Optimal. Leaf size=204 \[ -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}+e^8 \left (-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {125}{128} e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4} \]
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Rubi [A] time = 0.30, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1807, 811, 844, 217, 203, 266, 63, 208} \[ -\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}+e^8 \left (-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+\frac {125}{128} e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 811
Rule 844
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-24 d^4 e-25 d^3 e^2 x-8 d^2 e^3 x^2\right )}{x^8} \, dx}{8 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}+\frac {\int \frac {\left (175 d^5 e^2+56 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx}{56 d^4}\\ &=-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {\int \frac {\left (1750 d^7 e^4+672 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{672 d^6}\\ &=\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}+\frac {\int \frac {\left (10500 d^9 e^6+5376 d^8 e^7 x\right ) \sqrt {d^2-e^2 x^2}}{x^3} \, dx}{5376 d^8}\\ &=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {\int \frac {21000 d^{11} e^8+21504 d^{10} e^9 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{21504 d^{10}}\\ &=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {1}{128} \left (125 d e^8\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-e^9 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {1}{256} \left (125 d e^8\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-e^9 \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-e^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{128} \left (125 d e^6\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 {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-e^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {125}{128} e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [C] time = 0.15, size = 245, normalized size = 1.20 \[ -\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac {e^8 \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )}{7 d^7}-\frac {d^4 e^3 \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{5 x^5 \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {-8 d^7 e^2+34 d^5 e^4 x^2-59 d^3 e^6 x^4+15 d e^8 x^6 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )+33 d e^8 x^6}{16 x^6 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 163, normalized size = 0.80 \[ \frac {26880 \, e^{8} x^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 13125 \, e^{8} x^{8} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (14848 \, e^{7} x^{7} + 27195 \, d e^{6} x^{6} + 7424 \, d^{2} e^{5} x^{5} - 17710 \, d^{3} e^{4} x^{4} - 14592 \, d^{4} e^{3} x^{3} + 1960 \, d^{5} e^{2} x^{2} + 5760 \, d^{6} e x + 1680 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{13440 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 538, normalized size = 2.64 \[ -\arcsin \left (\frac {x e}{d}\right ) e^{8} \mathrm {sgn}\relax (d) + \frac {x^{8} {\left (\frac {720 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{16}}{x} + \frac {1120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{14}}{x^{2}} - \frac {3696 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{12}}{x^{3}} - \frac {14280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{10}}{x^{4}} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{8}}{x^{5}} + \frac {77280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{6}}{x^{6}} + \frac {122640 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{4}}{x^{7}} + 105 \, e^{18}\right )} e^{6}}{215040 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8}} - \frac {1}{215040} \, {\left (\frac {122640 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{86}}{x} + \frac {77280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{84}}{x^{2}} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{82}}{x^{3}} - \frac {14280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{80}}{x^{4}} - \frac {3696 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{78}}{x^{5}} + \frac {1120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{76}}{x^{6}} + \frac {720 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{74}}{x^{7}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{72}}{x^{8}}\right )} e^{\left (-80\right )} + \frac {125}{128} \, e^{8} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 402, normalized size = 1.97 \[ \frac {125 d \,e^{8} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 \sqrt {d^{2}}}-\frac {e^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{9} x}{d^{2}}-\frac {125 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{8}}{128 d}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{9} x}{3 d^{4}}-\frac {125 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{8}}{384 d^{3}}-\frac {8 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{9} x}{15 d^{6}}-\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{8}}{128 d^{5}}-\frac {8 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{7}}{15 d^{6} x}-\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{6}}{128 d^{5} x^{2}}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{15 d^{4} x^{3}}+\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{192 d^{3} x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{5 d^{2} x^{5}}-\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{48 d \,x^{6}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{7 x^{7}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{8 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 352, normalized size = 1.73 \[ -e^{8} \arcsin \left (\frac {e x}{d}\right ) + \frac {125}{128} \, e^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{9} x}{d^{2}} - \frac {125 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8}}{128 \, d} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{9} x}{3 \, d^{4}} - \frac {125 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8}}{384 \, d^{3}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{8}}{128 \, d^{5}} - \frac {8 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7}}{15 \, d^{4} x} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{6}}{128 \, d^{5} x^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{15 \, d^{4} x^{3}} + \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{192 \, d^{3} x^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{5 \, d^{2} x^{5}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{48 \, d x^{6}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{7 \, x^{7}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{8 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^9} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 31.40, size = 1719, normalized size = 8.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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