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

Optimal. Leaf size=14 \[ -\frac{1}{8 b (a+b x)^8} \]

[Out]

-1/(8*b*(a + b*x)^8)

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Rubi [A]  time = 0.0184617, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{1}{8 b (a+b x)^8} \]

Antiderivative was successfully verified.

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

[Out]

-1/(8*b*(a + b*x)^8)

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Rubi in Sympy [A]  time = 15.6879, size = 12, normalized size = 0.86 \[ - \frac{1}{8 b \left (a + b x\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(8*b*(a + b*x)**8)

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Mathematica [A]  time = 0.00506053, size = 14, normalized size = 1. \[ -\frac{1}{8 b (a+b x)^8} \]

Antiderivative was successfully verified.

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

[Out]

-1/(8*b*(a + b*x)^8)

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Maple [A]  time = 0.004, size = 13, normalized size = 0.9 \[ -{\frac{1}{8\,b \left ( bx+a \right ) ^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/b/(b*x+a)^8

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Maxima [A]  time = 0.787831, size = 122, normalized size = 8.71 \[ -\frac{1}{8 \,{\left (b^{9} x^{8} + 8 \, a b^{8} x^{7} + 28 \, a^{2} b^{7} x^{6} + 56 \, a^{3} b^{6} x^{5} + 70 \, a^{4} b^{5} x^{4} + 56 \, a^{5} b^{4} x^{3} + 28 \, a^{6} b^{3} x^{2} + 8 \, a^{7} b^{2} x + a^{8} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/(b^9*x^8 + 8*a*b^8*x^7 + 28*a^2*b^7*x^6 + 56*a^3*b^6*x^5 + 70*a^4*b^5*x^4 +
56*a^5*b^4*x^3 + 28*a^6*b^3*x^2 + 8*a^7*b^2*x + a^8*b)

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Fricas [A]  time = 0.268892, size = 122, normalized size = 8.71 \[ -\frac{1}{8 \,{\left (b^{9} x^{8} + 8 \, a b^{8} x^{7} + 28 \, a^{2} b^{7} x^{6} + 56 \, a^{3} b^{6} x^{5} + 70 \, a^{4} b^{5} x^{4} + 56 \, a^{5} b^{4} x^{3} + 28 \, a^{6} b^{3} x^{2} + 8 \, a^{7} b^{2} x + a^{8} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/(b^9*x^8 + 8*a*b^8*x^7 + 28*a^2*b^7*x^6 + 56*a^3*b^6*x^5 + 70*a^4*b^5*x^4 +
56*a^5*b^4*x^3 + 28*a^6*b^3*x^2 + 8*a^7*b^2*x + a^8*b)

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Sympy [A]  time = 2.70059, size = 97, normalized size = 6.93 \[ - \frac{1}{8 a^{8} b + 64 a^{7} b^{2} x + 224 a^{6} b^{3} x^{2} + 448 a^{5} b^{4} x^{3} + 560 a^{4} b^{5} x^{4} + 448 a^{3} b^{6} x^{5} + 224 a^{2} b^{7} x^{6} + 64 a b^{8} x^{7} + 8 b^{9} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(8*a**8*b + 64*a**7*b**2*x + 224*a**6*b**3*x**2 + 448*a**5*b**4*x**3 + 560*a*
*4*b**5*x**4 + 448*a**3*b**6*x**5 + 224*a**2*b**7*x**6 + 64*a*b**8*x**7 + 8*b**9
*x**8)

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GIAC/XCAS [A]  time = 0.259044, size = 16, normalized size = 1.14 \[ -\frac{1}{8 \,{\left (b x + a\right )}^{8} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/((b*x + a)^8*b)