Integrand size = 25, antiderivative size = 121 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=-\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {3}{2} d^2 e^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d^2 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {827, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=-\frac {3}{2} d^2 e^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d^2 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2} \]
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 827
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {3}{8} \int \frac {\left (-4 d^2 e+4 d e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx \\ & = -\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+\frac {3}{16} \int \frac {-8 d^3 e^2-8 d^2 e^3 x}{x \sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {1}{2} \left (3 d^3 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{2} \left (3 d^2 e^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {1}{4} \left (3 d^3 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{2} \left (3 d^2 e^3\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 {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {3}{2} d^2 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{2} \left (3 d^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 ) \\ & = -\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {3}{2} d^2 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d^2 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=\frac {1}{2} \left (-\frac {\sqrt {d^2-e^2 x^2} \left (d^3+2 d^2 e x+2 d e^2 x^2+e^3 x^3\right )}{x^2}+6 d^2 e^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+3 d \sqrt {d^2} e^2 \log (x)-3 d \sqrt {d^2} e^2 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )\right ) \]
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Time = 0.36 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24
method | result | size |
risch | \(-\frac {d^{2} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (2 e x +d \right )}{2 x^{2}}-\frac {3 e^{3} d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {e^{3} x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {3 e^{2} d^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}-d \,e^{2} \sqrt {-e^{2} x^{2}+d^{2}}\) | \(150\) |
default | \(d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )+e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{d^{2} x}-\frac {4 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{d^{2}}\right )\) | \(219\) |
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Time = 0.34 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=\frac {6 \, d^{2} e^{2} x^{2} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 3 \, d^{2} e^{2} x^{2} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - 2 \, d^{2} e^{2} x^{2} - {\left (e^{3} x^{3} + 2 \, d e^{2} x^{2} + 2 \, d^{2} e x + d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2 \, x^{2}} \]
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Result contains complex when optimal does not.
Time = 3.22 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.67 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=d^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 x} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} \frac {d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {d^{2} - e^{2} x^{2}}}{2} & \text {for}\: e^{2} \neq 0 \\x \sqrt {d^{2}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=-\frac {3 \, d^{2} e^{3} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} + \frac {3}{2} \, d^{2} e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3} x - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{2 \, d} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{x} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{2 \, d x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (106) = 212\).
Time = 0.29 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.02 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=-\frac {3 \, d^{2} e^{3} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} + \frac {3 \, d^{2} e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, {\left | e \right |}} + \frac {{\left (d^{2} e^{3} + \frac {4 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} e}{x}\right )} e^{4} x^{2}}{8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} {\left | e \right |}} - \frac {1}{2} \, {\left (e^{3} x + 2 \, d e^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}} - \frac {\frac {4 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} e {\left | e \right |}}{x} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} {\left | e \right |}}{e x^{2}}}{8 \, e^{2}} \]
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Time = 12.53 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=\frac {3\,d^2\,e^2\,\mathrm {atanh}\left (\frac {\sqrt {d^2-e^2\,x^2}}{d}\right )}{2}-\frac {d^3\,\sqrt {d^2-e^2\,x^2}}{2\,x^2}-d\,e^2\,\sqrt {d^2-e^2\,x^2}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ \frac {e^2\,x^2}{d^2}\right )}{x\,{\left (1-\frac {e^2\,x^2}{d^2}\right )}^{3/2}} \]
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