3.7.13 \(\int \frac {(1+x) (1+2 x+x^2)^5}{x^{14}} \, dx\) [613]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, 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, 47, 37} \begin {gather*} \frac {(x+1)^{12}}{156 x^{12}}-\frac {(x+1)^{12}}{13 x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/13*(1 + x)^12/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 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 47

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^{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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(25)=50\).
time = 0.00, size = 77, normalized size = 3.08 \begin {gather*} -\frac {1}{13 x^{13}}-\frac {11}{12 x^{12}}-\frac {5}{x^{11}}-\frac {33}{2 x^{10}}-\frac {110}{3 x^9}-\frac {231}{4 x^8}-\frac {66}{x^7}-\frac {55}{x^6}-\frac {33}{x^5}-\frac {55}{4 x^4}-\frac {11}{3 x^3}-\frac {1}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/13*1/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] Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(21)=42\).
time = 0.60, size = 62, normalized size = 2.48

method	result	size
norman	\(\frac {-\frac {1}{2} x^{11}-\frac {11}{3} x^{10}-\frac {55}{4} x^{9}-33 x^{8}-55 x^{7}-66 x^{6}-\frac {231}{4} x^{5}-\frac {110}{3} x^{4}-\frac {33}{2} x^{3}-5 x^{2}-\frac {11}{12} x -\frac {1}{13}}{x^{13}}\)	\(60\)
risch	\(\frac {-\frac {1}{2} x^{11}-\frac {11}{3} x^{10}-\frac {55}{4} x^{9}-33 x^{8}-55 x^{7}-66 x^{6}-\frac {231}{4} x^{5}-\frac {110}{3} x^{4}-\frac {33}{2} x^{3}-5 x^{2}-\frac {11}{12} x -\frac {1}{13}}{x^{13}}\)	\(60\)
gosper	\(-\frac {78 x^{11}+572 x^{10}+2145 x^{9}+5148 x^{8}+8580 x^{7}+10296 x^{6}+9009 x^{5}+5720 x^{4}+2574 x^{3}+780 x^{2}+143 x +12}{156 x^{13}}\)	\(61\)
default	\(-\frac {5}{x^{11}}-\frac {33}{x^{5}}-\frac {110}{3 x^{9}}-\frac {55}{4 x^{4}}-\frac {11}{12 x^{12}}-\frac {55}{x^{6}}-\frac {1}{13 x^{13}}-\frac {1}{2 x^{2}}-\frac {231}{4 x^{8}}-\frac {66}{x^{7}}-\frac {33}{2 x^{10}}-\frac {11}{3 x^{3}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(x^2+2*x+1)^5/x^14,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).
time = 0.28, size = 60, normalized size = 2.40 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).
time = 2.89, size = 60, normalized size = 2.40 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).
time = 0.07, size = 61, normalized size = 2.44 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).
time = 0.61, size = 60, normalized size = 2.40 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.03, size = 60, normalized size = 2.40 \begin {gather*} -\frac {\frac {x^{11}}{2}+\frac {11\,x^{10}}{3}+\frac {55\,x^9}{4}+33\,x^8+55\,x^7+66\,x^6+\frac {231\,x^5}{4}+\frac {110\,x^4}{3}+\frac {33\,x^3}{2}+5\,x^2+\frac {11\,x}{12}+\frac {1}{13}}{x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)*(2*x + x^2 + 1)^5)/x^14,x)

[Out]

-((11*x)/12 + 5*x^2 + (33*x^3)/2 + (110*x^4)/3 + (231*x^5)/4 + 66*x^6 + 55*x^7 + 33*x^8 + (55*x^9)/4 + (11*x^1
0)/3 + x^11/2 + 1/13)/x^13

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