3.68 \(\int \frac{1}{(a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5)^3} \, dx\)

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

[Out]

-1/(14*b*(a + b*x)^14)

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Rubi [A]  time = 0.0181846, 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} \[ -\frac{1}{14 b (a+b x)^{14}} \]

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)^(-3),x]

[Out]

-1/(14*b*(a + b*x)^14)

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^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\right )^3} \, dx &=\int \frac{1}{(a+b x)^{15}} \, dx\\ &=-\frac{1}{14 b (a+b x)^{14}}\\ \end{align*}

Mathematica [A]  time = 0.0039681, size = 14, normalized size = 1. \[ -\frac{1}{14 b (a+b x)^{14}} \]

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)^(-3),x]

[Out]

-1/(14*b*(a + b*x)^14)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{14\,b \left ( bx+a \right ) ^{14}}} \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)^3,x)

[Out]

-1/14/b/(b*x+a)^14

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Maxima [B]  time = 1.66979, size = 211, normalized size = 15.07 \begin{align*} -\frac{1}{14 \,{\left (b^{15} x^{14} + 14 \, a b^{14} x^{13} + 91 \, a^{2} b^{13} x^{12} + 364 \, a^{3} b^{12} x^{11} + 1001 \, a^{4} b^{11} x^{10} + 2002 \, a^{5} b^{10} x^{9} + 3003 \, a^{6} b^{9} x^{8} + 3432 \, a^{7} b^{8} x^{7} + 3003 \, a^{8} b^{7} x^{6} + 2002 \, a^{9} b^{6} x^{5} + 1001 \, a^{10} b^{5} x^{4} + 364 \, a^{11} b^{4} x^{3} + 91 \, a^{12} b^{3} x^{2} + 14 \, a^{13} b^{2} x + a^{14} 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)^3,x, algorithm="maxima")

[Out]

-1/14/(b^15*x^14 + 14*a*b^14*x^13 + 91*a^2*b^13*x^12 + 364*a^3*b^12*x^11 + 1001*a^4*b^11*x^10 + 2002*a^5*b^10*
x^9 + 3003*a^6*b^9*x^8 + 3432*a^7*b^8*x^7 + 3003*a^8*b^7*x^6 + 2002*a^9*b^6*x^5 + 1001*a^10*b^5*x^4 + 364*a^11
*b^4*x^3 + 91*a^12*b^3*x^2 + 14*a^13*b^2*x + a^14*b)

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Fricas [B]  time = 1.76439, size = 370, normalized size = 26.43 \begin{align*} -\frac{1}{14 \,{\left (b^{15} x^{14} + 14 \, a b^{14} x^{13} + 91 \, a^{2} b^{13} x^{12} + 364 \, a^{3} b^{12} x^{11} + 1001 \, a^{4} b^{11} x^{10} + 2002 \, a^{5} b^{10} x^{9} + 3003 \, a^{6} b^{9} x^{8} + 3432 \, a^{7} b^{8} x^{7} + 3003 \, a^{8} b^{7} x^{6} + 2002 \, a^{9} b^{6} x^{5} + 1001 \, a^{10} b^{5} x^{4} + 364 \, a^{11} b^{4} x^{3} + 91 \, a^{12} b^{3} x^{2} + 14 \, a^{13} b^{2} x + a^{14} 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)^3,x, algorithm="fricas")

[Out]

-1/14/(b^15*x^14 + 14*a*b^14*x^13 + 91*a^2*b^13*x^12 + 364*a^3*b^12*x^11 + 1001*a^4*b^11*x^10 + 2002*a^5*b^10*
x^9 + 3003*a^6*b^9*x^8 + 3432*a^7*b^8*x^7 + 3003*a^8*b^7*x^6 + 2002*a^9*b^6*x^5 + 1001*a^10*b^5*x^4 + 364*a^11
*b^4*x^3 + 91*a^12*b^3*x^2 + 14*a^13*b^2*x + a^14*b)

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Sympy [B]  time = 1.55131, size = 168, normalized size = 12. \begin{align*} - \frac{1}{14 a^{14} b + 196 a^{13} b^{2} x + 1274 a^{12} b^{3} x^{2} + 5096 a^{11} b^{4} x^{3} + 14014 a^{10} b^{5} x^{4} + 28028 a^{9} b^{6} x^{5} + 42042 a^{8} b^{7} x^{6} + 48048 a^{7} b^{8} x^{7} + 42042 a^{6} b^{9} x^{8} + 28028 a^{5} b^{10} x^{9} + 14014 a^{4} b^{11} x^{10} + 5096 a^{3} b^{12} x^{11} + 1274 a^{2} b^{13} x^{12} + 196 a b^{14} x^{13} + 14 b^{15} x^{14}} \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)**3,x)

[Out]

-1/(14*a**14*b + 196*a**13*b**2*x + 1274*a**12*b**3*x**2 + 5096*a**11*b**4*x**3 + 14014*a**10*b**5*x**4 + 2802
8*a**9*b**6*x**5 + 42042*a**8*b**7*x**6 + 48048*a**7*b**8*x**7 + 42042*a**6*b**9*x**8 + 28028*a**5*b**10*x**9
+ 14014*a**4*b**11*x**10 + 5096*a**3*b**12*x**11 + 1274*a**2*b**13*x**12 + 196*a*b**14*x**13 + 14*b**15*x**14)

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Giac [A]  time = 1.14627, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{14 \,{\left (b x + a\right )}^{14} 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)^3,x, algorithm="giac")

[Out]

-1/14/((b*x + a)^14*b)