Integrand size = 25, antiderivative size = 120 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=\frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+d e^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {825, 827, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=d e^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3} \]
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 825
Rule 827
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}-\frac {\int \frac {\left (4 d^3 e^2+6 d^2 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx}{4 d^2} \\ & = \frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+\frac {\int \frac {-12 d^4 e^3+8 d^3 e^4 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{8 d^2} \\ & = \frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}-\frac {1}{2} \left (3 d^2 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\left (d e^4\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}-\frac {1}{4} \left (3 d^2 e^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\left (d e^4\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 {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+d e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{2} \left (3 d^2 e\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 {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+d e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^3-3 d^2 e x+8 d e^2 x^2-6 e^3 x^3\right )}{6 x^3}-2 d e^3 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} \sqrt {d^2} e^3 \log (x)-\frac {3}{2} \sqrt {d^2} e^3 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]
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Time = 0.37 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.14
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d \left (-8 e^{2} x^{2}+3 d e x +2 d^{2}\right )}{6 x^{3}}+\frac {e^{4} d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-e^{3} \sqrt {-e^{2} x^{2}+d^{2}}+\frac {3 e^{3} d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}\) | \(137\) |
default | \(e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )+d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{d^{2} x}-\frac {4 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{d^{2}}\right )}{3 d^{2}}\right )\) | \(250\) |
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Time = 0.35 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=-\frac {12 \, d e^{3} x^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 9 \, d e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 6 \, d e^{3} x^{3} + {\left (6 \, e^{3} x^{3} - 8 \, d e^{2} x^{2} + 3 \, d^{2} e x + 2 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 3.46 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.81 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=d^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 x} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=\frac {d e^{4} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} + \frac {3}{2} \, d e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{4} x}{d} - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{2 \, d^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{3 \, d x} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{2 \, d^{2} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{3 \, d x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (106) = 212\).
Time = 0.28 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.38 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=\frac {d e^{4} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} + \frac {{\left (d e^{4} + \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e^{2}}{x} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d}{x^{2}}\right )} e^{6} x^{3}}{24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} {\left | e \right |}} + \frac {3 \, d e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, {\left | e \right |}} - \sqrt {-e^{2} x^{2} + d^{2}} e^{3} + \frac {\frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e^{4}}{x} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d e^{2}}{x^{2}} - \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d}{x^{3}}}{24 \, e^{2} {\left | e \right |}} \]
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Timed out. \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )}{x^4} \,d x \]
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