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

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

[Out]

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

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Rubi [A]  time = 0.0124213, antiderivative size = 45, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0063717, 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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Maple [A]  time = 0.005, size = 30, normalized size = 0.7 \begin{align*} -{\frac{3}{10\, \left ( -1+x \right ) ^{10}}}-{\frac{1}{11\, \left ( -1+x \right ) ^{11}}}-{\frac{1}{8\, \left ( -1+x \right ) ^{8}}}-{\frac{1}{3\, \left ( -1+x \right ) ^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.936681, size = 97, normalized size = 2.16 \begin{align*} -\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 )}} \end{align*}

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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Fricas [B]  time = 1.89486, size = 196, normalized size = 4.36 \begin{align*} -\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 )}} \end{align*}

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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Sympy [B]  time = 0.161188, size = 71, normalized size = 1.58 \begin{align*} - \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} \end{align*}

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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Giac [A]  time = 1.05528, size = 30, normalized size = 0.67 \begin{align*} -\frac{165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1}{1320 \,{\left (x - 1\right )}^{11}} \end{align*}

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