3.772 \(\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3} \]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(3*a*b*(a + b*x)^3)

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Rubi [A]  time = 0.0361296, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{\left (a^2-b^2 x^2\right )^{3/2}}{3 a b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 - b^2*x^2]/(a + b*x)^3,x]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(3*a*b*(a + b*x)^3)

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Rubi in Sympy [A]  time = 4.91411, size = 26, normalized size = 0.79 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3 a b \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**3,x)

[Out]

-(a**2 - b**2*x**2)**(3/2)/(3*a*b*(a + b*x)**3)

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Mathematica [A]  time = 0.0282372, size = 39, normalized size = 1.18 \[ -\frac{(a-b x) \sqrt{a^2-b^2 x^2}}{3 a b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^3,x]

[Out]

-((a - b*x)*Sqrt[a^2 - b^2*x^2])/(3*a*b*(a + b*x)^2)

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Maple [A]  time = 0.006, size = 36, normalized size = 1.1 \[ -{\frac{-bx+a}{3\, \left ( bx+a \right ) ^{2}ba}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21491, size = 126, normalized size = 3.82 \[ -\frac{2 \,{\left (b^{2} x^{3} - 3 \, a^{2} x + 3 \, \sqrt{-b^{2} x^{2} + a^{2}} a x\right )}}{3 \,{\left (a b^{3} x^{3} - 3 \, a^{3} b x - 2 \, a^{4} +{\left (a b^{2} x^{2} + 3 \, a^{2} b x + 2 \, a^{3}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^3,x, algorithm="fricas")

[Out]

-2/3*(b^2*x^3 - 3*a^2*x + 3*sqrt(-b^2*x^2 + a^2)*a*x)/(a*b^3*x^3 - 3*a^3*b*x - 2
*a^4 + (a*b^2*x^2 + 3*a^2*b*x + 2*a^3)*sqrt(-b^2*x^2 + a^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**3,x)

[Out]

Integral(sqrt(-(-a + b*x)*(a + b*x))/(a + b*x)**3, x)

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GIAC/XCAS [A]  time = 0.232948, size = 100, normalized size = 3.03 \[ \frac{2 \,{\left (\frac{3 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + 1\right )}}{3 \, a{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^3,x, algorithm="giac")

[Out]

2/3*(3*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^2/(b^4*x^2) + 1)/(a*((a*b + sqrt(-b^2
*x^2 + a^2)*abs(b))/(b^2*x) + 1)^3*abs(b))