3.614 \(\int \frac{(1+x) (1+2 x+x^2)^5}{x^{15}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{(x+1)^{12}}{1092 x^{12}}+\frac{(x+1)^{12}}{91 x^{13}}-\frac{(x+1)^{12}}{14 x^{14}} \]

[Out]

-(1 + x)^12/(14*x^14) + (1 + x)^12/(91*x^13) - (1 + x)^12/(1092*x^12)

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Rubi [A]  time = 0.0052638, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {27, 45, 37} \[ -\frac{(x+1)^{12}}{1092 x^{12}}+\frac{(x+1)^{12}}{91 x^{13}}-\frac{(x+1)^{12}}{14 x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^15,x]

[Out]

-(1 + x)^12/(14*x^14) + (1 + x)^12/(91*x^13) - (1 + x)^12/(1092*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^{15}} \, dx &=\int \frac{(1+x)^{11}}{x^{15}} \, dx\\ &=-\frac{(1+x)^{12}}{14 x^{14}}-\frac{1}{7} \int \frac{(1+x)^{11}}{x^{14}} \, dx\\ &=-\frac{(1+x)^{12}}{14 x^{14}}+\frac{(1+x)^{12}}{91 x^{13}}+\frac{1}{91} \int \frac{(1+x)^{11}}{x^{13}} \, dx\\ &=-\frac{(1+x)^{12}}{14 x^{14}}+\frac{(1+x)^{12}}{91 x^{13}}-\frac{(1+x)^{12}}{1092 x^{12}}\\ \end{align*}

Mathematica [B]  time = 0.0022057, size = 79, normalized size = 2.14 \[ -\frac{1}{3 x^3}-\frac{11}{4 x^4}-\frac{11}{x^5}-\frac{55}{2 x^6}-\frac{330}{7 x^7}-\frac{231}{4 x^8}-\frac{154}{3 x^9}-\frac{33}{x^{10}}-\frac{15}{x^{11}}-\frac{55}{12 x^{12}}-\frac{11}{13 x^{13}}-\frac{1}{14 x^{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^15,x]

[Out]

-1/(14*x^14) - 11/(13*x^13) - 55/(12*x^12) - 15/x^11 - 33/x^10 - 154/(3*x^9) - 231/(4*x^8) - 330/(7*x^7) - 55/
(2*x^6) - 11/x^5 - 11/(4*x^4) - 1/(3*x^3)

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Maple [A]  time = 0.007, size = 62, normalized size = 1.7 \begin{align*} -{\frac{154}{3\,{x}^{9}}}-{\frac{1}{3\,{x}^{3}}}-{\frac{55}{12\,{x}^{12}}}-{\frac{231}{4\,{x}^{8}}}-{\frac{330}{7\,{x}^{7}}}-15\,{x}^{-11}-33\,{x}^{-10}-11\,{x}^{-5}-{\frac{11}{4\,{x}^{4}}}-{\frac{1}{14\,{x}^{14}}}-{\frac{55}{2\,{x}^{6}}}-{\frac{11}{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^15,x)

[Out]

-154/3/x^9-1/3/x^3-55/12/x^12-231/4/x^8-330/7/x^7-15/x^11-33/x^10-11/x^5-11/4/x^4-1/14/x^14-55/2/x^6-11/13/x^1
3

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Maxima [A]  time = 1.00065, size = 81, normalized size = 2.19 \begin{align*} -\frac{364 \, x^{11} + 3003 \, x^{10} + 12012 \, x^{9} + 30030 \, x^{8} + 51480 \, x^{7} + 63063 \, x^{6} + 56056 \, x^{5} + 36036 \, x^{4} + 16380 \, x^{3} + 5005 \, x^{2} + 924 \, x + 78}{1092 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^15,x, algorithm="maxima")

[Out]

-1/1092*(364*x^11 + 3003*x^10 + 12012*x^9 + 30030*x^8 + 51480*x^7 + 63063*x^6 + 56056*x^5 + 36036*x^4 + 16380*
x^3 + 5005*x^2 + 924*x + 78)/x^14

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Fricas [A]  time = 1.19419, size = 196, normalized size = 5.3 \begin{align*} -\frac{364 \, x^{11} + 3003 \, x^{10} + 12012 \, x^{9} + 30030 \, x^{8} + 51480 \, x^{7} + 63063 \, x^{6} + 56056 \, x^{5} + 36036 \, x^{4} + 16380 \, x^{3} + 5005 \, x^{2} + 924 \, x + 78}{1092 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^15,x, algorithm="fricas")

[Out]

-1/1092*(364*x^11 + 3003*x^10 + 12012*x^9 + 30030*x^8 + 51480*x^7 + 63063*x^6 + 56056*x^5 + 36036*x^4 + 16380*
x^3 + 5005*x^2 + 924*x + 78)/x^14

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Sympy [B]  time = 0.196712, size = 61, normalized size = 1.65 \begin{align*} - \frac{364 x^{11} + 3003 x^{10} + 12012 x^{9} + 30030 x^{8} + 51480 x^{7} + 63063 x^{6} + 56056 x^{5} + 36036 x^{4} + 16380 x^{3} + 5005 x^{2} + 924 x + 78}{1092 x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**15,x)

[Out]

-(364*x**11 + 3003*x**10 + 12012*x**9 + 30030*x**8 + 51480*x**7 + 63063*x**6 + 56056*x**5 + 36036*x**4 + 16380
*x**3 + 5005*x**2 + 924*x + 78)/(1092*x**14)

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Giac [A]  time = 1.1181, size = 81, normalized size = 2.19 \begin{align*} -\frac{364 \, x^{11} + 3003 \, x^{10} + 12012 \, x^{9} + 30030 \, x^{8} + 51480 \, x^{7} + 63063 \, x^{6} + 56056 \, x^{5} + 36036 \, x^{4} + 16380 \, x^{3} + 5005 \, x^{2} + 924 \, x + 78}{1092 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^15,x, algorithm="giac")

[Out]

-1/1092*(364*x^11 + 3003*x^10 + 12012*x^9 + 30030*x^8 + 51480*x^7 + 63063*x^6 + 56056*x^5 + 36036*x^4 + 16380*
x^3 + 5005*x^2 + 924*x + 78)/x^14