3.1510 \(\int \frac{1}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{3 b (a+b x)^3} \]

[Out]

-1/(3*b*(a + b*x)^3)

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Rubi [A]  time = 0.0103761, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{1}{3 b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*b*(a + b*x)^3)

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Rubi in Sympy [A]  time = 1.90443, size = 31, normalized size = 2.21 \[ - \frac{2 a + 2 b x}{6 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a + 2*b*x)/(6*b*(a**2 + 2*a*b*x + b**2*x**2)**2)

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Mathematica [A]  time = 0.00543875, size = 14, normalized size = 1. \[ -\frac{1}{3 b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*b*(a + b*x)^3)

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Maple [A]  time = 0.004, size = 13, normalized size = 0.9 \[ -{\frac{1}{3\,b \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.684117, size = 47, normalized size = 3.36 \[ -\frac{1}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.192914, size = 47, normalized size = 3.36 \[ -\frac{1}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.46466, size = 37, normalized size = 2.64 \[ - \frac{1}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*a**3*b + 9*a**2*b**2*x + 9*a*b**3*x**2 + 3*b**4*x**3)

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GIAC/XCAS [A]  time = 0.20822, size = 16, normalized size = 1.14 \[ -\frac{1}{3 \,{\left (b x + a\right )}^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/((b*x + a)^3*b)