Optimal. Leaf size=108 \[ -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d}+\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{8 x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {807, 266, 47, 63, 208} \[ \frac {3 e^3 \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}-\frac {3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}+e \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}+\frac {1}{2} e \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}-\frac {1}{8} \left (3 e^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )\\ &=\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}+\frac {1}{16} \left (3 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )\\ &=\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}-\frac {1}{8} \left (3 e^3\right ) \operatorname {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 e^3 \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}-\frac {3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 133, normalized size = 1.23 \[ -\frac {8 d^6+10 d^5 e x-24 d^4 e^2 x^2-35 d^3 e^3 x^3+24 d^2 e^4 x^4+15 d e^5 x^5 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )+25 d e^5 x^5-8 e^6 x^6}{40 d x^5 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 98, normalized size = 0.91 \[ \frac {15 \, e^{5} x^{5} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (8 \, e^{4} x^{4} - 25 \, d e^{3} x^{3} - 16 \, d^{2} e^{2} x^{2} + 10 \, d^{3} e x + 8 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{40 \, d x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 368, normalized size = 3.41 \[ \frac {x^{5} {\left (\frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{10}}{x} - \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{8}}{x^{2}} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{6}}{x^{3}} + \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{4}}{x^{4}} + 2 \, e^{12}\right )} e^{3}}{320 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d} - \frac {3 \, e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{8 \, d} - \frac {{\left (\frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{38}}{x} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e^{36}}{x^{2}} - \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{34}}{x^{3}} + \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} e^{32}}{x^{4}} + \frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{4} e^{30}}{x^{5}}\right )} e^{\left (-35\right )}}{320 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 158, normalized size = 1.46 \[ -\frac {3 e^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}+\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{5}}{8 d^{2}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5}}{8 d^{4}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}{8 d^{4} x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}{4 d^{2} x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 d \,x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 155, normalized size = 1.44 \[ -\frac {3 \, e^{5} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{5}}{8 \, d^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}}{8 \, d^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{8 \, d^{4} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{4 \, d^{2} x^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{5 \, d x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.26, size = 93, normalized size = 0.86 \[ \frac {3\,d^2\,e\,\sqrt {d^2-e^2\,x^2}}{8\,x^4}-\frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{5\,d\,x^5}-\frac {3\,e^5\,\mathrm {atanh}\left (\frac {\sqrt {d^2-e^2\,x^2}}{d}\right )}{8\,d}-\frac {5\,e\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{8\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 11.36, size = 774, normalized size = 7.17 \[ d^{3} \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) + d^{2} e \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 e^{2} \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 ) - e^{3} \left (\begin {cases} - \frac {d^{2}}{2 e x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e}{2 x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \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 e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{2 x} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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