Optimal. Leaf size=46 \[ \frac{\sqrt{d^2-e^2 x^2}}{e}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
[Out]
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Rubi [A] time = 0.0446184, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{d^2-e^2 x^2}}{e}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d^2 - e^2*x^2]/(d + e*x),x]
[Out]
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Rubi in Sympy [A] time = 12.3079, size = 36, normalized size = 0.78 \[ \frac{d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e} + \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**(1/2)/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.0272258, size = 43, normalized size = 0.93 \[ \frac{\sqrt{d^2-e^2 x^2}+d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d^2 - e^2*x^2]/(d + e*x),x]
[Out]
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Maple [A] time = 0.005, size = 77, normalized size = 1.7 \[{\frac{1}{e}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{d\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(1/2)/(e*x+d),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(sqrt(-e^2*x^2 + d^2)/(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.28337, size = 111, normalized size = 2.41 \[ \frac{e^{2} x^{2} - 2 \,{\left (d^{2} - \sqrt{-e^{2} x^{2} + d^{2}} d\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right )}{d e - \sqrt{-e^{2} x^{2} + d^{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^2*x^2 + d^2)/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e**2*x**2+d**2)**(1/2)/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 0.2846, size = 42, normalized size = 0.91 \[ d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) + \sqrt{-x^{2} e^{2} + d^{2}} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-e^2*x^2 + d^2)/(e*x + d),x, algorithm="giac")
[Out]