3.616 \(\int \frac{(1+x) (1+2 x+x^2)^5}{x^{17}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{(x+1)^{12}}{21840 x^{12}}+\frac{(x+1)^{12}}{1820 x^{13}}-\frac{(x+1)^{12}}{280 x^{14}}+\frac{(x+1)^{12}}{60 x^{15}}-\frac{(x+1)^{12}}{16 x^{16}} \]

[Out]

-(1 + x)^12/(16*x^16) + (1 + x)^12/(60*x^15) - (1 + x)^12/(280*x^14) + (1 + x)^12/(1820*x^13) - (1 + x)^12/(21
840*x^12)

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Rubi [A]  time = 0.0114299, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {27, 45, 37} \[ -\frac{(x+1)^{12}}{21840 x^{12}}+\frac{(x+1)^{12}}{1820 x^{13}}-\frac{(x+1)^{12}}{280 x^{14}}+\frac{(x+1)^{12}}{60 x^{15}}-\frac{(x+1)^{12}}{16 x^{16}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^17,x]

[Out]

-(1 + x)^12/(16*x^16) + (1 + x)^12/(60*x^15) - (1 + x)^12/(280*x^14) + (1 + x)^12/(1820*x^13) - (1 + x)^12/(21
840*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^{17}} \, dx &=\int \frac{(1+x)^{11}}{x^{17}} \, dx\\ &=-\frac{(1+x)^{12}}{16 x^{16}}-\frac{1}{4} \int \frac{(1+x)^{11}}{x^{16}} \, dx\\ &=-\frac{(1+x)^{12}}{16 x^{16}}+\frac{(1+x)^{12}}{60 x^{15}}+\frac{1}{20} \int \frac{(1+x)^{11}}{x^{15}} \, dx\\ &=-\frac{(1+x)^{12}}{16 x^{16}}+\frac{(1+x)^{12}}{60 x^{15}}-\frac{(1+x)^{12}}{280 x^{14}}-\frac{1}{140} \int \frac{(1+x)^{11}}{x^{14}} \, dx\\ &=-\frac{(1+x)^{12}}{16 x^{16}}+\frac{(1+x)^{12}}{60 x^{15}}-\frac{(1+x)^{12}}{280 x^{14}}+\frac{(1+x)^{12}}{1820 x^{13}}+\frac{\int \frac{(1+x)^{11}}{x^{13}} \, dx}{1820}\\ &=-\frac{(1+x)^{12}}{16 x^{16}}+\frac{(1+x)^{12}}{60 x^{15}}-\frac{(1+x)^{12}}{280 x^{14}}+\frac{(1+x)^{12}}{1820 x^{13}}-\frac{(1+x)^{12}}{21840 x^{12}}\\ \end{align*}

Mathematica [A]  time = 0.0024011, size = 83, normalized size = 1.36 \[ -\frac{1}{5 x^5}-\frac{11}{6 x^6}-\frac{55}{7 x^7}-\frac{165}{8 x^8}-\frac{110}{3 x^9}-\frac{231}{5 x^{10}}-\frac{42}{x^{11}}-\frac{55}{2 x^{12}}-\frac{165}{13 x^{13}}-\frac{55}{14 x^{14}}-\frac{11}{15 x^{15}}-\frac{1}{16 x^{16}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^17,x]

[Out]

-1/(16*x^16) - 11/(15*x^15) - 55/(14*x^14) - 165/(13*x^13) - 55/(2*x^12) - 42/x^11 - 231/(5*x^10) - 110/(3*x^9
) - 165/(8*x^8) - 55/(7*x^7) - 11/(6*x^6) - 1/(5*x^5)

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Maple [A]  time = 0.007, size = 62, normalized size = 1. \begin{align*} -{\frac{110}{3\,{x}^{9}}}-42\,{x}^{-11}-{\frac{11}{15\,{x}^{15}}}-{\frac{165}{8\,{x}^{8}}}-{\frac{55}{7\,{x}^{7}}}-{\frac{231}{5\,{x}^{10}}}-{\frac{1}{5\,{x}^{5}}}-{\frac{55}{2\,{x}^{12}}}-{\frac{1}{16\,{x}^{16}}}-{\frac{55}{14\,{x}^{14}}}-{\frac{11}{6\,{x}^{6}}}-{\frac{165}{13\,{x}^{13}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)*(x^2+2*x+1)^5/x^17,x)

[Out]

-110/3/x^9-42/x^11-11/15/x^15-165/8/x^8-55/7/x^7-231/5/x^10-1/5/x^5-55/2/x^12-1/16/x^16-55/14/x^14-11/6/x^6-16
5/13/x^13

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Maxima [A]  time = 1.06031, size = 81, normalized size = 1.33 \begin{align*} -\frac{4368 \, x^{11} + 40040 \, x^{10} + 171600 \, x^{9} + 450450 \, x^{8} + 800800 \, x^{7} + 1009008 \, x^{6} + 917280 \, x^{5} + 600600 \, x^{4} + 277200 \, x^{3} + 85800 \, x^{2} + 16016 \, x + 1365}{21840 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^17,x, algorithm="maxima")

[Out]

-1/21840*(4368*x^11 + 40040*x^10 + 171600*x^9 + 450450*x^8 + 800800*x^7 + 1009008*x^6 + 917280*x^5 + 600600*x^
4 + 277200*x^3 + 85800*x^2 + 16016*x + 1365)/x^16

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Fricas [A]  time = 1.33908, size = 217, normalized size = 3.56 \begin{align*} -\frac{4368 \, x^{11} + 40040 \, x^{10} + 171600 \, x^{9} + 450450 \, x^{8} + 800800 \, x^{7} + 1009008 \, x^{6} + 917280 \, x^{5} + 600600 \, x^{4} + 277200 \, x^{3} + 85800 \, x^{2} + 16016 \, x + 1365}{21840 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^17,x, algorithm="fricas")

[Out]

-1/21840*(4368*x^11 + 40040*x^10 + 171600*x^9 + 450450*x^8 + 800800*x^7 + 1009008*x^6 + 917280*x^5 + 600600*x^
4 + 277200*x^3 + 85800*x^2 + 16016*x + 1365)/x^16

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Sympy [A]  time = 0.197808, size = 61, normalized size = 1. \begin{align*} - \frac{4368 x^{11} + 40040 x^{10} + 171600 x^{9} + 450450 x^{8} + 800800 x^{7} + 1009008 x^{6} + 917280 x^{5} + 600600 x^{4} + 277200 x^{3} + 85800 x^{2} + 16016 x + 1365}{21840 x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**17,x)

[Out]

-(4368*x**11 + 40040*x**10 + 171600*x**9 + 450450*x**8 + 800800*x**7 + 1009008*x**6 + 917280*x**5 + 600600*x**
4 + 277200*x**3 + 85800*x**2 + 16016*x + 1365)/(21840*x**16)

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Giac [A]  time = 1.10391, size = 81, normalized size = 1.33 \begin{align*} -\frac{4368 \, x^{11} + 40040 \, x^{10} + 171600 \, x^{9} + 450450 \, x^{8} + 800800 \, x^{7} + 1009008 \, x^{6} + 917280 \, x^{5} + 600600 \, x^{4} + 277200 \, x^{3} + 85800 \, x^{2} + 16016 \, x + 1365}{21840 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^17,x, algorithm="giac")

[Out]

-1/21840*(4368*x^11 + 40040*x^10 + 171600*x^9 + 450450*x^8 + 800800*x^7 + 1009008*x^6 + 917280*x^5 + 600600*x^
4 + 277200*x^3 + 85800*x^2 + 16016*x + 1365)/x^16