Optimal. Leaf size=113 \[ \frac{1}{8} d^2 (8 d-3 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{12} (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{3}{8} d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.357984, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ \frac{1}{8} d^2 (8 d-3 e x) \sqrt{d^2-e^2 x^2}+\frac{1}{12} (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{3}{8} d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^(5/2)/(x*(d + e*x)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 49.9324, size = 95, normalized size = 0.84 \[ - \frac{3 d^{4} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8} - d^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )} + \frac{d^{2} \left (8 d - 3 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{8} + \frac{\left (4 d - 3 e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{12} \]
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),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.101536, size = 108, normalized size = 0.96 \[ d^4 \log (x)-d^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )-\frac{3}{8} d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{1}{24} \sqrt{d^2-e^2 x^2} \left (32 d^3-15 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d^2 - e^2*x^2)^(5/2)/(x*(d + e*x)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.015, size = 245, normalized size = 2.2 \[{\frac{1}{5\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{d}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{d}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}-{{d}^{5}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{1}{5\,d} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{ex}{4} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{2}ex}{8}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{3\,{d}^{4}e}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(5/2)/x/(e*x+d),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)*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.291915, size = 524, normalized size = 4.64 \[ -\frac{24 \, d e^{7} x^{7} - 32 \, d^{2} e^{6} x^{6} - 132 \, d^{3} e^{5} x^{5} + 192 \, d^{4} e^{4} x^{4} + 228 \, d^{5} e^{3} x^{3} - 192 \, d^{6} e^{2} x^{2} - 120 \, d^{7} e x - 18 \,{\left (d^{4} e^{4} x^{4} - 8 \, d^{6} e^{2} x^{2} + 8 \, d^{8} + 4 \,{\left (d^{5} e^{2} x^{2} - 2 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 24 \,{\left (d^{4} e^{4} x^{4} - 8 \, d^{6} e^{2} x^{2} + 8 \, d^{8} + 4 \,{\left (d^{5} e^{2} x^{2} - 2 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (6 \, e^{7} x^{7} - 8 \, d e^{6} x^{6} - 63 \, d^{2} e^{5} x^{5} + 96 \, d^{3} e^{4} x^{4} + 168 \, d^{4} e^{3} x^{3} - 192 \, d^{5} e^{2} x^{2} - 120 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (e^{4} x^{4} - 8 \, d^{2} e^{2} x^{2} + 8 \, d^{4} + 4 \,{\left (d e^{2} x^{2} - 2 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 26.7394, size = 469, normalized size = 4.15 \[ d^{3} \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) - d^{2} e \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} - \frac{i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{3}} + \frac{i d^{3} x}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{3 i d x^{3}}{8 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{5}}{4 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{3}} - \frac{d^{3} x}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{3 d x^{3}}{8 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**(5/2)/x/(e*x+d),x)
[Out]
_______________________________________________________________________________________
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((-e^2*x^2 + d^2)^(5/2)/((e*x + d)*x),x, algorithm="giac")
[Out]