Integrand size = 25, antiderivative size = 143 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx=\frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {e^6 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^2} \]
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Time = 0.06 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {849, 821, 272, 43, 65, 214} \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx=-\frac {e^6 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^2}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}+\frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2} \]
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {\int \frac {\left (-6 d^2 e-d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{6 d^2} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}+\frac {e^2 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{6 d} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}+\frac {e^2 \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{12 d} \\ & = -\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {e^4 \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{16 d} \\ & = \frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}+\frac {e^6 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{32 d} \\ & = \frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {e^4 \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 )}{16 d} \\ & = \frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^2} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-40 d^5-48 d^4 e x+70 d^3 e^2 x^2+96 d^2 e^3 x^3-15 d e^4 x^4-48 e^5 x^5\right )}{240 d^2 x^6}-\frac {\sqrt {d^2} e^6 \log (x)}{16 d^3}+\frac {\sqrt {d^2} e^6 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{16 d^3} \]
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Time = 0.40 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (48 e^{5} x^{5}+15 d \,e^{4} x^{4}-96 d^{2} e^{3} x^{3}-70 d^{3} e^{2} x^{2}+48 d^{4} e x +40 d^{5}\right )}{240 x^{6} d^{2}}-\frac {e^{6} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 d \sqrt {d^{2}}}\) | \(121\) |
default | \(d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 d^{2} x^{6}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{4 d^{2} x^{4}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )-\frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 d^{2} x^{5}}\) | \(198\) |
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Time = 0.31 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx=\frac {15 \, e^{6} x^{6} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (48 \, e^{5} x^{5} + 15 \, d e^{4} x^{4} - 96 \, d^{2} e^{3} x^{3} - 70 \, d^{3} e^{2} x^{2} + 48 \, d^{4} e x + 40 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, d^{2} x^{6}} \]
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Result contains complex when optimal does not.
Time = 8.07 (sec) , antiderivative size = 918, normalized size of antiderivative = 6.42 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx=d^{3} \left (\begin {cases} - \frac {d^{2}}{6 e x^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {5 e}{24 x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{3}}{48 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{5}}{16 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{6} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{6 e x^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {5 i e}{24 x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{3}}{48 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{5}}{16 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{6} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.26 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx=-\frac {e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{2}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{6}}{16 \, d^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}}{48 \, d^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{48 \, d^{5} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}}{24 \, d^{3} x^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{5 \, d^{2} x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{6 \, d x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (123) = 246\).
Time = 0.30 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.24 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx=\frac {{\left (5 \, e^{7} + \frac {12 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{5}}{x} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{3}}{x^{2}} - \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e x^{4}} + \frac {120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{3} x^{5}}\right )} e^{12} x^{6}}{1920 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{2} {\left | e \right |}} - \frac {e^{7} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{16 \, d^{2} {\left | e \right |}} - \frac {\frac {120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{10} e^{9} {\left | e \right |}}{x} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{10} e^{7} {\left | e \right |}}{x^{2}} - \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{10} e^{5} {\left | e \right |}}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{10} e^{3} {\left | e \right |}}{x^{4}} + \frac {12 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{10} e {\left | e \right |}}{x^{5}} + \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{10} {\left | e \right |}}{e x^{6}}}{1920 \, d^{12} e^{6}} \]
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Time = 13.79 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx=\frac {d^3\,\sqrt {d^2-e^2\,x^2}}{16\,x^6}-\frac {d\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{6\,x^6}-\frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{16\,d\,x^6}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{5\,d^2\,x^5}+\frac {e^6\,\mathrm {atan}\left (\frac {\sqrt {d^2-e^2\,x^2}\,1{}\mathrm {i}}{d}\right )\,1{}\mathrm {i}}{16\,d^2} \]
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