Optimal. Leaf size=170 \[ -\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}+\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{95 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^3}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x} \]
[Out]
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Rubi [A] time = 0.648433, antiderivative size = 170, 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 \[ -\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}+\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{95 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^3}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x} \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)^4),x]
[Out]
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Rubi in Sympy [A] time = 54.6238, size = 148, normalized size = 0.87 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{4 x^{4}} + \frac{4 e \sqrt{d^{2} - e^{2} x^{2}}}{3 d x^{3}} - \frac{31 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{8 d^{2} x^{2}} - \frac{95 e^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8 d^{3}} + \frac{8 e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{d^{3} \left (d + e x\right )} + \frac{32 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**(5/2)/x**5/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.193004, size = 107, normalized size = 0.63 \[ \frac{-285 e^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (-6 d^4+26 d^3 e x-61 d^2 e^2 x^2+163 d e^3 x^3+448 e^4 x^4\right )}{x^4 (d+e x)}+285 e^4 \log (x)}{24 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)^4),x]
[Out]
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Maple [B] time = 0.023, size = 600, normalized size = 3.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(5/2)/x^5/(e*x+d)^4,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{4} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285901, size = 648, normalized size = 3.81 \[ \frac{640 \, e^{9} x^{9} - 1117 \, d e^{8} x^{8} - 5996 \, d^{2} e^{7} x^{7} + 4147 \, d^{3} e^{6} x^{6} + 8732 \, d^{4} e^{5} x^{5} - 4190 \, d^{5} e^{4} x^{4} - 2528 \, d^{6} e^{3} x^{3} + 1064 \, d^{7} e^{2} x^{2} - 464 \, d^{8} e x + 96 \, d^{9} + 285 \,{\left (e^{9} x^{9} + 5 \, d e^{8} x^{8} - 8 \, d^{2} e^{7} x^{7} - 20 \, d^{3} e^{6} x^{6} + 8 \, d^{4} e^{5} x^{5} + 16 \, d^{5} e^{4} x^{4} -{\left (e^{8} x^{8} - 4 \, d e^{7} x^{7} - 12 \, d^{2} e^{6} x^{6} + 8 \, d^{3} e^{5} x^{5} + 16 \, d^{4} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (256 \, e^{8} x^{8} + 2723 \, d e^{7} x^{7} - 2481 \, d^{2} e^{6} x^{6} - 7294 \, d^{3} e^{5} x^{5} + 3622 \, d^{4} e^{4} x^{4} + 2760 \, d^{5} e^{3} x^{3} - 1112 \, d^{6} e^{2} x^{2} + 464 \, d^{7} e x - 96 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (d^{3} e^{5} x^{9} + 5 \, d^{4} e^{4} x^{8} - 8 \, d^{5} e^{3} x^{7} - 20 \, d^{6} e^{2} x^{6} + 8 \, d^{7} e x^{5} + 16 \, d^{8} x^{4} -{\left (d^{3} e^{4} x^{8} - 4 \, d^{4} e^{3} x^{7} - 12 \, d^{5} e^{2} x^{6} + 8 \, d^{6} e x^{5} + 16 \, d^{7} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^5),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**(5/2)/x**5/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.353409, 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)^4*x^5),x, algorithm="giac")
[Out]