Optimal. Leaf size=117 \[ \frac{1}{2} d e (2 d-3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d^3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.293066, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ \frac{1}{2} d e (2 d-3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d^3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^2,x]
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Rubi in Sympy [A] time = 41.3694, size = 99, normalized size = 0.85 \[ - \frac{3 d^{3} e \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2} - d^{3} e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )} + \frac{d e \left (4 d - 6 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{4} - \frac{\left (3 d - e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(-e**2*x**2+d**2)**(3/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.161032, size = 114, normalized size = 0.97 \[ d^3 e \log (x)-d^3 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (-\frac{d^3}{x}+\frac{4 d^2 e}{3}-\frac{1}{2} d e^2 x-\frac{e^3 x^2}{3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^2,x]
[Out]
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Maple [A] time = 0.033, size = 182, normalized size = 1.6 \[ -{\frac{1}{dx} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{2}x}{d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{2}dx}{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{2}{d}^{3}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{e}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+e{d}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}-{e{d}^{4}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^2,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)^(3/2)*(e*x + d)/x^2,x, algorithm="maxima")
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Fricas [A] time = 0.288147, size = 532, normalized size = 4.55 \[ \frac{8 \, d e^{7} x^{7} + 12 \, d^{2} e^{6} x^{6} - 48 \, d^{3} e^{5} x^{5} - 12 \, d^{4} e^{4} x^{4} + 48 \, d^{5} e^{3} x^{3} - 48 \, d^{6} e^{2} x^{2} + 48 \, d^{8} + 18 \,{\left (d^{3} e^{5} x^{5} - 8 \, d^{5} e^{3} x^{3} + 8 \, d^{7} e x + 4 \,{\left (d^{4} e^{3} x^{3} - 2 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 6 \,{\left (d^{3} e^{5} x^{5} - 8 \, d^{5} e^{3} x^{3} + 8 \, d^{7} e x + 4 \,{\left (d^{4} e^{3} x^{3} - 2 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (2 \, e^{7} x^{7} + 3 \, d e^{6} x^{6} - 24 \, d^{2} e^{5} x^{5} - 18 \, d^{3} e^{4} x^{4} + 48 \, d^{4} e^{3} x^{3} - 24 \, d^{5} e^{2} x^{2} + 48 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (e^{4} x^{5} - 8 \, d^{2} e^{2} x^{3} + 8 \, d^{4} x + 4 \,{\left (d e^{2} x^{3} - 2 \, d^{3} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^2,x, algorithm="fricas")
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Sympy [A] time = 18.1014, size = 386, normalized size = 3.3 \[ d^{3} \left (\begin{cases} \frac{i d}{x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + i e \operatorname{acosh}{\left (\frac{e x}{d} \right )} - \frac{i e^{2} x}{d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{d}{x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - e \operatorname{asin}{\left (\frac{e x}{d} \right )} + \frac{e^{2} x}{d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + d^{2} e \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) - e^{3} \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(-e**2*x**2+d**2)**(3/2)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.319975, size = 212, normalized size = 1.81 \[ -\frac{3}{2} \, d^{3} \arcsin \left (\frac{x e}{d}\right ) e{\rm sign}\left (d\right ) - d^{3} e{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) + \frac{d^{3} x e^{3}}{2 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}} - \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{\left (-1\right )}}{2 \, x} + \frac{1}{6} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (8 \, d^{2} e -{\left (2 \, x e^{3} + 3 \, d e^{2}\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(3/2)*(e*x + d)/x^2,x, algorithm="giac")
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