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 + x)^12/(12*x^12)

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Rubi [A]  time = 0.0015824, antiderivative size = 12, normalized size of antiderivative = 1., 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*}

Mathematica [B]  time = 0.0023069, size = 75, normalized size = 6.25 \[ -\frac{11}{2 x^2}-\frac{55}{3 x^3}-\frac{165}{4 x^4}-\frac{66}{x^5}-\frac{77}{x^6}-\frac{66}{x^7}-\frac{165}{4 x^8}-\frac{55}{3 x^9}-\frac{11}{2 x^{10}}-\frac{1}{x^{11}}-\frac{1}{12 x^{12}}-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(12*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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Maple [B]  time = 0.006, size = 62, normalized size = 5.2 \begin{align*} -{\frac{55}{3\,{x}^{9}}}-{\frac{55}{3\,{x}^{3}}}-{\frac{11}{2\,{x}^{2}}}-{x}^{-1}-{\frac{165}{4\,{x}^{8}}}-{x}^{-11}-66\,{x}^{-7}-{\frac{11}{2\,{x}^{10}}}-66\,{x}^{-5}-{\frac{165}{4\,{x}^{4}}}-{\frac{1}{12\,{x}^{12}}}-77\,{x}^{-6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.03928, size = 81, normalized size = 6.75 \begin{align*} -\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}} \end{align*}

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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Fricas [B]  time = 1.21658, size = 165, normalized size = 13.75 \begin{align*} -\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}} \end{align*}

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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Sympy [B]  time = 0.168369, size = 61, normalized size = 5.08 \begin{align*} - \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}} \end{align*}

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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Giac [B]  time = 1.11349, size = 81, normalized size = 6.75 \begin{align*} -\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}} \end{align*}

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