Integrand size = 25, antiderivative size = 113 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {3}{8} d^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 113, 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 = {829, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {3}{8} d^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2} \]
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 829
Rule 858
Rubi steps \begin{align*} \text {integral}& = \frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int \frac {\left (-4 d^3 e^2-3 d^2 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{4 e^2} \\ & = \frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int \frac {8 d^5 e^4+3 d^4 e^5 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{8 e^4} \\ & = \frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+d^5 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{8} \left (3 d^4 e\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{2} d^5 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{8} \left (3 d^4 e\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}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {3}{8} d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d^5 \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^2} \\ & = \frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {3}{8} d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.26 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {1}{24} \sqrt {d^2-e^2 x^2} \left (32 d^3+15 d^2 e x-8 d e^2 x^2-6 e^3 x^3\right )-\frac {3}{4} d^4 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-d^3 \sqrt {d^2} \log (x)+d^3 \sqrt {d^2} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.39
method | result | size |
default | \(e \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )+d \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\) | \(157\) |
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Time = 0.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=-\frac {3}{4} \, d^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + d^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - \frac {1}{24} \, {\left (6 \, e^{3} x^{3} + 8 \, d e^{2} x^{2} - 15 \, d^{2} e x - 32 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]
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Result contains complex when optimal does not.
Time = 5.95 (sec) , antiderivative size = 400, normalized size of antiderivative = 3.54 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=d^{3} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} \frac {d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {d^{2} - e^{2} x^{2}}}{2} & \text {for}\: e^{2} \neq 0 \\x \sqrt {d^{2}} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} - \frac {d^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3 e^{2}} + \frac {x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{3} & \text {for}\: e^{2} \neq 0 \\\frac {x^{2} \sqrt {d^{2}}}{2} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} \frac {d^{4} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} - \frac {d^{2} x \sqrt {d^{2} - e^{2} x^{2}}}{8 e^{2}} + \frac {x^{3} \sqrt {d^{2} - e^{2} x^{2}}}{4} & \text {for}\: e^{2} \neq 0 \\\frac {x^{3} \sqrt {d^{2}}}{3} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {3 \, d^{4} e \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{8 \, \sqrt {e^{2}}} - d^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {3}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e x + \sqrt {-e^{2} x^{2} + d^{2}} d^{3} + \frac {1}{4} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e x + \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d \]
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Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {3 \, d^{4} e \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, {\left | e \right |}} - \frac {d^{4} e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{{\left | e \right |}} + \frac {1}{24} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (32 \, d^{3} + {\left (15 \, d^{2} e - 2 \, {\left (3 \, e^{3} x + 4 \, d e^{2}\right )} x\right )} x\right )} \]
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Time = 11.74 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx=\frac {d\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{3}-d^4\,\mathrm {atanh}\left (\frac {\sqrt {d^2-e^2\,x^2}}{d}\right )+d^3\,\sqrt {d^2-e^2\,x^2}+\frac {e\,x\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right )}{{\left (1-\frac {e^2\,x^2}{d^2}\right )}^{3/2}} \]
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