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

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]

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 = 0.0520132, 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]

[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 = 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]

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 = 0.0370224, 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]

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

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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]

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 = 0.219654, 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]

(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. \[ \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{a + b x}\, dx \]

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]

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

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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]

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