Optimal. Leaf size=201 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{3 e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d^4}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4} \]
[Out]
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Rubi [A] time = 0.442919, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac{3 e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{128 d^4}-\frac{2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac{3 e^6 \sqrt{d^2-e^2 x^2}}{128 d^3 x^2}-\frac{e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^9,x]
[Out]
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Rubi in Sympy [A] time = 59.2166, size = 175, normalized size = 0.87 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{8 d x^{8}} - \frac{e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{7 d^{2} x^{7}} + \frac{3 e^{6} \sqrt{d^{2} - e^{2} x^{2}}}{128 d^{3} x^{2}} - \frac{e^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{64 d^{3} x^{4}} - \frac{e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{16 d^{3} x^{6}} - \frac{3 e^{8} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{128 d^{4}} - \frac{2 e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{35 d^{4} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(-e**2*x**2+d**2)**(3/2)/x**9,x)
[Out]
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Mathematica [A] time = 0.20593, size = 139, normalized size = 0.69 \[ -\frac{105 e^8 x^8 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (560 d^7+640 d^6 e x-840 d^5 e^2 x^2-1024 d^4 e^3 x^3+70 d^3 e^4 x^4+128 d^2 e^5 x^5+105 d e^6 x^6+256 e^7 x^7\right )-105 e^8 x^8 \log (x)}{4480 d^4 x^8} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^9,x]
[Out]
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Maple [A] time = 0.075, size = 236, normalized size = 1.2 \[ -{\frac{1}{8\,d{x}^{8}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{2}}{16\,{d}^{3}{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{4}}{64\,{d}^{5}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{6}}{128\,{d}^{7}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{8}}{128\,{d}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{8}}{128\,{d}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{8}}{128\,{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{e}{7\,{d}^{2}{x}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{e}^{3}}{35\,{d}^{4}{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^9,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^9,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.523407, size = 799, normalized size = 3.98 \[ \frac{2048 \, d e^{15} x^{15} + 840 \, d^{2} e^{14} x^{14} - 21504 \, d^{3} e^{13} x^{13} - 8680 \, d^{4} e^{12} x^{12} + 50176 \, d^{5} e^{11} x^{11} + 15680 \, d^{6} e^{10} x^{10} + 48128 \, d^{7} e^{9} x^{9} + 63840 \, d^{8} e^{8} x^{8} - 343040 \, d^{9} e^{7} x^{7} - 286720 \, d^{10} e^{6} x^{6} + 518144 \, d^{11} e^{5} x^{5} + 430080 \, d^{12} e^{4} x^{4} - 335872 \, d^{13} e^{3} x^{3} - 286720 \, d^{14} e^{2} x^{2} + 81920 \, d^{15} e x + 71680 \, d^{16} + 105 \,{\left (e^{16} x^{16} - 32 \, d^{2} e^{14} x^{14} + 160 \, d^{4} e^{12} x^{12} - 256 \, d^{6} e^{10} x^{10} + 128 \, d^{8} e^{8} x^{8} + 8 \,{\left (d e^{14} x^{14} - 10 \, d^{3} e^{12} x^{12} + 24 \, d^{5} e^{10} x^{10} - 16 \, d^{7} e^{8} x^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (256 \, e^{15} x^{15} + 105 \, d e^{14} x^{14} - 8064 \, d^{2} e^{13} x^{13} - 3290 \, d^{3} e^{12} x^{12} + 35840 \, d^{4} e^{11} x^{11} + 13720 \, d^{5} e^{10} x^{10} - 11648 \, d^{6} e^{9} x^{9} + 11760 \, d^{7} e^{8} x^{8} - 184320 \, d^{8} e^{7} x^{7} - 156800 \, d^{9} e^{6} x^{6} + 380928 \, d^{10} e^{5} x^{5} + 313600 \, d^{11} e^{4} x^{4} - 294912 \, d^{12} e^{3} x^{3} - 250880 \, d^{13} e^{2} x^{2} + 81920 \, d^{14} e x + 71680 \, d^{15}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{4480 \,{\left (d^{4} e^{8} x^{16} - 32 \, d^{6} e^{6} x^{14} + 160 \, d^{8} e^{4} x^{12} - 256 \, d^{10} e^{2} x^{10} + 128 \, d^{12} x^{8} + 8 \,{\left (d^{5} e^{6} x^{14} - 10 \, d^{7} e^{4} x^{12} + 24 \, d^{9} e^{2} x^{10} - 16 \, d^{11} x^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^9,x, algorithm="fricas")
[Out]
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Sympy [A] time = 63.5172, size = 1159, normalized size = 5.77 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(-e**2*x**2+d**2)**(3/2)/x**9,x)
[Out]
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GIAC/XCAS [A] time = 0.298833, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^9,x, algorithm="giac")
[Out]