3.6.42 \(\int \frac {(1+x) (1+2 x+x^2)^5}{x^{16}} \, dx\)

Optimal. Leaf size=49 \[ -\frac {(x+1)^{12}}{15 x^{15}}+\frac {(x+1)^{12}}{70 x^{14}}-\frac {(x+1)^{12}}{455 x^{13}}+\frac {(x+1)^{12}}{5460 x^{12}} \]

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Rubi [A]  time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, 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}}{5460 x^{12}}-\frac {(x+1)^{12}}{455 x^{13}}+\frac {(x+1)^{12}}{70 x^{14}}-\frac {(x+1)^{12}}{15 x^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^16,x]

[Out]

-(1 + x)^12/(15*x^15) + (1 + x)^12/(70*x^14) - (1 + x)^12/(455*x^13) + (1 + x)^12/(5460*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^{16}} \, dx &=\int \frac {(1+x)^{11}}{x^{16}} \, dx\\ &=-\frac {(1+x)^{12}}{15 x^{15}}-\frac {1}{5} \int \frac {(1+x)^{11}}{x^{15}} \, dx\\ &=-\frac {(1+x)^{12}}{15 x^{15}}+\frac {(1+x)^{12}}{70 x^{14}}+\frac {1}{35} \int \frac {(1+x)^{11}}{x^{14}} \, dx\\ &=-\frac {(1+x)^{12}}{15 x^{15}}+\frac {(1+x)^{12}}{70 x^{14}}-\frac {(1+x)^{12}}{455 x^{13}}-\frac {1}{455} \int \frac {(1+x)^{11}}{x^{13}} \, dx\\ &=-\frac {(1+x)^{12}}{15 x^{15}}+\frac {(1+x)^{12}}{70 x^{14}}-\frac {(1+x)^{12}}{455 x^{13}}+\frac {(1+x)^{12}}{5460 x^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 83, normalized size = 1.69 \begin {gather*} -\frac {1}{15 x^{15}}-\frac {11}{14 x^{14}}-\frac {55}{13 x^{13}}-\frac {55}{4 x^{12}}-\frac {30}{x^{11}}-\frac {231}{5 x^{10}}-\frac {154}{3 x^9}-\frac {165}{4 x^8}-\frac {165}{7 x^7}-\frac {55}{6 x^6}-\frac {11}{5 x^5}-\frac {1}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^16,x]

[Out]

-1/15*1/x^15 - 11/(14*x^14) - 55/(13*x^13) - 55/(4*x^12) - 30/x^11 - 231/(5*x^10) - 154/(3*x^9) - 165/(4*x^8)
- 165/(7*x^7) - 55/(6*x^6) - 11/(5*x^5) - 1/(4*x^4)

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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^{16}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 + x)*(1 + 2*x + x^2)^5)/x^16,x]

[Out]

IntegrateAlgebraic[((1 + x)*(1 + 2*x + x^2)^5)/x^16, x]

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fricas [A]  time = 0.39, size = 60, normalized size = 1.22 \begin {gather*} -\frac {1365 \, x^{11} + 12012 \, x^{10} + 50050 \, x^{9} + 128700 \, x^{8} + 225225 \, x^{7} + 280280 \, x^{6} + 252252 \, x^{5} + 163800 \, x^{4} + 75075 \, x^{3} + 23100 \, x^{2} + 4290 \, x + 364}{5460 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^16,x, algorithm="fricas")

[Out]

-1/5460*(1365*x^11 + 12012*x^10 + 50050*x^9 + 128700*x^8 + 225225*x^7 + 280280*x^6 + 252252*x^5 + 163800*x^4 +
75075*x^3 + 23100*x^2 + 4290*x + 364)/x^15

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giac [A]  time = 0.15, size = 60, normalized size = 1.22 \begin {gather*} -\frac {1365 \, x^{11} + 12012 \, x^{10} + 50050 \, x^{9} + 128700 \, x^{8} + 225225 \, x^{7} + 280280 \, x^{6} + 252252 \, x^{5} + 163800 \, x^{4} + 75075 \, x^{3} + 23100 \, x^{2} + 4290 \, x + 364}{5460 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^16,x, algorithm="giac")

[Out]

-1/5460*(1365*x^11 + 12012*x^10 + 50050*x^9 + 128700*x^8 + 225225*x^7 + 280280*x^6 + 252252*x^5 + 163800*x^4 +
75075*x^3 + 23100*x^2 + 4290*x + 364)/x^15

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maple [A]  time = 0.05, size = 62, normalized size = 1.27 \begin {gather*} -\frac {1}{4 x^{4}}-\frac {11}{5 x^{5}}-\frac {55}{6 x^{6}}-\frac {165}{7 x^{7}}-\frac {165}{4 x^{8}}-\frac {154}{3 x^{9}}-\frac {231}{5 x^{10}}-\frac {30}{x^{11}}-\frac {55}{4 x^{12}}-\frac {55}{13 x^{13}}-\frac {11}{14 x^{14}}-\frac {1}{15 x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(x^2+2*x+1)^5/x^16,x)

[Out]

-11/5/x^5-1/4/x^4-11/14/x^14-165/4/x^8-231/5/x^10-1/15/x^15-154/3/x^9-165/7/x^7-55/6/x^6-55/13/x^13-30/x^11-55
/4/x^12

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maxima [A]  time = 0.56, size = 60, normalized size = 1.22 \begin {gather*} -\frac {1365 \, x^{11} + 12012 \, x^{10} + 50050 \, x^{9} + 128700 \, x^{8} + 225225 \, x^{7} + 280280 \, x^{6} + 252252 \, x^{5} + 163800 \, x^{4} + 75075 \, x^{3} + 23100 \, x^{2} + 4290 \, x + 364}{5460 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^16,x, algorithm="maxima")

[Out]

-1/5460*(1365*x^11 + 12012*x^10 + 50050*x^9 + 128700*x^8 + 225225*x^7 + 280280*x^6 + 252252*x^5 + 163800*x^4 +
75075*x^3 + 23100*x^2 + 4290*x + 364)/x^15

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mupad [B]  time = 1.07, size = 60, normalized size = 1.22 \begin {gather*} -\frac {\frac {x^{11}}{4}+\frac {11\,x^{10}}{5}+\frac {55\,x^9}{6}+\frac {165\,x^8}{7}+\frac {165\,x^7}{4}+\frac {154\,x^6}{3}+\frac {231\,x^5}{5}+30\,x^4+\frac {55\,x^3}{4}+\frac {55\,x^2}{13}+\frac {11\,x}{14}+\frac {1}{15}}{x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)*(2*x + x^2 + 1)^5)/x^16,x)

[Out]

-((11*x)/14 + (55*x^2)/13 + (55*x^3)/4 + 30*x^4 + (231*x^5)/5 + (154*x^6)/3 + (165*x^7)/4 + (165*x^8)/7 + (55*
x^9)/6 + (11*x^10)/5 + x^11/4 + 1/15)/x^15

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sympy [A]  time = 0.19, size = 61, normalized size = 1.24 \begin {gather*} \frac {- 1365 x^{11} - 12012 x^{10} - 50050 x^{9} - 128700 x^{8} - 225225 x^{7} - 280280 x^{6} - 252252 x^{5} - 163800 x^{4} - 75075 x^{3} - 23100 x^{2} - 4290 x - 364}{5460 x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**16,x)

[Out]

(-1365*x**11 - 12012*x**10 - 50050*x**9 - 128700*x**8 - 225225*x**7 - 280280*x**6 - 252252*x**5 - 163800*x**4
- 75075*x**3 - 23100*x**2 - 4290*x - 364)/(5460*x**15)

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