6} d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {25 e^{5} d^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}+\frac {3 e^{6} d x \sqrt {-e^{2} x^{2}+d^{2}}}{2}\) | \(210\) |
default | \(d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{2} x^{5}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{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 )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )}{5 d^{2}}\right )+e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{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 )\right )\right )}{2 d^{2}}\right )+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{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 )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{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 )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )\) | \(660\) |
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Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx=-\frac {1560 \, d^{3} e^{5} x^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 375 \, d^{3} e^{5} x^{5} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - 80 \, d^{3} e^{5} x^{5} - {\left (40 \, e^{7} x^{7} + 180 \, d e^{6} x^{6} + 80 \, d^{2} e^{5} x^{5} + 656 \, d^{3} e^{4} x^{4} + 345 \, d^{4} e^{3} x^{3} - 32 \, d^{5} e^{2} x^{2} - 90 \, d^{6} e x - 24 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{120 \, x^{5}} \]
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Result contains complex when optimal does not.
Time = 6.31 (sec) , antiderivative size = 1182, normalized size of antiderivative = 5.47 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx=\text {Too large to display} \]
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Time = 0.30 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.34 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx=\frac {13 \, d^{3} e^{6} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} - \frac {25}{8} \, d^{3} e^{5} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {13}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{6} x + \frac {25}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{5} + \frac {13 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6} x}{3 \, d} + \frac {25}{24} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}}{8 \, d^{2}} + \frac {52 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{15 \, d x} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{8 \, d^{2} x^{2}} - \frac {13 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{15 \, d x^{3}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{4 \, x^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{5 \, x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (188) = 376\).
Time = 0.30 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.13 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx=\frac {13 \, d^{3} e^{6} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} + \frac {{\left (6 \, d^{3} e^{6} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{3} e^{4}}{x} + \frac {50 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{3} e^{2}}{x^{2}} - \frac {600 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{3}}{x^{3}} - \frac {2580 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{3}}{e^{2} x^{4}}\right )} e^{10} x^{5}}{960 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} {\left | e \right |}} - \frac {25 \, d^{3} e^{6} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{8 \, {\left | e \right |}} + \frac {1}{6} \, {\left (4 \, d^{2} e^{5} + {\left (2 \, e^{7} x + 9 \, d e^{6}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} + \frac {\frac {2580 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{3} e^{8}}{x} + \frac {600 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{3} e^{6}}{x^{2}} - \frac {50 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{3} e^{4}}{x^{3}} - \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{3} e^{2}}{x^{4}} - \frac {6 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{3}}{x^{5}}}{960 \, e^{4} {\left | e \right |}} \]
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Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^6} \,d x \]
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