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/(b*x+a)^5

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ -\frac {1}{5 b (a+b x)^5} \]

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*1/(b*(a + b*x)^5)

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fricas [B]  time = 0.73, size = 57, normalized size = 4.07 \[ -\frac {1}{5 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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giac [A]  time = 0.29, size = 12, normalized size = 0.86 \[ -\frac {1}{5 \, {\left (b x + a\right )}^{5} b} \]

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)

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \[ -\frac {1}{5 \left (b x +a \right )^{5} b} \]

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 = 0.71, size = 57, normalized size = 4.07 \[ -\frac {1}{5 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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mupad [B]  time = 2.05, size = 59, normalized size = 4.21 \[ -\frac {1}{5\,a^5\,b+25\,a^4\,b^2\,x+50\,a^3\,b^3\,x^2+50\,a^2\,b^4\,x^3+25\,a\,b^5\,x^4+5\,b^6\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.35, 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**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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