Optimal. Leaf size=114 \[ \frac{e \sqrt{d^2-e^2 x^2}}{2 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{2 e^2 \sqrt{d^2-e^2 x^2}}{3 d^3 x}+\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.364858, antiderivative size = 114, 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 \[ \frac{e \sqrt{d^2-e^2 x^2}}{2 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{2 e^2 \sqrt{d^2-e^2 x^2}}{3 d^3 x}+\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d^2 - e^2*x^2]/(x^4*(d + e*x)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 38.3383, size = 94, normalized size = 0.82 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{3 d x^{3}} + \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{2 d^{2} x^{2}} + \frac{e^{3} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{3}} - \frac{2 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**(1/2)/x**4/(e*x+d),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0764567, size = 84, normalized size = 0.74 \[ \frac{\left (-2 d^2+3 d e x-4 e^2 x^2\right ) \sqrt{d^2-e^2 x^2}+3 e^3 x^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )-3 e^3 x^3 \log (x)}{6 d^3 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d^2 - e^2*x^2]/(x^4*(d + e*x)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.017, size = 280, normalized size = 2.5 \[ -{\frac{1}{3\,{d}^{3}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{3}}{{d}^{4}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{{e}^{4}}{{d}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{e}^{2}}{{d}^{5}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{4}x}{{d}^{5}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{e}^{4}}{{d}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{e}^{3}}{2\,{d}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{e}^{3}}{2\,{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{e}{2\,{d}^{4}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(1/2)/x^4/(e*x+d),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.284932, size = 355, normalized size = 3.11 \[ -\frac{4 \, e^{6} x^{6} - 3 \, d e^{5} x^{5} - 18 \, d^{2} e^{4} x^{4} + 15 \, d^{3} e^{3} x^{3} + 6 \, d^{4} e^{2} x^{2} - 12 \, d^{5} e x + 8 \, d^{6} + 3 \,{\left (3 \, d e^{5} x^{5} - 4 \, d^{3} e^{3} x^{3} -{\left (e^{5} x^{5} - 4 \, d^{2} e^{3} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (12 \, d e^{4} x^{4} - 9 \, d^{2} e^{3} x^{3} - 10 \, d^{3} e^{2} x^{2} + 12 \, d^{4} e x - 8 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (3 \, d^{4} e^{2} x^{5} - 4 \, d^{6} x^{3} -{\left (d^{3} e^{2} x^{5} - 4 \, d^{5} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{x^{4} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**(1/2)/x**4/(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(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^4),x, algorithm="giac")
[Out]