3.159 \(\int \frac {x^3}{(-1+x)^{12}} \, dx\)

Optimal. Leaf size=45 \[ -\frac {1}{8 (1-x)^8}+\frac {1}{3 (1-x)^9}-\frac {3}{10 (1-x)^{10}}+\frac {1}{11 (1-x)^{11}} \]

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \[ -\frac {1}{8 (1-x)^8}+\frac {1}{3 (1-x)^9}-\frac {3}{10 (1-x)^{10}}+\frac {1}{11 (1-x)^{11}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(-1 + x)^12,x]

[Out]

1/(11*(1 - x)^11) - 3/(10*(1 - x)^10) + 1/(3*(1 - x)^9) - 1/(8*(1 - x)^8)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {x^3}{(-1+x)^{12}} \, dx &=\int \left (\frac {1}{(-1+x)^{12}}+\frac {3}{(-1+x)^{11}}+\frac {3}{(-1+x)^{10}}+\frac {1}{(-1+x)^9}\right ) \, dx\\ &=\frac {1}{11 (1-x)^{11}}-\frac {3}{10 (1-x)^{10}}+\frac {1}{3 (1-x)^9}-\frac {1}{8 (1-x)^8}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.53 \[ \frac {-165 x^3+55 x^2-11 x+1}{1320 (x-1)^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(-1 + x)^12,x]

[Out]

(1 - 11*x + 55*x^2 - 165*x^3)/(1320*(-1 + x)^11)

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IntegrateAlgebraic [A]  time = 0.01, size = 24, normalized size = 0.53 \[ \frac {-165 x^3+55 x^2-11 x+1}{1320 (x-1)^{11}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3/(-1 + x)^12,x]

[Out]

(1 - 11*x + 55*x^2 - 165*x^3)/(1320*(-1 + x)^11)

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fricas [B]  time = 1.06, size = 72, normalized size = 1.60 \[ -\frac {165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1}{1320 \, {\left (x^{11} - 11 \, x^{10} + 55 \, x^{9} - 165 \, x^{8} + 330 \, x^{7} - 462 \, x^{6} + 462 \, x^{5} - 330 \, x^{4} + 165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-1+x)^12,x, algorithm="fricas")

[Out]

-1/1320*(165*x^3 - 55*x^2 + 11*x - 1)/(x^11 - 11*x^10 + 55*x^9 - 165*x^8 + 330*x^7 - 462*x^6 + 462*x^5 - 330*x
^4 + 165*x^3 - 55*x^2 + 11*x - 1)

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giac [A]  time = 1.00, size = 22, normalized size = 0.49 \[ -\frac {165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1}{1320 \, {\left (x - 1\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-1+x)^12,x, algorithm="giac")

[Out]

-1/1320*(165*x^3 - 55*x^2 + 11*x - 1)/(x - 1)^11

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maple [A]  time = 0.29, size = 22, normalized size = 0.49

method	result	size
norman	\(\frac {-\frac {1}{8} x^{3}+\frac {1}{24} x^{2}-\frac {1}{120} x +\frac {1}{1320}}{\left (-1+x \right )^{11}}\)	\(22\)
risch	\(\frac {-\frac {1}{8} x^{3}+\frac {1}{24} x^{2}-\frac {1}{120} x +\frac {1}{1320}}{\left (-1+x \right )^{11}}\)	\(22\)
gosper	\(-\frac {165 x^{3}-55 x^{2}+11 x -1}{1320 \left (-1+x \right )^{11}}\)	\(23\)
default	\(-\frac {1}{8 \left (-1+x \right )^{8}}-\frac {3}{10 \left (-1+x \right )^{10}}-\frac {1}{11 \left (-1+x \right )^{11}}-\frac {1}{3 \left (-1+x \right )^{9}}\)	\(30\)
meijerg	\(\frac {x^{4} \left (-x^{7}+11 x^{6}-55 x^{5}+165 x^{4}-330 x^{3}+462 x^{2}-462 x +330\right )}{1320 \left (1-x \right )^{11}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-1+x)^12,x,method=_RETURNVERBOSE)

[Out]

1/(-1+x)^11*(-1/8*x^3+1/24*x^2-1/120*x+1/1320)

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maxima [B]  time = 0.44, size = 72, normalized size = 1.60 \[ -\frac {165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1}{1320 \, {\left (x^{11} - 11 \, x^{10} + 55 \, x^{9} - 165 \, x^{8} + 330 \, x^{7} - 462 \, x^{6} + 462 \, x^{5} - 330 \, x^{4} + 165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-1+x)^12,x, algorithm="maxima")

[Out]

-1/1320*(165*x^3 - 55*x^2 + 11*x - 1)/(x^11 - 11*x^10 + 55*x^9 - 165*x^8 + 330*x^7 - 462*x^6 + 462*x^5 - 330*x
^4 + 165*x^3 - 55*x^2 + 11*x - 1)

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mupad [B]  time = 0.10, size = 29, normalized size = 0.64 \[ -\frac {1}{8\,{\left (x-1\right )}^8}-\frac {1}{3\,{\left (x-1\right )}^9}-\frac {3}{10\,{\left (x-1\right )}^{10}}-\frac {1}{11\,{\left (x-1\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x - 1)^12,x)

[Out]

- 1/(8*(x - 1)^8) - 1/(3*(x - 1)^9) - 3/(10*(x - 1)^10) - 1/(11*(x - 1)^11)

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sympy [B]  time = 0.17, size = 70, normalized size = 1.56 \[ \frac {- 165 x^{3} + 55 x^{2} - 11 x + 1}{1320 x^{11} - 14520 x^{10} + 72600 x^{9} - 217800 x^{8} + 435600 x^{7} - 609840 x^{6} + 609840 x^{5} - 435600 x^{4} + 217800 x^{3} - 72600 x^{2} + 14520 x - 1320} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(-1+x)**12,x)

[Out]

(-165*x**3 + 55*x**2 - 11*x + 1)/(1320*x**11 - 14520*x**10 + 72600*x**9 - 217800*x**8 + 435600*x**7 - 609840*x
**6 + 609840*x**5 - 435600*x**4 + 217800*x**3 - 72600*x**2 + 14520*x - 1320)

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