3.1.67 \(\int \frac {1}{(a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5)^2} \, dx\) [67]

Optimal. Leaf size=14 \[ -\frac {1}{9 b (a+b x)^9} \]

[Out]

-1/9/b/(b*x+a)^9

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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 = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2083, 32} \begin {gather*} -\frac {1}{9 b (a+b x)^9} \end {gather*}

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)^(-2),x]

[Out]

-1/9*1/(b*(a + b*x)^9)

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 2083

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^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 )^2} \, dx &=\int \frac {1}{(a+b x)^{10}} \, dx\\ &=-\frac {1}{9 b (a+b x)^9}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{9 b (a+b x)^9} \end {gather*}

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)^(-2),x]

[Out]

-1/9*1/(b*(a + b*x)^9)

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Maple [A]
time = 0.05, size = 13, normalized size = 0.93

method	result	size
default	\(-\frac {1}{9 b \left (b x +a \right )^{9}}\)	\(13\)
norman	\(-\frac {1}{9 b \left (b x +a \right )^{9}}\)	\(13\)
risch	\(-\frac {1}{9 b \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )^{2} \left (b x +a \right )}\)	\(53\)
gosper	\(-\frac {1}{9 \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right ) \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 a^{4} b x +a^{5}\right ) b}\)	\(97\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

-1/9/b/(b*x+a)^9

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (12) = 24\).
time = 0.31, size = 101, normalized size = 7.21 \begin {gather*} -\frac {1}{9 \, {\left (b^{10} x^{9} + 9 \, a b^{9} x^{8} + 36 \, a^{2} b^{8} x^{7} + 84 \, a^{3} b^{7} x^{6} + 126 \, a^{4} b^{6} x^{5} + 126 \, a^{5} b^{5} x^{4} + 84 \, a^{6} b^{4} x^{3} + 36 \, a^{7} b^{3} x^{2} + 9 \, a^{8} b^{2} x + a^{9} b\right )}} \end {gather*}

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)^2,x, algorithm="maxima")

[Out]

-1/9/(b^10*x^9 + 9*a*b^9*x^8 + 36*a^2*b^8*x^7 + 84*a^3*b^7*x^6 + 126*a^4*b^6*x^5 + 126*a^5*b^5*x^4 + 84*a^6*b^
4*x^3 + 36*a^7*b^3*x^2 + 9*a^8*b^2*x + a^9*b)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (12) = 24\).
time = 0.36, size = 101, normalized size = 7.21 \begin {gather*} -\frac {1}{9 \, {\left (b^{10} x^{9} + 9 \, a b^{9} x^{8} + 36 \, a^{2} b^{8} x^{7} + 84 \, a^{3} b^{7} x^{6} + 126 \, a^{4} b^{6} x^{5} + 126 \, a^{5} b^{5} x^{4} + 84 \, a^{6} b^{4} x^{3} + 36 \, a^{7} b^{3} x^{2} + 9 \, a^{8} b^{2} x + a^{9} b\right )}} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

-1/9/(b^10*x^9 + 9*a*b^9*x^8 + 36*a^2*b^8*x^7 + 84*a^3*b^7*x^6 + 126*a^4*b^6*x^5 + 126*a^5*b^5*x^4 + 84*a^6*b^
4*x^3 + 36*a^7*b^3*x^2 + 9*a^8*b^2*x + a^9*b)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (12) = 24\).
time = 0.29, size = 109, normalized size = 7.79 \begin {gather*} - \frac {1}{9 a^{9} b + 81 a^{8} b^{2} x + 324 a^{7} b^{3} x^{2} + 756 a^{6} b^{4} x^{3} + 1134 a^{5} b^{5} x^{4} + 1134 a^{4} b^{6} x^{5} + 756 a^{3} b^{7} x^{6} + 324 a^{2} b^{8} x^{7} + 81 a b^{9} x^{8} + 9 b^{10} x^{9}} \end {gather*}

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)**2,x)

[Out]

-1/(9*a**9*b + 81*a**8*b**2*x + 324*a**7*b**3*x**2 + 756*a**6*b**4*x**3 + 1134*a**5*b**5*x**4 + 1134*a**4*b**6
*x**5 + 756*a**3*b**7*x**6 + 324*a**2*b**8*x**7 + 81*a*b**9*x**8 + 9*b**10*x**9)

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Giac [A]
time = 6.01, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{9 \, {\left (b x + a\right )}^{9} b} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

-1/9/((b*x + a)^9*b)

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Mupad [B]
time = 2.10, size = 103, normalized size = 7.36 \begin {gather*} -\frac {1}{9\,a^9\,b+81\,a^8\,b^2\,x+324\,a^7\,b^3\,x^2+756\,a^6\,b^4\,x^3+1134\,a^5\,b^5\,x^4+1134\,a^4\,b^6\,x^5+756\,a^3\,b^7\,x^6+324\,a^2\,b^8\,x^7+81\,a\,b^9\,x^8+9\,b^{10}\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^5 + b^5*x^5 + 5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x)^2,x)

[Out]

-1/(9*a^9*b + 9*b^10*x^9 + 81*a^8*b^2*x + 81*a*b^9*x^8 + 324*a^7*b^3*x^2 + 756*a^6*b^4*x^3 + 1134*a^5*b^5*x^4
+ 1134*a^4*b^6*x^5 + 756*a^3*b^7*x^6 + 324*a^2*b^8*x^7)

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