3.6.46 \(\int \frac {(1+x) (1+2 x+x^2)^5}{x^{20}} \, dx\)

Optimal. Leaf size=97 \[ -\frac {(x+1)^{12}}{19 x^{19}}+\frac {7 (x+1)^{12}}{342 x^{18}}-\frac {7 (x+1)^{12}}{969 x^{17}}+\frac {35 (x+1)^{12}}{15504 x^{16}}-\frac {7 (x+1)^{12}}{11628 x^{15}}+\frac {(x+1)^{12}}{7752 x^{14}}-\frac {(x+1)^{12}}{50388 x^{13}}+\frac {(x+1)^{12}}{604656 x^{12}} \]

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Rubi [A]  time = 0.03, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {27, 45, 37} \begin {gather*} \frac {(x+1)^{12}}{604656 x^{12}}-\frac {(x+1)^{12}}{50388 x^{13}}+\frac {(x+1)^{12}}{7752 x^{14}}-\frac {7 (x+1)^{12}}{11628 x^{15}}+\frac {35 (x+1)^{12}}{15504 x^{16}}-\frac {7 (x+1)^{12}}{969 x^{17}}+\frac {7 (x+1)^{12}}{342 x^{18}}-\frac {(x+1)^{12}}{19 x^{19}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^20,x]

[Out]

-(1 + x)^12/(19*x^19) + (7*(1 + x)^12)/(342*x^18) - (7*(1 + x)^12)/(969*x^17) + (35*(1 + x)^12)/(15504*x^16) -
(7*(1 + x)^12)/(11628*x^15) + (1 + x)^12/(7752*x^14) - (1 + x)^12/(50388*x^13) + (1 + x)^12/(604656*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 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]

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])

Rubi steps

\begin {align*} \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx &=\int \frac {(1+x)^{11}}{x^{20}} \, dx\\ &=-\frac {(1+x)^{12}}{19 x^{19}}-\frac {7}{19} \int \frac {(1+x)^{11}}{x^{19}} \, dx\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}+\frac {7}{57} \int \frac {(1+x)^{11}}{x^{18}} \, dx\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}-\frac {35}{969} \int \frac {(1+x)^{11}}{x^{17}} \, dx\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}+\frac {35 \int \frac {(1+x)^{11}}{x^{16}} \, dx}{3876}\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}-\frac {7 (1+x)^{12}}{11628 x^{15}}-\frac {7 \int \frac {(1+x)^{11}}{x^{15}} \, dx}{3876}\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}-\frac {7 (1+x)^{12}}{11628 x^{15}}+\frac {(1+x)^{12}}{7752 x^{14}}+\frac {\int \frac {(1+x)^{11}}{x^{14}} \, dx}{3876}\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}-\frac {7 (1+x)^{12}}{11628 x^{15}}+\frac {(1+x)^{12}}{7752 x^{14}}-\frac {(1+x)^{12}}{50388 x^{13}}-\frac {\int \frac {(1+x)^{11}}{x^{13}} \, dx}{50388}\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}-\frac {7 (1+x)^{12}}{11628 x^{15}}+\frac {(1+x)^{12}}{7752 x^{14}}-\frac {(1+x)^{12}}{50388 x^{13}}+\frac {(1+x)^{12}}{604656 x^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 79, normalized size = 0.81 \begin {gather*} -\frac {1}{19 x^{19}}-\frac {11}{18 x^{18}}-\frac {55}{17 x^{17}}-\frac {165}{16 x^{16}}-\frac {22}{x^{15}}-\frac {33}{x^{14}}-\frac {462}{13 x^{13}}-\frac {55}{2 x^{12}}-\frac {15}{x^{11}}-\frac {11}{2 x^{10}}-\frac {11}{9 x^9}-\frac {1}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^20,x]

[Out]

-1/19*1/x^19 - 11/(18*x^18) - 55/(17*x^17) - 165/(16*x^16) - 22/x^15 - 33/x^14 - 462/(13*x^13) - 55/(2*x^12) -
15/x^11 - 11/(2*x^10) - 11/(9*x^9) - 1/(8*x^8)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 + x)*(1 + 2*x + x^2)^5)/x^20,x]

[Out]

IntegrateAlgebraic[((1 + x)*(1 + 2*x + x^2)^5)/x^20, x]

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fricas [A]  time = 0.39, size = 60, normalized size = 0.62 \begin {gather*} -\frac {75582 \, x^{11} + 739024 \, x^{10} + 3325608 \, x^{9} + 9069840 \, x^{8} + 16628040 \, x^{7} + 21488544 \, x^{6} + 19953648 \, x^{5} + 13302432 \, x^{4} + 6235515 \, x^{3} + 1956240 \, x^{2} + 369512 \, x + 31824}{604656 \, x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^20,x, algorithm="fricas")

[Out]

-1/604656*(75582*x^11 + 739024*x^10 + 3325608*x^9 + 9069840*x^8 + 16628040*x^7 + 21488544*x^6 + 19953648*x^5 +
13302432*x^4 + 6235515*x^3 + 1956240*x^2 + 369512*x + 31824)/x^19

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giac [A]  time = 0.16, size = 60, normalized size = 0.62 \begin {gather*} -\frac {75582 \, x^{11} + 739024 \, x^{10} + 3325608 \, x^{9} + 9069840 \, x^{8} + 16628040 \, x^{7} + 21488544 \, x^{6} + 19953648 \, x^{5} + 13302432 \, x^{4} + 6235515 \, x^{3} + 1956240 \, x^{2} + 369512 \, x + 31824}{604656 \, x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^20,x, algorithm="giac")

[Out]

-1/604656*(75582*x^11 + 739024*x^10 + 3325608*x^9 + 9069840*x^8 + 16628040*x^7 + 21488544*x^6 + 19953648*x^5 +
13302432*x^4 + 6235515*x^3 + 1956240*x^2 + 369512*x + 31824)/x^19

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maple [A]  time = 0.05, size = 62, normalized size = 0.64 \begin {gather*} -\frac {1}{8 x^{8}}-\frac {11}{9 x^{9}}-\frac {11}{2 x^{10}}-\frac {15}{x^{11}}-\frac {55}{2 x^{12}}-\frac {462}{13 x^{13}}-\frac {33}{x^{14}}-\frac {22}{x^{15}}-\frac {165}{16 x^{16}}-\frac {55}{17 x^{17}}-\frac {11}{18 x^{18}}-\frac {1}{19 x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(x^2+2*x+1)^5/x^20,x)

[Out]

-462/13/x^13-1/8/x^8-11/18/x^18-11/2/x^10-22/x^15-55/2/x^12-33/x^14-11/9/x^9-165/16/x^16-55/17/x^17-1/19/x^19-
15/x^11

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maxima [A]  time = 0.58, size = 60, normalized size = 0.62 \begin {gather*} -\frac {75582 \, x^{11} + 739024 \, x^{10} + 3325608 \, x^{9} + 9069840 \, x^{8} + 16628040 \, x^{7} + 21488544 \, x^{6} + 19953648 \, x^{5} + 13302432 \, x^{4} + 6235515 \, x^{3} + 1956240 \, x^{2} + 369512 \, x + 31824}{604656 \, x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^20,x, algorithm="maxima")

[Out]

-1/604656*(75582*x^11 + 739024*x^10 + 3325608*x^9 + 9069840*x^8 + 16628040*x^7 + 21488544*x^6 + 19953648*x^5 +
13302432*x^4 + 6235515*x^3 + 1956240*x^2 + 369512*x + 31824)/x^19

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mupad [B]  time = 0.03, size = 60, normalized size = 0.62 \begin {gather*} -\frac {\frac {x^{11}}{8}+\frac {11\,x^{10}}{9}+\frac {11\,x^9}{2}+15\,x^8+\frac {55\,x^7}{2}+\frac {462\,x^6}{13}+33\,x^5+22\,x^4+\frac {165\,x^3}{16}+\frac {55\,x^2}{17}+\frac {11\,x}{18}+\frac {1}{19}}{x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)*(2*x + x^2 + 1)^5)/x^20,x)

[Out]

-((11*x)/18 + (55*x^2)/17 + (165*x^3)/16 + 22*x^4 + 33*x^5 + (462*x^6)/13 + (55*x^7)/2 + 15*x^8 + (11*x^9)/2 +
(11*x^10)/9 + x^11/8 + 1/19)/x^19

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sympy [A]  time = 0.21, size = 61, normalized size = 0.63 \begin {gather*} \frac {- 75582 x^{11} - 739024 x^{10} - 3325608 x^{9} - 9069840 x^{8} - 16628040 x^{7} - 21488544 x^{6} - 19953648 x^{5} - 13302432 x^{4} - 6235515 x^{3} - 1956240 x^{2} - 369512 x - 31824}{604656 x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**20,x)

[Out]

(-75582*x**11 - 739024*x**10 - 3325608*x**9 - 9069840*x**8 - 16628040*x**7 - 21488544*x**6 - 19953648*x**5 - 1
3302432*x**4 - 6235515*x**3 - 1956240*x**2 - 369512*x - 31824)/(604656*x**19)

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