3.66 \(\int \frac{1}{a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{4 b (a+b x)^4} \]

[Out]

-1/(4*b*(a + b*x)^4)

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Rubi [A]  time = 0.035304, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.039 \[ -\frac{1}{4 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a^5 + 5*a^4*b*x + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a*b^4*x^4 + b^5*x^5)^(-1),x]

[Out]

-1/(4*b*(a + b*x)^4)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b**5*x**5+5*a*b**4*x**4+10*a**2*b**3*x**3+10*a**3*b**2*x**2+5*a**4*b*x+a**5),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0060112, size = 14, normalized size = 1. \[ -\frac{1}{4 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^5 + 5*a^4*b*x + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a*b^4*x^4 + b^5*x^5)^(-1),x]

[Out]

-1/(4*b*(a + b*x)^4)

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Maple [A]  time = 0.006, size = 13, normalized size = 0.9 \[ -{\frac{1}{4\,b \left ( bx+a \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5),x)

[Out]

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

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Maxima [A]  time = 0.832262, size = 62, normalized size = 4.43 \[ -\frac{1}{4 \,{\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.270872, size = 62, normalized size = 4.43 \[ -\frac{1}{4 \,{\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.68389, size = 49, normalized size = 3.5 \[ - \frac{1}{4 a^{4} b + 16 a^{3} b^{2} x + 24 a^{2} b^{3} x^{2} + 16 a b^{4} x^{3} + 4 b^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b**5*x**5+5*a*b**4*x**4+10*a**2*b**3*x**3+10*a**3*b**2*x**2+5*a**4*b*x+a**5),x)

[Out]

-1/(4*a**4*b + 16*a**3*b**2*x + 24*a**2*b**3*x**2 + 16*a*b**4*x**3 + 4*b**5*x**4
)

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GIAC/XCAS [A]  time = 0.259136, size = 16, normalized size = 1.14 \[ -\frac{1}{4 \,{\left (b x + a\right )}^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5),x, algorithm="giac")

[Out]

-1/4/((b*x + a)^4*b)