3.194 \(\int \frac{1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=34 \[ -\frac{1}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

-1/(2*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.0181075, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{1}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 1.98816, size = 32, normalized size = 0.94 \[ - \frac{2 a + 2 b x}{4 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.016426, size = 23, normalized size = 0.68 \[ -\frac{a+b x}{2 b \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(a + b*x)/(2*b*((a + b*x)^2)^(3/2))

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Maple [A]  time = 0.004, size = 20, normalized size = 0.6 \[ -{\frac{bx+a}{2\,b} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.724247, size = 22, normalized size = 0.65 \[ -\frac{1}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/((b^2)^(3/2)*(x + a/b)^2)

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Fricas [A]  time = 0.217467, size = 32, normalized size = 0.94 \[ -\frac{1}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(b^3*x^2 + 2*a*b^2*x + a^2*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.548649, size = 4, normalized size = 0.12 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x