Integrand size = 25, antiderivative size = 117 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^3 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 117, 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 = {827, 829, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx=-\frac {3}{2} d^3 e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^3 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x} \]
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 827
Rule 829
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {1}{2} \int \frac {\left (-2 d^2 e+6 d e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx \\ & = \frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}+\frac {\int \frac {4 d^4 e^3-6 d^3 e^4 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{4 e^2} \\ & = \frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}+\left (d^4 e\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{2} \left (3 d^3 e^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}+\frac {1}{2} \left (d^4 e\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^3 e^2\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 {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d^4 \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 )}{e} \\ & = \frac {1}{2} d e (2 d-3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-6 d^3+8 d^2 e x-3 d e^2 x^2-2 e^3 x^3\right )}{6 x}+2 d^3 e \text {arctanh}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {3}{2} d^3 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right ) \]
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Time = 0.36 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.41
method | result | size |
risch | \(-\frac {d^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{x}-\frac {e^{3} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3}+\frac {4 e \,d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3}-\frac {e \,d^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}-\frac {3 d^{3} e^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {d \,e^{2} x \sqrt {-e^{2} x^{2}+d^{2}}}{2}\) | \(165\) |
default | \(e \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )+d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{d^{2} x}-\frac {4 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{d^{2}}\right )\) | \(188\) |
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Time = 0.40 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {18 \, d^{3} e x \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 6 \, d^{3} e x \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 8 \, d^{3} e x - {\left (2 \, e^{3} x^{3} + 3 \, d e^{2} x^{2} - 8 \, d^{2} e x + 6 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, x} \]
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Result contains complex when optimal does not.
Time = 2.58 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.32 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx=d^{3} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \frac {d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {d^{2} - e^{2} x^{2}}}{2} & \text {for}\: e^{2} \neq 0 \\x \sqrt {d^{2}} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} - \frac {d^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{2}} + \frac {x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3} & \text {for}\: e^{2} \neq 0 \\\frac {x^{2} \sqrt {d^{2}}}{2} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx=-\frac {3 \, d^{3} e^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} - d^{3} e \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e + \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{x} \]
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Time = 0.29 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.52 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx=-\frac {3 \, d^{3} e^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} + \frac {d^{3} e^{4} x}{2 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} {\left | e \right |}} - \frac {d^{3} e^{2} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{{\left | e \right |}} - \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{3}}{2 \, x {\left | e \right |}} + \frac {1}{6} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (8 \, d^{2} e - {\left (2 \, e^{3} x + 3 \, d e^{2}\right )} x\right )} \]
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Time = 12.36 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx=\frac {e\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{3}+d^2\,e\,\sqrt {d^2-e^2\,x^2}-d^3\,e\,\mathrm {atanh}\left (\frac {\sqrt {d^2-e^2\,x^2}}{d}\right )-\frac {d^3\,\sqrt {d^2-e^2\,x^2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ \frac {e^2\,x^2}{d^2}\right )}{x\,\sqrt {1-\frac {e^2\,x^2}{d^2}}} \]
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