Optimal. Leaf size=108 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{8 x^2} \]
[Out]
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Rubi [A] time = 0.26445, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}+\frac{e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{8 x^2} \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)),x]
[Out]
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Rubi in Sympy [A] time = 25.8976, size = 88, normalized size = 0.81 \[ - \frac{3 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{8 x^{2}} + \frac{e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{4 x^{4}} + \frac{3 e^{5} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8 d} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{5 d x^{5}} \]
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),x)
[Out]
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Mathematica [A] time = 0.108551, size = 106, normalized size = 0.98 \[ \frac{15 e^5 x^5 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (-8 d^4+10 d^3 e x+16 d^2 e^2 x^2-25 d e^3 x^3-8 e^4 x^4\right )-15 e^5 x^5 \log (x)}{40 d x^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)),x]
[Out]
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Maple [B] time = 0.021, size = 493, normalized size = 4.6 \[ -{\frac{1}{5\,{d}^{3}{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{2}}{5\,{d}^{5}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{4}}{5\,{d}^{7}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{6}x}{5\,{d}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{6}x}{4\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{6}x}{8\,{d}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{6}}{8\,d}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{{e}^{5}}{5\,{d}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{6}x}{4\,{d}^{5}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{6}x}{8\,{d}^{3}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{3\,{e}^{6}}{8\,d}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{3\,{e}^{5}}{40\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{5}}{8\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{5}}{8\,{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{3\,{e}^{5}}{8}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{{e}^{3}}{8\,{d}^{6}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{e}{4\,{d}^{4}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]
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),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^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.307, size = 531, normalized size = 4.92 \[ -\frac{8 \, e^{10} x^{10} + 25 \, d e^{9} x^{9} - 120 \, d^{2} e^{8} x^{8} - 335 \, d^{3} e^{7} x^{7} + 440 \, d^{4} e^{6} x^{6} + 830 \, d^{5} e^{5} x^{5} - 680 \, d^{6} e^{4} x^{4} - 680 \, d^{7} e^{3} x^{3} + 480 \, d^{8} e^{2} x^{2} + 160 \, d^{9} e x - 128 \, d^{10} + 15 \,{\left (5 \, d e^{9} x^{9} - 20 \, d^{3} e^{7} x^{7} + 16 \, d^{5} e^{5} x^{5} -{\left (e^{9} x^{9} - 12 \, d^{2} e^{7} x^{7} + 16 \, d^{4} e^{5} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (40 \, d e^{8} x^{8} + 125 \, d^{2} e^{7} x^{7} - 240 \, d^{3} e^{6} x^{6} - 550 \, d^{4} e^{5} x^{5} + 488 \, d^{5} e^{4} x^{4} + 600 \, d^{6} e^{3} x^{3} - 416 \, d^{7} e^{2} x^{2} - 160 \, d^{8} e x + 128 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{40 \,{\left (5 \, d^{2} e^{4} x^{9} - 20 \, d^{4} e^{2} x^{7} + 16 \, d^{6} x^{5} -{\left (d e^{4} x^{9} - 12 \, d^{3} e^{2} x^{7} + 16 \, d^{5} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 30.5357, size = 774, normalized size = 7.17 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**(5/2)/x**6/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 0.294825, 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^6),x, algorithm="giac")
[Out]