Optimal. Leaf size=110 \[ -\frac{4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d-8 e x}{5 d^3 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^3} \]
[Out]
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Rubi [A] time = 0.424799, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d-8 e x}{5 d^3 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d^2 - e^2*x^2]/(x*(d + e*x)^4),x]
[Out]
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Rubi in Sympy [A] time = 48.2365, size = 95, normalized size = 0.86 \[ \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 d^{2} \left (d + e x\right )^{4}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{3}} + \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{d^{3} \left (d + e x\right )} + \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 d^{3} \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**(1/2)/x/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.123273, size = 76, normalized size = 0.69 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (13 d^2+19 d e x+8 e^2 x^2\right )}{(d+e x)^3}-5 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+5 \log (x)}{5 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d^2 - e^2*x^2]/(x*(d + e*x)^4),x]
[Out]
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Maple [B] time = 0.017, size = 196, normalized size = 1.8 \[{\frac{1}{{d}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{1}{{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{1}{{d}^{4}{e}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}}+{\frac{2}{5\,{e}^{3}{d}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-3}}+{\frac{1}{5\,{d}^{2}{e}^{4}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(1/2)/x/(e*x+d)^4,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )}^{4} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)^4*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287757, size = 448, normalized size = 4.07 \[ \frac{21 \, e^{5} x^{5} + 60 \, d e^{4} x^{4} + 5 \, d^{2} e^{3} x^{3} - 110 \, d^{3} e^{2} x^{2} - 80 \, d^{4} e x + 5 \,{\left (e^{5} x^{5} + 5 \, d e^{4} x^{4} + 5 \, d^{2} e^{3} x^{3} - 5 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x - 4 \, d^{5} -{\left (e^{4} x^{4} - 7 \, d^{2} e^{2} x^{2} - 10 \, d^{3} e x - 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) - 5 \,{\left (e^{4} x^{4} - 7 \, d e^{3} x^{3} - 22 \, d^{2} e^{2} x^{2} - 16 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d^{3} e^{5} x^{5} + 5 \, d^{4} e^{4} x^{4} + 5 \, d^{5} e^{3} x^{3} - 5 \, d^{6} e^{2} x^{2} - 10 \, d^{7} e x - 4 \, d^{8} -{\left (d^{3} e^{4} x^{4} - 7 \, d^{5} e^{2} x^{2} - 10 \, d^{6} e x - 4 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)^4*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{x \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**(1/2)/x/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.300639, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)^4*x),x, algorithm="giac")
[Out]