3.617 \(\int \frac{(1+x) (1+2 x+x^2)^5}{x^{18}} \, dx\)

Optimal. Leaf size=73 \[ \frac{(x+1)^{12}}{74256 x^{12}}-\frac{(x+1)^{12}}{6188 x^{13}}+\frac{(x+1)^{12}}{952 x^{14}}-\frac{(x+1)^{12}}{204 x^{15}}+\frac{5 (x+1)^{12}}{272 x^{16}}-\frac{(x+1)^{12}}{17 x^{17}} \]

[Out]

-(1 + x)^12/(17*x^17) + (5*(1 + x)^12)/(272*x^16) - (1 + x)^12/(204*x^15) + (1 + x)^12/(952*x^14) - (1 + x)^12
/(6188*x^13) + (1 + x)^12/(74256*x^12)

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Rubi [A]  time = 0.0157929, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {27, 45, 37} \[ \frac{(x+1)^{12}}{74256 x^{12}}-\frac{(x+1)^{12}}{6188 x^{13}}+\frac{(x+1)^{12}}{952 x^{14}}-\frac{(x+1)^{12}}{204 x^{15}}+\frac{5 (x+1)^{12}}{272 x^{16}}-\frac{(x+1)^{12}}{17 x^{17}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^18,x]

[Out]

-(1 + x)^12/(17*x^17) + (5*(1 + x)^12)/(272*x^16) - (1 + x)^12/(204*x^15) + (1 + x)^12/(952*x^14) - (1 + x)^12
/(6188*x^13) + (1 + x)^12/(74256*x^12)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(1+x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx &=\int \frac{(1+x)^{11}}{x^{18}} \, dx\\ &=-\frac{(1+x)^{12}}{17 x^{17}}-\frac{5}{17} \int \frac{(1+x)^{11}}{x^{17}} \, dx\\ &=-\frac{(1+x)^{12}}{17 x^{17}}+\frac{5 (1+x)^{12}}{272 x^{16}}+\frac{5}{68} \int \frac{(1+x)^{11}}{x^{16}} \, dx\\ &=-\frac{(1+x)^{12}}{17 x^{17}}+\frac{5 (1+x)^{12}}{272 x^{16}}-\frac{(1+x)^{12}}{204 x^{15}}-\frac{1}{68} \int \frac{(1+x)^{11}}{x^{15}} \, dx\\ &=-\frac{(1+x)^{12}}{17 x^{17}}+\frac{5 (1+x)^{12}}{272 x^{16}}-\frac{(1+x)^{12}}{204 x^{15}}+\frac{(1+x)^{12}}{952 x^{14}}+\frac{1}{476} \int \frac{(1+x)^{11}}{x^{14}} \, dx\\ &=-\frac{(1+x)^{12}}{17 x^{17}}+\frac{5 (1+x)^{12}}{272 x^{16}}-\frac{(1+x)^{12}}{204 x^{15}}+\frac{(1+x)^{12}}{952 x^{14}}-\frac{(1+x)^{12}}{6188 x^{13}}-\frac{\int \frac{(1+x)^{11}}{x^{13}} \, dx}{6188}\\ &=-\frac{(1+x)^{12}}{17 x^{17}}+\frac{5 (1+x)^{12}}{272 x^{16}}-\frac{(1+x)^{12}}{204 x^{15}}+\frac{(1+x)^{12}}{952 x^{14}}-\frac{(1+x)^{12}}{6188 x^{13}}+\frac{(1+x)^{12}}{74256 x^{12}}\\ \end{align*}

Mathematica [A]  time = 0.002379, size = 81, normalized size = 1.11 \[ -\frac{1}{6 x^6}-\frac{11}{7 x^7}-\frac{55}{8 x^8}-\frac{55}{3 x^9}-\frac{33}{x^{10}}-\frac{42}{x^{11}}-\frac{77}{2 x^{12}}-\frac{330}{13 x^{13}}-\frac{165}{14 x^{14}}-\frac{11}{3 x^{15}}-\frac{11}{16 x^{16}}-\frac{1}{17 x^{17}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^18,x]

[Out]

-1/(17*x^17) - 11/(16*x^16) - 11/(3*x^15) - 165/(14*x^14) - 330/(13*x^13) - 77/(2*x^12) - 42/x^11 - 33/x^10 -
55/(3*x^9) - 55/(8*x^8) - 11/(7*x^7) - 1/(6*x^6)

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Maple [A]  time = 0.007, size = 62, normalized size = 0.9 \begin{align*} -{\frac{55}{3\,{x}^{9}}}-{\frac{55}{8\,{x}^{8}}}-{\frac{11}{7\,{x}^{7}}}-33\,{x}^{-10}-42\,{x}^{-11}-{\frac{165}{14\,{x}^{14}}}-{\frac{11}{16\,{x}^{16}}}-{\frac{77}{2\,{x}^{12}}}-{\frac{1}{6\,{x}^{6}}}-{\frac{11}{3\,{x}^{15}}}-{\frac{1}{17\,{x}^{17}}}-{\frac{330}{13\,{x}^{13}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)*(x^2+2*x+1)^5/x^18,x)

[Out]

-55/3/x^9-55/8/x^8-11/7/x^7-33/x^10-42/x^11-165/14/x^14-11/16/x^16-77/2/x^12-1/6/x^6-11/3/x^15-1/17/x^17-330/1
3/x^13

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Maxima [A]  time = 1.02722, size = 81, normalized size = 1.11 \begin{align*} -\frac{12376 \, x^{11} + 116688 \, x^{10} + 510510 \, x^{9} + 1361360 \, x^{8} + 2450448 \, x^{7} + 3118752 \, x^{6} + 2858856 \, x^{5} + 1884960 \, x^{4} + 875160 \, x^{3} + 272272 \, x^{2} + 51051 \, x + 4368}{74256 \, x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^18,x, algorithm="maxima")

[Out]

-1/74256*(12376*x^11 + 116688*x^10 + 510510*x^9 + 1361360*x^8 + 2450448*x^7 + 3118752*x^6 + 2858856*x^5 + 1884
960*x^4 + 875160*x^3 + 272272*x^2 + 51051*x + 4368)/x^17

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Fricas [A]  time = 1.06807, size = 227, normalized size = 3.11 \begin{align*} -\frac{12376 \, x^{11} + 116688 \, x^{10} + 510510 \, x^{9} + 1361360 \, x^{8} + 2450448 \, x^{7} + 3118752 \, x^{6} + 2858856 \, x^{5} + 1884960 \, x^{4} + 875160 \, x^{3} + 272272 \, x^{2} + 51051 \, x + 4368}{74256 \, x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^18,x, algorithm="fricas")

[Out]

-1/74256*(12376*x^11 + 116688*x^10 + 510510*x^9 + 1361360*x^8 + 2450448*x^7 + 3118752*x^6 + 2858856*x^5 + 1884
960*x^4 + 875160*x^3 + 272272*x^2 + 51051*x + 4368)/x^17

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Sympy [A]  time = 0.204097, size = 61, normalized size = 0.84 \begin{align*} - \frac{12376 x^{11} + 116688 x^{10} + 510510 x^{9} + 1361360 x^{8} + 2450448 x^{7} + 3118752 x^{6} + 2858856 x^{5} + 1884960 x^{4} + 875160 x^{3} + 272272 x^{2} + 51051 x + 4368}{74256 x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**18,x)

[Out]

-(12376*x**11 + 116688*x**10 + 510510*x**9 + 1361360*x**8 + 2450448*x**7 + 3118752*x**6 + 2858856*x**5 + 18849
60*x**4 + 875160*x**3 + 272272*x**2 + 51051*x + 4368)/(74256*x**17)

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Giac [A]  time = 1.1246, size = 81, normalized size = 1.11 \begin{align*} -\frac{12376 \, x^{11} + 116688 \, x^{10} + 510510 \, x^{9} + 1361360 \, x^{8} + 2450448 \, x^{7} + 3118752 \, x^{6} + 2858856 \, x^{5} + 1884960 \, x^{4} + 875160 \, x^{3} + 272272 \, x^{2} + 51051 \, x + 4368}{74256 \, x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^18,x, algorithm="giac")

[Out]

-1/74256*(12376*x^11 + 116688*x^10 + 510510*x^9 + 1361360*x^8 + 2450448*x^7 + 3118752*x^6 + 2858856*x^5 + 1884
960*x^4 + 875160*x^3 + 272272*x^2 + 51051*x + 4368)/x^17