3.618 \(\int \frac{(1+x) (1+2 x+x^2)^5}{x^{19}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{(x+1)^{12}}{222768 x^{12}}+\frac{(x+1)^{12}}{18564 x^{13}}-\frac{(x+1)^{12}}{2856 x^{14}}+\frac{(x+1)^{12}}{612 x^{15}}-\frac{5 (x+1)^{12}}{816 x^{16}}+\frac{(x+1)^{12}}{51 x^{17}}-\frac{(x+1)^{12}}{18 x^{18}} \]

[Out]

-(1 + x)^12/(18*x^18) + (1 + x)^12/(51*x^17) - (5*(1 + x)^12)/(816*x^16) + (1 + x)^12/(612*x^15) - (1 + x)^12/
(2856*x^14) + (1 + x)^12/(18564*x^13) - (1 + x)^12/(222768*x^12)

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Rubi [A]  time = 0.019389, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {27, 45, 37} \[ -\frac{(x+1)^{12}}{222768 x^{12}}+\frac{(x+1)^{12}}{18564 x^{13}}-\frac{(x+1)^{12}}{2856 x^{14}}+\frac{(x+1)^{12}}{612 x^{15}}-\frac{5 (x+1)^{12}}{816 x^{16}}+\frac{(x+1)^{12}}{51 x^{17}}-\frac{(x+1)^{12}}{18 x^{18}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^19,x]

[Out]

-(1 + x)^12/(18*x^18) + (1 + x)^12/(51*x^17) - (5*(1 + x)^12)/(816*x^16) + (1 + x)^12/(612*x^15) - (1 + x)^12/
(2856*x^14) + (1 + x)^12/(18564*x^13) - (1 + x)^12/(222768*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^{19}} \, dx &=\int \frac{(1+x)^{11}}{x^{19}} \, dx\\ &=-\frac{(1+x)^{12}}{18 x^{18}}-\frac{1}{3} \int \frac{(1+x)^{11}}{x^{18}} \, dx\\ &=-\frac{(1+x)^{12}}{18 x^{18}}+\frac{(1+x)^{12}}{51 x^{17}}+\frac{5}{51} \int \frac{(1+x)^{11}}{x^{17}} \, dx\\ &=-\frac{(1+x)^{12}}{18 x^{18}}+\frac{(1+x)^{12}}{51 x^{17}}-\frac{5 (1+x)^{12}}{816 x^{16}}-\frac{5}{204} \int \frac{(1+x)^{11}}{x^{16}} \, dx\\ &=-\frac{(1+x)^{12}}{18 x^{18}}+\frac{(1+x)^{12}}{51 x^{17}}-\frac{5 (1+x)^{12}}{816 x^{16}}+\frac{(1+x)^{12}}{612 x^{15}}+\frac{1}{204} \int \frac{(1+x)^{11}}{x^{15}} \, dx\\ &=-\frac{(1+x)^{12}}{18 x^{18}}+\frac{(1+x)^{12}}{51 x^{17}}-\frac{5 (1+x)^{12}}{816 x^{16}}+\frac{(1+x)^{12}}{612 x^{15}}-\frac{(1+x)^{12}}{2856 x^{14}}-\frac{\int \frac{(1+x)^{11}}{x^{14}} \, dx}{1428}\\ &=-\frac{(1+x)^{12}}{18 x^{18}}+\frac{(1+x)^{12}}{51 x^{17}}-\frac{5 (1+x)^{12}}{816 x^{16}}+\frac{(1+x)^{12}}{612 x^{15}}-\frac{(1+x)^{12}}{2856 x^{14}}+\frac{(1+x)^{12}}{18564 x^{13}}+\frac{\int \frac{(1+x)^{11}}{x^{13}} \, dx}{18564}\\ &=-\frac{(1+x)^{12}}{18 x^{18}}+\frac{(1+x)^{12}}{51 x^{17}}-\frac{5 (1+x)^{12}}{816 x^{16}}+\frac{(1+x)^{12}}{612 x^{15}}-\frac{(1+x)^{12}}{2856 x^{14}}+\frac{(1+x)^{12}}{18564 x^{13}}-\frac{(1+x)^{12}}{222768 x^{12}}\\ \end{align*}

Mathematica [A]  time = 0.0022143, size = 81, normalized size = 0.95 \[ -\frac{1}{7 x^7}-\frac{11}{8 x^8}-\frac{55}{9 x^9}-\frac{33}{2 x^{10}}-\frac{30}{x^{11}}-\frac{77}{2 x^{12}}-\frac{462}{13 x^{13}}-\frac{165}{7 x^{14}}-\frac{11}{x^{15}}-\frac{55}{16 x^{16}}-\frac{11}{17 x^{17}}-\frac{1}{18 x^{18}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^19,x]

[Out]

-1/(18*x^18) - 11/(17*x^17) - 55/(16*x^16) - 11/x^15 - 165/(7*x^14) - 462/(13*x^13) - 77/(2*x^12) - 30/x^11 -
33/(2*x^10) - 55/(9*x^9) - 11/(8*x^8) - 1/(7*x^7)

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Maple [A]  time = 0.008, size = 62, normalized size = 0.7 \begin{align*} -{\frac{1}{18\,{x}^{18}}}-{\frac{55}{9\,{x}^{9}}}-30\,{x}^{-11}-{\frac{77}{2\,{x}^{12}}}-{\frac{11}{8\,{x}^{8}}}-{\frac{1}{7\,{x}^{7}}}-11\,{x}^{-15}-{\frac{33}{2\,{x}^{10}}}-{\frac{11}{17\,{x}^{17}}}-{\frac{55}{16\,{x}^{16}}}-{\frac{165}{7\,{x}^{14}}}-{\frac{462}{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^19,x)

[Out]

-1/18/x^18-55/9/x^9-30/x^11-77/2/x^12-11/8/x^8-1/7/x^7-11/x^15-33/2/x^10-11/17/x^17-55/16/x^16-165/7/x^14-462/
13/x^13

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Maxima [A]  time = 1.00258, size = 81, normalized size = 0.95 \begin{align*} -\frac{31824 \, x^{11} + 306306 \, x^{10} + 1361360 \, x^{9} + 3675672 \, x^{8} + 6683040 \, x^{7} + 8576568 \, x^{6} + 7916832 \, x^{5} + 5250960 \, x^{4} + 2450448 \, x^{3} + 765765 \, x^{2} + 144144 \, x + 12376}{222768 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^19,x, algorithm="maxima")

[Out]

-1/222768*(31824*x^11 + 306306*x^10 + 1361360*x^9 + 3675672*x^8 + 6683040*x^7 + 8576568*x^6 + 7916832*x^5 + 52
50960*x^4 + 2450448*x^3 + 765765*x^2 + 144144*x + 12376)/x^18

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Fricas [A]  time = 1.16503, size = 234, normalized size = 2.75 \begin{align*} -\frac{31824 \, x^{11} + 306306 \, x^{10} + 1361360 \, x^{9} + 3675672 \, x^{8} + 6683040 \, x^{7} + 8576568 \, x^{6} + 7916832 \, x^{5} + 5250960 \, x^{4} + 2450448 \, x^{3} + 765765 \, x^{2} + 144144 \, x + 12376}{222768 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^19,x, algorithm="fricas")

[Out]

-1/222768*(31824*x^11 + 306306*x^10 + 1361360*x^9 + 3675672*x^8 + 6683040*x^7 + 8576568*x^6 + 7916832*x^5 + 52
50960*x^4 + 2450448*x^3 + 765765*x^2 + 144144*x + 12376)/x^18

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Sympy [A]  time = 0.213139, size = 61, normalized size = 0.72 \begin{align*} - \frac{31824 x^{11} + 306306 x^{10} + 1361360 x^{9} + 3675672 x^{8} + 6683040 x^{7} + 8576568 x^{6} + 7916832 x^{5} + 5250960 x^{4} + 2450448 x^{3} + 765765 x^{2} + 144144 x + 12376}{222768 x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**19,x)

[Out]

-(31824*x**11 + 306306*x**10 + 1361360*x**9 + 3675672*x**8 + 6683040*x**7 + 8576568*x**6 + 7916832*x**5 + 5250
960*x**4 + 2450448*x**3 + 765765*x**2 + 144144*x + 12376)/(222768*x**18)

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Giac [A]  time = 1.17066, size = 81, normalized size = 0.95 \begin{align*} -\frac{31824 \, x^{11} + 306306 \, x^{10} + 1361360 \, x^{9} + 3675672 \, x^{8} + 6683040 \, x^{7} + 8576568 \, x^{6} + 7916832 \, x^{5} + 5250960 \, x^{4} + 2450448 \, x^{3} + 765765 \, x^{2} + 144144 \, x + 12376}{222768 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^19,x, algorithm="giac")

[Out]

-1/222768*(31824*x^11 + 306306*x^10 + 1361360*x^9 + 3675672*x^8 + 6683040*x^7 + 8576568*x^6 + 7916832*x^5 + 52
50960*x^4 + 2450448*x^3 + 765765*x^2 + 144144*x + 12376)/x^18