3.805 \(\int \frac{\sqrt{a^2-b^2 x^2}}{a-b x} \, dx\)

Optimal. Leaf size=47 \[ \frac{a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b}-\frac{\sqrt{a^2-b^2 x^2}}{b} \]

[Out]

-(Sqrt[a^2 - b^2*x^2]/b) + (a*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/b

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Rubi [A]  time = 0.0562476, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b}-\frac{\sqrt{a^2-b^2 x^2}}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 - b^2*x^2]/(a - b*x),x]

[Out]

-(Sqrt[a^2 - b^2*x^2]/b) + (a*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/b

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Rubi in Sympy [A]  time = 13.939, size = 36, normalized size = 0.77 \[ \frac{a \operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{b} - \frac{\sqrt{a^{2} - b^{2} x^{2}}}{b} \]

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]

a*atan(b*x/sqrt(a**2 - b**2*x**2))/b - sqrt(a**2 - b**2*x**2)/b

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Mathematica [A]  time = 0.0395115, size = 47, normalized size = 1. \[ \frac{a \tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b}-\frac{\sqrt{a^2-b^2 x^2}}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 - b^2*x^2]/(a - b*x),x]

[Out]

-(Sqrt[a^2 - b^2*x^2]/b) + (a*ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/b

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Maple [A]  time = 0.013, size = 82, 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]

-1/b*(-(x-a/b)^2*b^2-2*a*b*(x-a/b))^(1/2)+a/(b^2)^(1/2)*arctan((b^2)^(1/2)*x/(-(
x-a/b)^2*b^2-2*a*b*(x-a/b))^(1/2))

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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]

Exception raised: ValueError

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Fricas [A]  time = 0.223301, size = 112, normalized size = 2.38 \[ -\frac{b^{2} x^{2} + 2 \,{\left (a^{2} - \sqrt{-b^{2} x^{2} + a^{2}} a\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right )}{a b - \sqrt{-b^{2} x^{2} + a^{2}} b} \]

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]

-(b^2*x^2 + 2*(a^2 - sqrt(-b^2*x^2 + a^2)*a)*arctan(-(a - sqrt(-b^2*x^2 + a^2))/
(b*x)))/(a*b - sqrt(-b^2*x^2 + a^2)*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{a^{2} - b^{2} x^{2}}}{- a + b x}\, dx \]

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]

-Integral(sqrt(a**2 - b**2*x**2)/(-a + b*x), x)

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GIAC/XCAS [A]  time = 0.255328, size = 50, normalized size = 1.06 \[ \frac{a \arcsin \left (\frac{b x}{a}\right ){\rm sign}\left (a\right ){\rm sign}\left (b\right )}{{\left | b \right |}} - \frac{\sqrt{-b^{2} x^{2} + a^{2}}}{b} \]

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]

a*arcsin(b*x/a)*sign(a)*sign(b)/abs(b) - sqrt(-b^2*x^2 + a^2)/b