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)

________________________________________________________________________________________

Rubi [A]  time = 0.0176062, 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, Rules used = {2058, 32} \[ -\frac{1}{4 b (a+b x)^4} \]

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)

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}{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 &=\int \frac{1}{(a+b x)^5} \, dx\\ &=-\frac{1}{4 b (a+b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0037967, 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)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,b \left ( bx+a \right ) ^{4}}} \end{align*}

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

________________________________________________________________________________________

Maxima [B]  time = 1.20622, size = 62, normalized size = 4.43 \begin{align*} -\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 )}} \end{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 1.69806, size = 92, normalized size = 6.57 \begin{align*} -\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 )}} \end{align*}

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)

________________________________________________________________________________________

Sympy [B]  time = 0.443923, size = 49, normalized size = 3.5 \begin{align*} - \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}} \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.09632, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{4 \,{\left (b x + a\right )}^{4} b} \end{align*}

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)