Optimal. Leaf size=143 \[ -\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^4}+\frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2} \]
[Out]
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Rubi [A] time = 0.455178, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^4}+\frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d^2 - e^2*x^2]/(x^5*(d + e*x)),x]
[Out]
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Rubi in Sympy [A] time = 50.6556, size = 122, normalized size = 0.85 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{4 d x^{4}} + \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{2} x^{3}} - \frac{3 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{8 d^{3} x^{2}} - \frac{3 e^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8 d^{4}} + \frac{2 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**(1/2)/x**5/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.0947802, size = 95, normalized size = 0.66 \[ \frac{-9 e^4 x^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (-6 d^3+8 d^2 e x-9 d e^2 x^2+16 e^3 x^3\right )+9 e^4 x^4 \log (x)}{24 d^4 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d^2 - e^2*x^2]/(x^5*(d + e*x)),x]
[Out]
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Maple [B] time = 0.018, size = 304, normalized size = 2.1 \[ -{\frac{1}{4\,{d}^{3}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{e}^{2}}{8\,{d}^{5}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{4}}{8\,{d}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{4}}{8\,{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{{e}^{4}}{{d}^{5}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{{e}^{5}}{{d}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{{e}^{3}}{{d}^{6}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{5}x}{{d}^{6}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{e}^{5}}{{d}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{e}{3\,{d}^{4}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(1/2)/x^5/(e*x+d),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289201, size = 428, normalized size = 2.99 \[ -\frac{64 \, d e^{7} x^{7} - 36 \, d^{2} e^{6} x^{6} - 160 \, d^{3} e^{5} x^{5} + 84 \, d^{4} e^{4} x^{4} + 32 \, d^{5} e^{3} x^{3} + 64 \, d^{7} e x - 48 \, d^{8} - 9 \,{\left (e^{8} x^{8} - 8 \, d^{2} e^{6} x^{6} + 8 \, d^{4} e^{4} x^{4} + 4 \,{\left (d e^{6} x^{6} - 2 \, d^{3} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (16 \, e^{7} x^{7} - 9 \, d e^{6} x^{6} - 120 \, d^{2} e^{5} x^{5} + 66 \, d^{3} e^{4} x^{4} + 64 \, d^{4} e^{3} x^{3} - 24 \, d^{5} e^{2} x^{2} + 64 \, d^{6} e x - 48 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (d^{4} e^{4} x^{8} - 8 \, d^{6} e^{2} x^{6} + 8 \, d^{8} x^{4} + 4 \,{\left (d^{5} e^{2} x^{6} - 2 \, d^{7} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^5),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{x^{5} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**(1/2)/x**5/(e*x+d),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^5),x, algorithm="giac")
[Out]