3.612 \(\int \frac {(1+x) (1+2 x+x^2)^5}{x^{13}} \, dx\)

Optimal. Leaf size=12 \[ -\frac {(x+1)^{12}}{12 x^{12}} \]

[Out]

-1/12*(1+x)^12/x^12

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {27, 37} \[ -\frac {(x+1)^{12}}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^13,x]

[Out]

-(1 + x)^12/(12*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]

Rubi steps

\begin {align*} \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{13}} \, dx &=\int \frac {(1+x)^{11}}{x^{13}} \, dx\\ &=-\frac {(1+x)^{12}}{12 x^{12}}\\ \end {align*}

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Mathematica [B]  time = 0.00, size = 75, normalized size = 6.25 \[ -\frac {1}{12 x^{12}}-\frac {1}{x^{11}}-\frac {11}{2 x^{10}}-\frac {55}{3 x^9}-\frac {165}{4 x^8}-\frac {66}{x^7}-\frac {77}{x^6}-\frac {66}{x^5}-\frac {165}{4 x^4}-\frac {55}{3 x^3}-\frac {11}{2 x^2}-\frac {1}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^13,x]

[Out]

-1/12*1/x^12 - x^(-11) - 11/(2*x^10) - 55/(3*x^9) - 165/(4*x^8) - 66/x^7 - 77/x^6 - 66/x^5 - 165/(4*x^4) - 55/
(3*x^3) - 11/(2*x^2) - x^(-1)

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fricas [B]  time = 0.85, size = 60, normalized size = 5.00 \[ -\frac {12 \, x^{11} + 66 \, x^{10} + 220 \, x^{9} + 495 \, x^{8} + 792 \, x^{7} + 924 \, x^{6} + 792 \, x^{5} + 495 \, x^{4} + 220 \, x^{3} + 66 \, x^{2} + 12 \, x + 1}{12 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^13,x, algorithm="fricas")

[Out]

-1/12*(12*x^11 + 66*x^10 + 220*x^9 + 495*x^8 + 792*x^7 + 924*x^6 + 792*x^5 + 495*x^4 + 220*x^3 + 66*x^2 + 12*x
+ 1)/x^12

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giac [B]  time = 0.17, size = 60, normalized size = 5.00 \[ -\frac {12 \, x^{11} + 66 \, x^{10} + 220 \, x^{9} + 495 \, x^{8} + 792 \, x^{7} + 924 \, x^{6} + 792 \, x^{5} + 495 \, x^{4} + 220 \, x^{3} + 66 \, x^{2} + 12 \, x + 1}{12 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^13,x, algorithm="giac")

[Out]

-1/12*(12*x^11 + 66*x^10 + 220*x^9 + 495*x^8 + 792*x^7 + 924*x^6 + 792*x^5 + 495*x^4 + 220*x^3 + 66*x^2 + 12*x
+ 1)/x^12

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maple [B]  time = 0.05, size = 62, normalized size = 5.17 \[ -\frac {1}{x}-\frac {11}{2 x^{2}}-\frac {55}{3 x^{3}}-\frac {165}{4 x^{4}}-\frac {66}{x^{5}}-\frac {77}{x^{6}}-\frac {66}{x^{7}}-\frac {165}{4 x^{8}}-\frac {55}{3 x^{9}}-\frac {11}{2 x^{10}}-\frac {1}{x^{11}}-\frac {1}{12 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(x^2+2*x+1)^5/x^13,x)

[Out]

-66/x^5-165/4/x^4-55/3/x^3-165/4/x^8-11/2/x^10-11/2/x^2-55/3/x^9-66/x^7-77/x^6-1/x-1/12/x^12-1/x^11

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maxima [B]  time = 0.47, size = 60, normalized size = 5.00 \[ -\frac {12 \, x^{11} + 66 \, x^{10} + 220 \, x^{9} + 495 \, x^{8} + 792 \, x^{7} + 924 \, x^{6} + 792 \, x^{5} + 495 \, x^{4} + 220 \, x^{3} + 66 \, x^{2} + 12 \, x + 1}{12 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^13,x, algorithm="maxima")

[Out]

-1/12*(12*x^11 + 66*x^10 + 220*x^9 + 495*x^8 + 792*x^7 + 924*x^6 + 792*x^5 + 495*x^4 + 220*x^3 + 66*x^2 + 12*x
+ 1)/x^12

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mupad [B]  time = 0.03, size = 56, normalized size = 4.67 \[ -\frac {x^{11}+\frac {11\,x^{10}}{2}+\frac {55\,x^9}{3}+\frac {165\,x^8}{4}+66\,x^7+77\,x^6+66\,x^5+\frac {165\,x^4}{4}+\frac {55\,x^3}{3}+\frac {11\,x^2}{2}+x+\frac {1}{12}}{x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)*(2*x + x^2 + 1)^5)/x^13,x)

[Out]

-(x + (11*x^2)/2 + (55*x^3)/3 + (165*x^4)/4 + 66*x^5 + 77*x^6 + 66*x^7 + (165*x^8)/4 + (55*x^9)/3 + (11*x^10)/
2 + x^11 + 1/12)/x^12

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sympy [B]  time = 0.17, size = 61, normalized size = 5.08 \[ \frac {- 12 x^{11} - 66 x^{10} - 220 x^{9} - 495 x^{8} - 792 x^{7} - 924 x^{6} - 792 x^{5} - 495 x^{4} - 220 x^{3} - 66 x^{2} - 12 x - 1}{12 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**13,x)

[Out]

(-12*x**11 - 66*x**10 - 220*x**9 - 495*x**8 - 792*x**7 - 924*x**6 - 792*x**5 - 495*x**4 - 220*x**3 - 66*x**2 -
12*x - 1)/(12*x**12)

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