Optimal. Leaf size=96 \[ d (d-e x) \sqrt{d^2-e^2 x^2}-\frac{1}{3} \left (d^2-e^2 x^2\right )^{3/2}+d^3 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )-d^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.390941, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ d (d-e x) \sqrt{d^2-e^2 x^2}-\frac{1}{3} \left (d^2-e^2 x^2\right )^{3/2}+d^3 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )-d^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^(5/2)/(x*(d + e*x)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 34.9221, size = 92, normalized size = 0.96 \[ - d^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} - d^{3} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )} + d^{2} \sqrt{d^{2} - e^{2} x^{2}} - d e x \sqrt{d^{2} - e^{2} x^{2}} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**(5/2)/x/(e*x+d)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0855334, size = 96, normalized size = 1. \[ d^3 \log (x)+\sqrt{d^2-e^2 x^2} \left (\frac{2 d^2}{3}-d e x+\frac{e^2 x^2}{3}\right )-d^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+d^3 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d^2 - e^2*x^2)^(5/2)/(x*(d + e*x)^2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.017, size = 290, normalized size = 3. \[{\frac{1}{5\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{1}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{d}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}-{{d}^{4}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{8}{15\,{d}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{2\,ex}{3\,d} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}-de\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }x-{{d}^{3}e\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{3\,{d}^{2}{e}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(5/2)/x/(e*x+d)^2,x)
[Out]
_______________________________________________________________________________________
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)^2*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.300798, size = 392, normalized size = 4.08 \[ \frac{e^{6} x^{6} - 3 \, d e^{5} x^{5} - 3 \, d^{2} e^{4} x^{4} + 15 \, d^{3} e^{3} x^{3} - 12 \, d^{5} e x + 6 \,{\left (3 \, d^{4} e^{2} x^{2} - 4 \, d^{6} -{\left (d^{3} e^{2} x^{2} - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \,{\left (3 \, d^{4} e^{2} x^{2} - 4 \, d^{6} -{\left (d^{3} e^{2} x^{2} - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + 3 \,{\left (d e^{4} x^{4} - 3 \, d^{2} e^{3} x^{3} + 4 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (3 \, d e^{2} x^{2} - 4 \, d^{3} -{\left (e^{2} x^{2} - 4 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^2*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 19.0629, size = 267, normalized size = 2.78 \[ d^{2} \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) - 2 d e \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**(5/2)/x/(e*x+d)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 4.91639, size = 1, normalized size = 0.01 \[ +\infty \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^2*x),x, algorithm="giac")
[Out]