Integrand size = 27, antiderivative size = 80 \[ \int \frac {(d+e x)^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {3 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d} \]
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Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1821, 821, 272, 65, 214} \[ \int \frac {(d+e x)^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {3 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1821
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {\int \frac {-4 d^3 e-3 d^2 e^2 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{2 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}+\frac {1}{2} \left (3 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}+\frac {1}{4} \left (3 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {3}{2} \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 {\sqrt {d^2-e^2 x^2}}{2 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d x}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.45 \[ \int \frac {(d+e x)^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {(d+4 e x) \sqrt {d^2-e^2 x^2}-3 e^2 x^2 \log \left (d \left (-d-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )\right )+3 e^2 x^2 \log \left (d-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{2 d x^2} \]
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Time = 0.37 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (4 e x +d \right )}{2 d \,x^{2}}-\frac {3 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}\) | \(72\) |
default | \(-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}+d^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\right )-\frac {2 e \sqrt {-e^{2} x^{2}+d^{2}}}{d x}\) | \(139\) |
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Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx=\frac {3 \, e^{2} x^{2} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - \sqrt {-e^{2} x^{2} + d^{2}} {\left (4 \, e x + d\right )}}{2 \, d x^{2}} \]
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Result contains complex when optimal does not.
Time = 2.42 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.68 \[ \int \frac {(d+e x)^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx=d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 d^{2} x} - \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{d^{2}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {\operatorname {acosh}{\left (\frac {d}{e x} \right )}}{d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i \operatorname {asin}{\left (\frac {d}{e x} \right )}}{d} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {3 \, e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} e}{d x} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{2 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (70) = 140\).
Time = 0.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.35 \[ \int \frac {(d+e x)^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx=\frac {{\left (e^{3} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e}{x}\right )} e^{4} x^{2}}{8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d {\left | e \right |}} - \frac {3 \, e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, d {\left | e \right |}} - \frac {\frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e {\left | e \right |}}{x} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d {\left | e \right |}}{e x^{2}}}{8 \, d^{2} e^{2}} \]
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Timed out. \[ \int \frac {(d+e x)^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{x^3\,\sqrt {d^2-e^2\,x^2}} \,d x \]
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