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

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 = 0.0086401, 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} \[ -\frac{1}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(5*b*(a + b*x)^5)

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 )^2} \, dx &=\int \frac{1}{(a+b x)^6} \, dx\\ &=-\frac{1}{5 b (a+b x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0034624, size = 14, normalized size = 1. \[ -\frac{1}{5 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(5*b*(a + b*x)^5)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,b \left ( bx+a \right ) ^{5}}} \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)^2,x)

[Out]

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

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Maxima [B]  time = 1.19541, size = 77, normalized size = 5.5 \begin{align*} -\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 )}} \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)^2,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 [B]  time = 1.18818, size = 116, normalized size = 8.29 \begin{align*} -\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 )}} \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)^2,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 [B]  time = 0.494435, size = 61, normalized size = 4.36 \begin{align*} - \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}} \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)**2,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 [A]  time = 1.08833, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{5 \,{\left (b x + a\right )}^{5} 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)^2,x, algorithm="giac")

[Out]

-1/5/((b*x + a)^5*b)