Optimal. Leaf size=47 \[ \frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}-\frac{\sqrt{d^2-e^2 x^2}}{e} \]
[Out]
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Rubi [A] time = 0.0452245, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}-\frac{\sqrt{d^2-e^2 x^2}}{e} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/Sqrt[d^2 - e^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 12.1172, size = 36, normalized size = 0.77 \[ \frac{d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(-e**2*x**2+d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.03071, size = 47, normalized size = 1. \[ \frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}-\frac{\sqrt{d^2-e^2 x^2}}{e} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/Sqrt[d^2 - e^2*x^2],x]
[Out]
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Maple [A] time = 0.007, size = 50, normalized size = 1.1 \[{d\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(-e^2*x^2+d^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.783295, size = 57, normalized size = 1.21 \[ \frac{d \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/sqrt(-e^2*x^2 + d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221844, size = 112, normalized size = 2.38 \[ -\frac{e^{2} x^{2} + 2 \,{\left (d^{2} - \sqrt{-e^{2} x^{2} + d^{2}} d\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right )}{d e - \sqrt{-e^{2} x^{2} + d^{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/sqrt(-e^2*x^2 + d^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.32923, size = 42, normalized size = 0.89 \[ \begin{cases} \frac{d \left (\begin{cases} \operatorname{asin}{\left (e x \sqrt{\frac{1}{d^{2}}} \right )} & \text{for}\: d^{2} > 0 \end{cases}\right ) - \sqrt{d^{2} - e^{2} x^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{d x}{\sqrt{d^{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(-e**2*x**2+d**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.225764, size = 43, 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((e*x + d)/sqrt(-e^2*x^2 + d^2),x, algorithm="giac")
[Out]