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

Optimal. Leaf size=54 \[ -\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}-\frac{\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0486688, antiderivative size = 54, 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 \[ -\frac{2 \sqrt{a^2-b^2 x^2}}{b (a+b x)}-\frac{\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.9767, size = 42, normalized size = 0.78 \[ - \frac{\operatorname{atan}{\left (\frac{b x}{\sqrt{a^{2} - b^{2} x^{2}}} \right )}}{b} - \frac{2 \sqrt{a^{2} - b^{2} x^{2}}}{b \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0721766, size = 51, normalized size = 0.94 \[ -\frac{\frac{2 \sqrt{a^2-b^2 x^2}}{a+b x}+\tan ^{-1}\left (\frac{b x}{\sqrt{a^2-b^2 x^2}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 126, normalized size = 2.3 \[ -{\frac{1}{a{b}^{3}} \left ( - \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{a}{b}} \right ) ^{-2}}-{\frac{1}{ab}\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) }}-{1\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+2\,ab \left ( x+{\frac{a}{b}} \right ) }}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^2,x)

[Out]

-1/b^3/a/(x+a/b)^2*(-(x+a/b)^2*b^2+2*a*b*(x+a/b))^(3/2)-1/b/a*(-(x+a/b)^2*b^2+2*
a*b*(x+a/b))^(1/2)-1/(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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222556, size = 115, normalized size = 2.13 \[ -\frac{2 \,{\left (2 \, b x -{\left (b x + a - \sqrt{-b^{2} x^{2} + a^{2}}\right )} \arctan \left (-\frac{a - \sqrt{-b^{2} x^{2} + a^{2}}}{b x}\right )\right )}}{b^{2} x + a b - \sqrt{-b^{2} x^{2} + a^{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.227524, size = 97, normalized size = 1.8 \[ 2 \,{\left (\frac{{\left (i - \arctan \left (i\right )\right )}{\rm sign}\left (\frac{1}{b x + a}\right ){\rm sign}\left (b\right )}{b^{2}} - \frac{{\left (\sqrt{\frac{2 \, a}{b x + a} - 1} - \arctan \left (\sqrt{\frac{2 \, a}{b x + a} - 1}\right )\right )}{\rm sign}\left (\frac{1}{b x + a}\right ){\rm sign}\left (b\right )}{b^{2}}\right )}{\left | b \right |} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^2,x, algorithm="giac")

[Out]

2*((i - arctan(i))*sign(1/(b*x + a))*sign(b)/b^2 - (sqrt(2*a/(b*x + a) - 1) - ar
ctan(sqrt(2*a/(b*x + a) - 1)))*sign(1/(b*x + a))*sign(b)/b^2)*abs(b)