Integrand size = 27, antiderivative size = 169 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {e \sqrt {d^2-e^2 x^2}}{2 d x^4}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{4 d^3 x^2}-\frac {6 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {3 e^5 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^4} \]
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Time = 0.13 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1821, 849, 821, 272, 65, 214} \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {3 e^5 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^4}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {e \sqrt {d^2-e^2 x^2}}{2 d x^4}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {6 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{4 d^3 x^2} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 1821
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {\int \frac {-10 d^3 e-9 d^2 e^2 x}{x^5 \sqrt {d^2-e^2 x^2}} \, dx}{5 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {e \sqrt {d^2-e^2 x^2}}{2 d x^4}+\frac {\int \frac {36 d^4 e^2+30 d^3 e^3 x}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{20 d^4} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {e \sqrt {d^2-e^2 x^2}}{2 d x^4}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {\int \frac {-90 d^5 e^3-72 d^4 e^4 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{60 d^6} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {e \sqrt {d^2-e^2 x^2}}{2 d x^4}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{4 d^3 x^2}+\frac {\int \frac {144 d^6 e^4+90 d^5 e^5 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{120 d^8} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {e \sqrt {d^2-e^2 x^2}}{2 d x^4}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{4 d^3 x^2}-\frac {6 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {\left (3 e^5\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{4 d^3} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {e \sqrt {d^2-e^2 x^2}}{2 d x^4}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{4 d^3 x^2}-\frac {6 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {\left (3 e^5\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{8 d^3} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {e \sqrt {d^2-e^2 x^2}}{2 d x^4}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{4 d^3 x^2}-\frac {6 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {\left (3 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 )}{4 d^3} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}-\frac {e \sqrt {d^2-e^2 x^2}}{2 d x^4}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{4 d^3 x^2}-\frac {6 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^4} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-4 d^4-10 d^3 e x-12 d^2 e^2 x^2-15 d e^3 x^3-24 e^4 x^4\right )}{20 d^4 x^5}-\frac {3 \sqrt {d^2} e^5 \log (x)}{4 d^5}+\frac {3 \sqrt {d^2} e^5 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{4 d^5} \]
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Time = 0.40 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.65
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (24 e^{4} x^{4}+15 d \,e^{3} x^{3}+12 d^{2} e^{2} x^{2}+10 d^{3} e x +4 d^{4}\right )}{20 x^{5} d^{4}}-\frac {3 e^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{4 d^{3} \sqrt {d^{2}}}\) | \(110\) |
default | \(d^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{2} x^{5}}+\frac {4 e^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{4} x}\right )}{5 d^{2}}\right )+e^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{4} x}\right )+2 d e \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{4 d^{2} x^{4}}+\frac {3 e^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\right )}{4 d^{2}}\right )\) | \(240\) |
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Time = 0.34 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.58 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=\frac {15 \, e^{5} x^{5} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (24 \, e^{4} x^{4} + 15 \, d e^{3} x^{3} + 12 \, d^{2} e^{2} x^{2} + 10 \, d^{3} e x + 4 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{20 \, d^{4} x^{5}} \]
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Result contains complex when optimal does not.
Time = 3.96 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.02 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{5 d^{2} x^{4}} - \frac {4 e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{15 d^{4} x^{2}} - \frac {8 e^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{15 d^{6}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{5 d^{2} x^{4}} - \frac {4 i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{15 d^{4} x^{2}} - \frac {8 i e^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{15 d^{6}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {1}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e}{8 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e^{3}}{8 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {3 e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e}{8 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e^{3}}{8 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {3 i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{5}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac {2 e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac {2 i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {3 \, e^{5} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{4 \, d^{4}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{4}}{5 \, d^{4} x} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3}}{4 \, d^{3} x^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{2}}{5 \, d^{2} x^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e}{2 \, d x^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{5 \, x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (145) = 290\).
Time = 0.29 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.28 \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=\frac {{\left (e^{6} + \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{4}}{x} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{2}}{x^{2}} + \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{x^{3}} + \frac {110 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{2} x^{4}}\right )} e^{10} x^{5}}{160 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{4} {\left | e \right |}} - \frac {3 \, e^{6} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{4 \, d^{4} {\left | e \right |}} - \frac {\frac {110 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{16} e^{8}}{x} + \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{16} e^{6}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{16} e^{4}}{x^{3}} + \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{16} e^{2}}{x^{4}} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{16}}{x^{5}}}{160 \, d^{20} e^{4} {\left | e \right |}} \]
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Timed out. \[ \int \frac {(d+e x)^2}{x^6 \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{x^6\,\sqrt {d^2-e^2\,x^2}} \,d x \]
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