Optimal. Leaf size=196 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}+\frac{27 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2} \]
[Out]
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Rubi [A] time = 0.796705, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}+\frac{27 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2} \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^4),x]
[Out]
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Rubi in Sympy [A] time = 67.506, size = 172, normalized size = 0.88 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 x^{5}} + \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{d x^{4}} - \frac{13 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{2} x^{3}} + \frac{11 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{2 d^{3} x^{2}} + \frac{27 e^{5} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{4}} - \frac{8 e^{5} \sqrt{d^{2} - e^{2} x^{2}}}{d^{4} \left (d + e x\right )} - \frac{66 e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**(5/2)/x**6/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.298614, size = 118, normalized size = 0.6 \[ -\frac{-135 e^5 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (2 d^5-8 d^4 e x+16 d^3 e^2 x^2-29 d^2 e^3 x^3+77 d e^4 x^4+212 e^5 x^5\right )}{x^5 (d+e x)}+135 e^5 \log (x)}{10 d^4} \]
Antiderivative was successfully verified.
[In] Integrate[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^4),x]
[Out]
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Maple [B] time = 0.024, size = 628, 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^6/(e*x+d)^4,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{4} x^{6}}\,{d x} \]
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^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285192, size = 765, normalized size = 3.9 \[ \frac{132 \, e^{11} x^{11} + 1537 \, d e^{10} x^{10} - 2020 \, d^{2} e^{9} x^{9} - 7355 \, d^{3} e^{8} x^{8} + 5390 \, d^{4} e^{7} x^{7} + 8622 \, d^{5} e^{6} x^{6} - 4550 \, d^{6} e^{5} x^{5} - 2036 \, d^{7} e^{4} x^{4} + 760 \, d^{8} e^{3} x^{3} - 544 \, d^{9} e^{2} x^{2} + 288 \, d^{10} e x - 64 \, d^{11} - 135 \,{\left (e^{11} x^{11} - 5 \, d e^{10} x^{10} - 18 \, d^{2} e^{9} x^{9} + 20 \, d^{3} e^{8} x^{8} + 48 \, d^{4} e^{7} x^{7} - 16 \, d^{5} e^{6} x^{6} - 32 \, d^{6} e^{5} x^{5} +{\left (e^{10} x^{10} + 6 \, d e^{9} x^{9} - 12 \, d^{2} e^{8} x^{8} - 32 \, d^{3} e^{7} x^{7} + 16 \, d^{4} e^{6} x^{6} + 32 \, d^{5} e^{5} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (292 \, e^{10} x^{10} - 715 \, d e^{9} x^{9} - 3995 \, d^{2} e^{8} x^{8} + 3490 \, d^{3} e^{7} x^{7} + 7380 \, d^{4} e^{6} x^{6} - 4062 \, d^{5} e^{5} x^{5} - 2332 \, d^{6} e^{4} x^{4} + 904 \, d^{7} e^{3} x^{3} - 576 \, d^{8} e^{2} x^{2} + 288 \, d^{9} e x - 64 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{10 \,{\left (d^{4} e^{6} x^{11} - 5 \, d^{5} e^{5} x^{10} - 18 \, d^{6} e^{4} x^{9} + 20 \, d^{7} e^{3} x^{8} + 48 \, d^{8} e^{2} x^{7} - 16 \, d^{9} e x^{6} - 32 \, d^{10} x^{5} +{\left (d^{4} e^{5} x^{10} + 6 \, d^{5} e^{4} x^{9} - 12 \, d^{6} e^{3} x^{8} - 32 \, d^{7} e^{2} x^{7} + 16 \, d^{8} e x^{6} + 32 \, d^{9} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
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^6),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**6/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.34795, 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^6),x, algorithm="giac")
[Out]