3.1522 \(\int \frac{1}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{5 b (a+b x)^5} \]

[Out]

-1/(5*b*(a + b*x)^5)

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Rubi [A]  time = 0.0102097, 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 \[ -\frac{1}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-1/(5*b*(a + b*x)^5)

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Rubi in Sympy [A]  time = 2.04068, size = 31, normalized size = 2.21 \[ - \frac{2 a + 2 b x}{10 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{3}} \]

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,x)

[Out]

-(2*a + 2*b*x)/(10*b*(a**2 + 2*a*b*x + b**2*x**2)**3)

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Mathematica [A]  time = 0.00653021, size = 14, normalized size = 1. \[ -\frac{1}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

-1/(5*b*(a + b*x)^5)

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Maple [A]  time = 0.005, size = 13, normalized size = 0.9 \[ -{\frac{1}{5\,b \left ( bx+a \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5/b/(b*x+a)^5

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Maxima [A]  time = 0.687665, size = 77, normalized size = 5.5 \[ -\frac{1}{5 \,{\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5/(b^6*x^5 + 5*a*b^5*x^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3*x^2 + 5*a^4*b^2*x + a^
5*b)

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Fricas [A]  time = 0.19361, size = 77, normalized size = 5.5 \[ -\frac{1}{5 \,{\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5/(b^6*x^5 + 5*a*b^5*x^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3*x^2 + 5*a^4*b^2*x + a^
5*b)

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Sympy [A]  time = 1.87065, size = 61, normalized size = 4.36 \[ - \frac{1}{5 a^{5} b + 25 a^{4} b^{2} x + 50 a^{3} b^{3} x^{2} + 50 a^{2} b^{4} x^{3} + 25 a b^{5} x^{4} + 5 b^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(5*a**5*b + 25*a**4*b**2*x + 50*a**3*b**3*x**2 + 50*a**2*b**4*x**3 + 25*a*b**
5*x**4 + 5*b**6*x**5)

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GIAC/XCAS [A]  time = 0.209009, size = 16, normalized size = 1.14 \[ -\frac{1}{5 \,{\left (b x + a\right )}^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5/((b*x + a)^5*b)