Integrand size = 25, antiderivative size = 118 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=\frac {e^2 (3 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}+e^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {3}{8} e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 118, 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 = {825, 858, 223, 209, 272, 65, 214} \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=e^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {3}{8} e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {e^2 (3 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4} \]
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 825
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}-\frac {\int \frac {\left (6 d^3 e^2+8 d^2 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^3} \, dx}{8 d^2} \\ & = \frac {e^2 (3 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}+\frac {\int \frac {12 d^5 e^4+32 d^4 e^5 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{32 d^4} \\ & = \frac {e^2 (3 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}+\frac {1}{8} \left (3 d e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+e^5 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {e^2 (3 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}+\frac {1}{16} \left (3 d e^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+e^5 \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 (3 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}+e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{8} \left (3 d e^2\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 (3 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}+e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {3}{8} e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=\frac {1}{24} \left (\frac {\sqrt {d^2-e^2 x^2} \left (-6 d^3-8 d^2 e x+15 d e^2 x^2+32 e^3 x^3\right )}{x^4}-48 e^4 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-\frac {9 \sqrt {d^2} e^4 \log (x)}{d}+\frac {9 \sqrt {d^2} e^4 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{d}\right ) \]
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Time = 0.38 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.06
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-32 e^{3} x^{3}-15 d \,e^{2} x^{2}+8 d^{2} e x +6 d^{3}\right )}{24 x^{4}}+\frac {e^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {3 e^{4} d \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}\) | \(125\) |
default | \(d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{4 d^{2} x^{4}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )+e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{d^{2} x}-\frac {4 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{d^{2}}\right )}{3 d^{2}}\right )\) | \(281\) |
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Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=-\frac {48 \, e^{4} x^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 9 \, e^{4} x^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (32 \, e^{3} x^{3} + 15 \, d e^{2} x^{2} - 8 \, d^{2} e x - 6 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{24 \, x^{4}} \]
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Result contains complex when optimal does not.
Time = 4.59 (sec) , antiderivative size = 541, normalized size of antiderivative = 4.58 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=d^{3} \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) + d^{2} e \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 e^{2} \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 ) - e^{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 ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (104) = 208\).
Time = 0.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=\frac {e^{5} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} - \frac {3}{8} \, e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{5} x}{d^{2}} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{4}}{8 \, d} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{8 \, d^{3}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{3 \, d^{2} x} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}}{8 \, d^{3} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{3 \, d^{2} x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{4 \, d x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (104) = 208\).
Time = 0.30 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.77 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=\frac {{\left (3 \, e^{5} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{3}}{x} - \frac {24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e}{x^{2}} - \frac {120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e x^{3}}\right )} e^{8} x^{4}}{192 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} {\left | e \right |}} + \frac {e^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {3 \, e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{8 \, {\left | e \right |}} + \frac {\frac {120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{5} {\left | e \right |}}{x} + \frac {24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{3} {\left | e \right |}}{x^{2}} - \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e {\left | e \right |}}{x^{3}} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} {\left | e \right |}}{e x^{4}}}{192 \, e^{4}} \]
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Timed out. \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )}{x^5} \,d x \]
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