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

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 = 0.0081347, 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, Rules used = {2058, 32} \[ -\frac{1}{8 b (a+b x)^8} \]

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)

Rule 2058

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{\left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^3} \, dx &=\int \frac{1}{(a+b x)^9} \, dx\\ &=-\frac{1}{8 b (a+b x)^8}\\ \end{align*}

Mathematica [A]  time = 0.0029336, 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 = 0.003, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,b \left ( bx+a \right ) ^{8}}} \end{align*}

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 [B]  time = 1.13839, size = 122, normalized size = 8.71 \begin{align*} -\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 )}} \end{align*}

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, 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 [B]  time = 1.21306, size = 185, normalized size = 13.21 \begin{align*} -\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 )}} \end{align*}

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, 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 [B]  time = 0.73029, size = 97, normalized size = 6.93 \begin{align*} - \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}} \end{align*}

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 [A]  time = 1.07468, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{8 \,{\left (b x + a\right )}^{8} b} \end{align*}

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, algorithm="giac")

[Out]

-1/8/((b*x + a)^8*b)