Optimal. Leaf size=172 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}+\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}-\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
[Out]
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Rubi [A] time = 0.344622, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}+\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}-\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^8,x]
[Out]
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Rubi in Sympy [A] time = 45.3999, size = 146, normalized size = 0.85 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{7 d x^{7}} + \frac{e^{5} \sqrt{d^{2} - e^{2} x^{2}}}{16 d^{2} x^{2}} - \frac{e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{24 d^{2} x^{4}} - \frac{e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{6 d^{2} x^{6}} - \frac{e^{7} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{16 d^{3}} - \frac{2 e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{35 d^{3} x^{5}} \]
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**8,x)
[Out]
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Mathematica [A] time = 0.17817, size = 128, normalized size = 0.74 \[ -\frac{105 e^7 x^7 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (240 d^6+280 d^5 e x-384 d^4 e^2 x^2-490 d^3 e^3 x^3+48 d^2 e^4 x^4+105 d e^5 x^5+96 e^6 x^6\right )-105 e^7 x^7 \log (x)}{1680 d^3 x^7} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^8,x]
[Out]
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Maple [A] time = 0.053, size = 211, normalized size = 1.2 \[ -{\frac{1}{7\,d{x}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{e}^{2}}{35\,{d}^{3}{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{e}{6\,{d}^{2}{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{3}}{24\,{d}^{4}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{5}}{48\,{d}^{6}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{7}}{48\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{7}}{16\,{d}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{e}^{7}}{16\,{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^8,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^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.37202, size = 711, normalized size = 4.13 \[ -\frac{96 \, e^{14} x^{14} + 105 \, d e^{13} x^{13} - 2352 \, d^{2} e^{12} x^{12} - 3115 \, d^{3} e^{11} x^{11} + 8400 \, d^{4} e^{10} x^{10} + 23450 \, d^{5} e^{9} x^{9} + 1008 \, d^{6} e^{8} x^{8} - 73080 \, d^{7} e^{7} x^{7} - 46704 \, d^{8} e^{6} x^{6} + 106400 \, d^{9} e^{5} x^{5} + 83328 \, d^{10} e^{4} x^{4} - 71680 \, d^{11} e^{3} x^{3} - 59136 \, d^{12} e^{2} x^{2} + 17920 \, d^{13} e x + 15360 \, d^{14} - 105 \,{\left (7 \, d e^{13} x^{13} - 56 \, d^{3} e^{11} x^{11} + 112 \, d^{5} e^{9} x^{9} - 64 \, d^{7} e^{7} x^{7} -{\left (e^{13} x^{13} - 24 \, d^{2} e^{11} x^{11} + 80 \, d^{4} e^{9} x^{9} - 64 \, d^{6} e^{7} x^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (672 \, d e^{12} x^{12} + 735 \, d^{2} e^{11} x^{11} - 5040 \, d^{3} e^{10} x^{10} - 9310 \, d^{4} e^{9} x^{9} + 5376 \, d^{5} e^{8} x^{8} + 41160 \, d^{6} e^{7} x^{7} + 22416 \, d^{7} e^{6} x^{6} - 77280 \, d^{8} e^{5} x^{5} - 59520 \, d^{9} e^{4} x^{4} + 62720 \, d^{10} e^{3} x^{3} + 51456 \, d^{11} e^{2} x^{2} - 17920 \, d^{12} e x - 15360 \, d^{13}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1680 \,{\left (7 \, d^{4} e^{6} x^{13} - 56 \, d^{6} e^{4} x^{11} + 112 \, d^{8} e^{2} x^{9} - 64 \, d^{10} x^{7} -{\left (d^{3} e^{6} x^{13} - 24 \, d^{5} e^{4} x^{11} + 80 \, d^{7} e^{2} x^{9} - 64 \, d^{9} x^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^8,x, algorithm="fricas")
[Out]
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Sympy [A] time = 43.5264, size = 1037, normalized size = 6.03 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(-e**2*x**2+d**2)**(3/2)/x**8,x)
[Out]
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GIAC/XCAS [A] time = 0.293522, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^8,x, algorithm="giac")
[Out]