Integrand size = 27, antiderivative size = 107 \[ \int \frac {(d+e x)^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}-\frac {e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \]
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Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1821, 849, 821, 272, 65, 214} \[ \int \frac {(d+e x)^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2}-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3} \]
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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}}{3 x^3}-\frac {\int \frac {-6 d^3 e-5 d^2 e^2 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}+\frac {\int \frac {10 d^4 e^2+6 d^3 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{6 d^4} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}+\frac {e^3 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}+\frac {e^3 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}-\frac {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} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {5 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}-\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (d^2+3 d e x+5 e^2 x^2\right )}{x^3}+3 \sqrt {d^2} e^3 \log (x)-3 \sqrt {d^2} e^3 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{3 d^3} \]
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Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (5 e^{2} x^{2}+3 d e x +d^{2}\right )}{3 x^{3} d^{2}}-\frac {e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d \sqrt {d^{2}}}\) | \(86\) |
default | \(-\frac {e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{d^{2} x}+d^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{4} x}\right )+2 d e \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\right )\) | \(151\) |
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Time = 0.43 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.69 \[ \int \frac {(d+e x)^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx=\frac {3 \, e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (5 \, e^{2} x^{2} + 3 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, d^{2} x^{3}} \]
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Result contains complex when optimal does not.
Time = 2.17 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.83 \[ \int \frac {(d+e x)^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx=d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac {2 e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac {2 i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 d^{2} x} - \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{d^{2}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{2}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{2}}{3 \, d^{2} x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e}{d x^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{3 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (95) = 190\).
Time = 0.29 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.38 \[ \int \frac {(d+e x)^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx=\frac {{\left (e^{4} + \frac {6 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{2}}{x} + \frac {21 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{x^{2}}\right )} e^{6} x^{3}}{24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{2} {\left | e \right |}} - \frac {e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{d^{2} {\left | e \right |}} - \frac {\frac {21 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{4} e^{4}}{x} + \frac {6 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{4} e^{2}}{x^{2}} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{4}}{x^{3}}}{24 \, d^{6} e^{2} {\left | e \right |}} \]
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Timed out. \[ \int \frac {(d+e x)^2}{x^4 \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{x^4\,\sqrt {d^2-e^2\,x^2}} \,d x \]
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