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 = 0.439908, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac{e^5 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}+\frac{e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac{2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac{e^7 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)),x]
[Out]
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Rubi in Sympy [A] time = 53.7654, 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**2*x**2+d**2)**(5/2)/x**8/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.128306, 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^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)),x]
[Out]
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Maple [B] time = 0.022, size = 546, normalized size = 3.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d),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)^(5/2)/((e*x + d)*x^8),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.381634, 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)^(5/2)/((e*x + d)*x^8),x, algorithm="fricas")
[Out]
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Sympy [A] time = 49.1006, 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**2*x**2+d**2)**(5/2)/x**8/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 0.313949, 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)^(5/2)/((e*x + d)*x^8),x, algorithm="giac")
[Out]