Optimal. Leaf size=172 \[ \frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {849, 821, 272,
43, 65, 214} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}+\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {\int \frac {\left (-7 d^2 e-2 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx}{7 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}+\frac {\int \frac {\left (12 d^3 e^2+7 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{42 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac {e^3 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{6 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac {e^3 \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{12 d^2}\\ &=-\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e^5 \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{16 d^2}\\ &=\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac {e^7 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{32 d^2}\\ &=\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e^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 )}{16 d^2}\\ &=\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 132, normalized size = 0.77 \begin {gather*} \frac {-\frac {\sqrt {d^2-e^2 x^2} \left (240 d^6+280 d^5 e x-384 d^4 e^2 x^2-490 d^3 e^3 x^3+48 d^2 e^4 x^4+105 d e^5 x^5+96 e^6 x^6\right )}{x^7}+210 e^7 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{1680 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 225, normalized size = 1.31
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (96 e^{6} x^{6}+105 d \,e^{5} x^{5}+48 d^{2} e^{4} x^{4}-490 d^{3} x^{3} e^{3}-384 d^{4} e^{2} x^{2}+280 d^{5} e x +240 d^{6}\right )}{1680 x^{7} d^{3}}-\frac {e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 d^{2} \sqrt {d^{2}}}\) | \(132\) |
default | \(e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 d^{2} x^{6}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{4 d^{2} x^{4}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )+d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 d^{2} x^{7}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 d^{4} x^{5}}\right )\) | \(225\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 192, normalized size = 1.12 \begin {gather*} -\frac {e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{3}} + \frac {\sqrt {-x^{2} e^{2} + d^{2}} e^{7}}{16 \, d^{4}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}}{48 \, d^{6}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}}{48 \, d^{6} x^{2}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{24 \, d^{4} x^{4}} - \frac {2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}}{35 \, d^{3} x^{5}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e}{6 \, d^{2} x^{6}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}}{7 \, d x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.50, size = 113, normalized size = 0.66 \begin {gather*} \frac {105 \, x^{7} e^{7} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - {\left (96 \, x^{6} e^{6} + 105 \, d x^{5} e^{5} + 48 \, d^{2} x^{4} e^{4} - 490 \, d^{3} x^{3} e^{3} - 384 \, d^{4} x^{2} e^{2} + 280 \, d^{5} x e + 240 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{1680 \, d^{3} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 9.76, size = 1037, normalized size = 6.03 \begin {gather*} d^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{7 x^{6}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{35 d^{2} x^{4}} + \frac {4 e^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{105 d^{4} x^{2}} + \frac {8 e^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{105 d^{6}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{7 x^{6}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{35 d^{2} x^{4}} + \frac {4 i e^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{105 d^{4} x^{2}} + \frac {8 i e^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{105 d^{6}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} - \frac {d^{2}}{6 e x^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {5 e}{24 x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{3}}{48 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{5}}{16 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{6} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{6 e x^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {5 i e}{24 x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{3}}{48 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{5}}{16 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{6} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {otherwise} \end {cases}\right ) - d e^{2} \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 ) - e^{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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 492 vs.
\(2 (139) = 278\).
time = 1.07, size = 492, normalized size = 2.86 \begin {gather*} \frac {x^{7} {\left (\frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{5}}{x} - \frac {21 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{3}}{x^{2}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e}{x^{3}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-1\right )}}{x^{4}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-3\right )}}{x^{5}} + \frac {315 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-5\right )}}{x^{6}} + 15 \, e^{7}\right )} e^{14}}{13440 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{3}} - \frac {e^{7} \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 )}{16 \, d^{3}} - \frac {\frac {315 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{18} e^{5}}{x} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{18} e^{3}}{x^{2}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{18} e}{x^{3}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{18} e^{\left (-1\right )}}{x^{4}} - \frac {21 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{18} e^{\left (-3\right )}}{x^{5}} + \frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{18} e^{\left (-5\right )}}{x^{6}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{18} e^{\left (-7\right )}}{x^{7}}}{13440 \, d^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.33, size = 192, normalized size = 1.12 \begin {gather*} \frac {8\,d\,e^2\,\sqrt {d^2-e^2\,x^2}}{35\,x^5}-\frac {d^3\,\sqrt {d^2-e^2\,x^2}}{7\,x^7}-\frac {e^4\,\sqrt {d^2-e^2\,x^2}}{35\,d\,x^3}-\frac {2\,e^6\,\sqrt {d^2-e^2\,x^2}}{35\,d^3\,x}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{6\,x^6}+\frac {d^2\,e\,\sqrt {d^2-e^2\,x^2}}{16\,x^6}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{16\,d^2\,x^6}+\frac {e^7\,\mathrm {atan}\left (\frac {\sqrt {d^2-e^2\,x^2}\,1{}\mathrm {i}}{d}\right )\,1{}\mathrm {i}}{16\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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