3.613 \(\int \frac{(1+x) (1+2 x+x^2)^5}{x^{14}} \, dx\)

Optimal. Leaf size=25 \[ \frac{(x+1)^{12}}{156 x^{12}}-\frac{(x+1)^{12}}{13 x^{13}} \]

[Out]

-(1 + x)^12/(13*x^13) + (1 + x)^12/(156*x^12)

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Rubi [A]  time = 0.0032369, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {27, 45, 37} \[ \frac{(x+1)^{12}}{156 x^{12}}-\frac{(x+1)^{12}}{13 x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^14,x]

[Out]

-(1 + x)^12/(13*x^13) + (1 + x)^12/(156*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^{14}} \, dx &=\int \frac{(1+x)^{11}}{x^{14}} \, dx\\ &=-\frac{(1+x)^{12}}{13 x^{13}}-\frac{1}{13} \int \frac{(1+x)^{11}}{x^{13}} \, dx\\ &=-\frac{(1+x)^{12}}{13 x^{13}}+\frac{(1+x)^{12}}{156 x^{12}}\\ \end{align*}

Mathematica [B]  time = 0.002501, size = 77, normalized size = 3.08 \[ -\frac{1}{2 x^2}-\frac{11}{3 x^3}-\frac{55}{4 x^4}-\frac{33}{x^5}-\frac{55}{x^6}-\frac{66}{x^7}-\frac{231}{4 x^8}-\frac{110}{3 x^9}-\frac{33}{2 x^{10}}-\frac{5}{x^{11}}-\frac{11}{12 x^{12}}-\frac{1}{13 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^14,x]

[Out]

-1/(13*x^13) - 11/(12*x^12) - 5/x^11 - 33/(2*x^10) - 110/(3*x^9) - 231/(4*x^8) - 66/x^7 - 55/x^6 - 33/x^5 - 55
/(4*x^4) - 11/(3*x^3) - 1/(2*x^2)

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Maple [B]  time = 0.007, size = 62, normalized size = 2.5 \begin{align*} -{\frac{1}{13\,{x}^{13}}}-{\frac{110}{3\,{x}^{9}}}-{\frac{11}{3\,{x}^{3}}}-{\frac{1}{2\,{x}^{2}}}-{\frac{231}{4\,{x}^{8}}}-66\,{x}^{-7}-{\frac{33}{2\,{x}^{10}}}-33\,{x}^{-5}-{\frac{55}{4\,{x}^{4}}}-{\frac{11}{12\,{x}^{12}}}-5\,{x}^{-11}-55\,{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^14,x)

[Out]

-1/13/x^13-110/3/x^9-11/3/x^3-1/2/x^2-231/4/x^8-66/x^7-33/2/x^10-33/x^5-55/4/x^4-11/12/x^12-5/x^11-55/x^6

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Maxima [B]  time = 0.999433, size = 81, normalized size = 3.24 \begin{align*} -\frac{78 \, x^{11} + 572 \, x^{10} + 2145 \, x^{9} + 5148 \, x^{8} + 8580 \, x^{7} + 10296 \, x^{6} + 9009 \, x^{5} + 5720 \, x^{4} + 2574 \, x^{3} + 780 \, x^{2} + 143 \, x + 12}{156 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^14,x, algorithm="maxima")

[Out]

-1/156*(78*x^11 + 572*x^10 + 2145*x^9 + 5148*x^8 + 8580*x^7 + 10296*x^6 + 9009*x^5 + 5720*x^4 + 2574*x^3 + 780
*x^2 + 143*x + 12)/x^13

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Fricas [B]  time = 1.18778, size = 182, normalized size = 7.28 \begin{align*} -\frac{78 \, x^{11} + 572 \, x^{10} + 2145 \, x^{9} + 5148 \, x^{8} + 8580 \, x^{7} + 10296 \, x^{6} + 9009 \, x^{5} + 5720 \, x^{4} + 2574 \, x^{3} + 780 \, x^{2} + 143 \, x + 12}{156 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^14,x, algorithm="fricas")

[Out]

-1/156*(78*x^11 + 572*x^10 + 2145*x^9 + 5148*x^8 + 8580*x^7 + 10296*x^6 + 9009*x^5 + 5720*x^4 + 2574*x^3 + 780
*x^2 + 143*x + 12)/x^13

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Sympy [B]  time = 0.183225, size = 61, normalized size = 2.44 \begin{align*} - \frac{78 x^{11} + 572 x^{10} + 2145 x^{9} + 5148 x^{8} + 8580 x^{7} + 10296 x^{6} + 9009 x^{5} + 5720 x^{4} + 2574 x^{3} + 780 x^{2} + 143 x + 12}{156 x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**14,x)

[Out]

-(78*x**11 + 572*x**10 + 2145*x**9 + 5148*x**8 + 8580*x**7 + 10296*x**6 + 9009*x**5 + 5720*x**4 + 2574*x**3 +
780*x**2 + 143*x + 12)/(156*x**13)

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Giac [B]  time = 1.11297, size = 81, normalized size = 3.24 \begin{align*} -\frac{78 \, x^{11} + 572 \, x^{10} + 2145 \, x^{9} + 5148 \, x^{8} + 8580 \, x^{7} + 10296 \, x^{6} + 9009 \, x^{5} + 5720 \, x^{4} + 2574 \, x^{3} + 780 \, x^{2} + 143 \, x + 12}{156 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^14,x, algorithm="giac")

[Out]

-1/156*(78*x^11 + 572*x^10 + 2145*x^9 + 5148*x^8 + 8580*x^7 + 10296*x^6 + 9009*x^5 + 5720*x^4 + 2574*x^3 + 780
*x^2 + 143*x + 12)/x^13