Optimal. Leaf size=46 \[ \frac{\sqrt{a^2-b^2 x^2}}{b}+\frac{a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]
[Out]
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Rubi [A] time = 0.0520132, 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 \[ \frac{\sqrt{a^2-b^2 x^2}}{b}+\frac{a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 - b^2*x^2]/(a + b*x),x]
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Rubi in Sympy [A] time = 12.3894, size = 36, normalized size = 0.78 \[ \frac{a \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{b} + \frac{\sqrt{a^{2} - b^{2} x^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0370224, size = 43, normalized size = 0.93 \[ \frac{\sqrt{a^2-b^2 x^2}+a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x),x]
[Out]
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Maple [A] time = 0.016, size = 77, normalized size = 1.7 \[{\frac{1}{b}\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) }}+{a\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) }}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b^2*x^2+a^2)^(1/2)/(b*x+a),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(-b^2*x^2 + a^2)/(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219654, size = 111, normalized size = 2.41 \[ \frac{b^{2} x^{2} - 2 \,{\left (a^{2} - \sqrt{-b^{2} x^{2} + a^{2}} a\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right )}{a b - \sqrt{-b^{2} x^{2} + a^{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.228, size = 49, normalized size = 1.07 \[ \frac{a \arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{{\left | b \right |}} + \frac{\sqrt{-b^{2} x^{2} + a^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a),x, algorithm="giac")
[Out]