Optimal. Leaf size=172 \[ -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}+\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
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Rubi [A] time = 0.13, 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 = {835, 807, 266, 47, 63, 208} \[ \frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
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 \operatorname {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 \operatorname {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 \operatorname {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 \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 )}{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 [C] time = 0.02, size = 72, normalized size = 0.42 \[ -\frac {\left (d^2-e^2 x^2\right )^{5/2} \left (5 d^7+2 d^5 e^2 x^2+7 e^7 x^7 \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};1-\frac {e^2 x^2}{d^2}\right )\right )}{35 d^8 x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 120, normalized size = 0.70 \[ \frac {105 \, e^{7} x^{7} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (96 \, e^{6} x^{6} + 105 \, d e^{5} x^{5} + 48 \, d^{2} e^{4} x^{4} - 490 \, d^{3} e^{3} x^{3} - 384 \, d^{4} e^{2} x^{2} + 280 \, d^{5} e x + 240 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1680 \, d^{3} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 494, normalized size = 2.87 \[ \frac {x^{7} {\left (\frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{14}}{x} - \frac {21 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{12}}{x^{2}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{10}}{x^{3}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{8}}{x^{4}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{6}}{x^{5}} + \frac {315 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{4}}{x^{6}} + 15 \, e^{16}\right )} e^{5}}{13440 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{3}} - \frac {e^{7} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{16 \, d^{3}} - \frac {{\left (\frac {315 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{18} e^{68}}{x} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{18} e^{66}}{x^{2}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{18} e^{64}}{x^{3}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{18} e^{62}}{x^{4}} - \frac {21 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{18} e^{60}}{x^{5}} + \frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{18} e^{58}}{x^{6}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{18} e^{56}}{x^{7}}\right )} e^{\left (-63\right )}}{13440 \, d^{21}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 211, normalized size = 1.23 \[ -\frac {e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 \sqrt {d^{2}}\, d^{2}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{7}}{16 d^{4}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7}}{48 d^{6}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{5}}{48 d^{6} x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}{24 d^{4} x^{4}}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}{35 d^{3} x^{5}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}{6 d^{2} x^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 d \,x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 205, normalized size = 1.19 \[ -\frac {e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{3}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{7}}{16 \, d^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}}{48 \, d^{6}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}}{48 \, d^{6} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{24 \, d^{4} x^{4}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}}{35 \, d^{3} x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{6 \, d^{2} x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{7 \, d x^{7}} \]
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 \[ \frac {8\,d\,e^2\,\sqrt {d^2-e^2\,x^2}}{35\,x^5}-\frac {d^3\,\sqrt {d^2-e^2\,x^2}}{7\,x^7}-\frac {e^4\,\sqrt {d^2-e^2\,x^2}}{35\,d\,x^3}-\frac {2\,e^6\,\sqrt {d^2-e^2\,x^2}}{35\,d^3\,x}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{6\,x^6}+\frac {d^2\,e\,\sqrt {d^2-e^2\,x^2}}{16\,x^6}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{16\,d^2\,x^6}+\frac {e^7\,\mathrm {atan}\left (\frac {\sqrt {d^2-e^2\,x^2}\,1{}\mathrm {i}}{d}\right )\,1{}\mathrm {i}}{16\,d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 16.62, size = 1037, normalized size = 6.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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