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

Optimal. Leaf size=71 \[ \frac{(x+1)^{11} (3 d-14 e)}{12012 x^{11}}-\frac{(x+1)^{11} (3 d-14 e)}{1092 x^{12}}+\frac{(x+1)^{11} (3 d-14 e)}{182 x^{13}}-\frac{d (x+1)^{11}}{14 x^{14}} \]

[Out]

-(d*(1 + x)^11)/(14*x^14) + ((3*d - 14*e)*(1 + x)^11)/(182*x^13) - ((3*d - 14*e)*(1 + x)^11)/(1092*x^12) + ((3
*d - 14*e)*(1 + x)^11)/(12012*x^11)

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Rubi [A]  time = 0.0171321, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {27, 78, 45, 37} \[ \frac{(x+1)^{11} (3 d-14 e)}{12012 x^{11}}-\frac{(x+1)^{11} (3 d-14 e)}{1092 x^{12}}+\frac{(x+1)^{11} (3 d-14 e)}{182 x^{13}}-\frac{d (x+1)^{11}}{14 x^{14}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(1 + x)^11)/(14*x^14) + ((3*d - 14*e)*(1 + x)^11)/(182*x^13) - ((3*d - 14*e)*(1 + x)^11)/(1092*x^12) + ((3
*d - 14*e)*(1 + x)^11)/(12012*x^11)

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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{(d+e x) \left (1+2 x+x^2\right )^5}{x^{15}} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^{15}} \, dx\\ &=-\frac{d (1+x)^{11}}{14 x^{14}}-\frac{1}{14} (3 d-14 e) \int \frac{(1+x)^{10}}{x^{14}} \, dx\\ &=-\frac{d (1+x)^{11}}{14 x^{14}}+\frac{(3 d-14 e) (1+x)^{11}}{182 x^{13}}-\frac{1}{91} (-3 d+14 e) \int \frac{(1+x)^{10}}{x^{13}} \, dx\\ &=-\frac{d (1+x)^{11}}{14 x^{14}}+\frac{(3 d-14 e) (1+x)^{11}}{182 x^{13}}-\frac{(3 d-14 e) (1+x)^{11}}{1092 x^{12}}-\frac{(3 d-14 e) \int \frac{(1+x)^{10}}{x^{12}} \, dx}{1092}\\ &=-\frac{d (1+x)^{11}}{14 x^{14}}+\frac{(3 d-14 e) (1+x)^{11}}{182 x^{13}}-\frac{(3 d-14 e) (1+x)^{11}}{1092 x^{12}}+\frac{(3 d-14 e) (1+x)^{11}}{12012 x^{11}}\\ \end{align*}

Mathematica [B]  time = 0.0359747, size = 149, normalized size = 2.1 \[ -\frac{d+10 e}{4 x^4}-\frac{2 d+9 e}{x^5}-\frac{5 (3 d+8 e)}{2 x^6}-\frac{30 (4 d+7 e)}{7 x^7}-\frac{21 (5 d+6 e)}{4 x^8}-\frac{14 (6 d+5 e)}{3 x^9}-\frac{3 (7 d+4 e)}{x^{10}}-\frac{15 (8 d+3 e)}{11 x^{11}}-\frac{5 (9 d+2 e)}{12 x^{12}}-\frac{10 d+e}{13 x^{13}}-\frac{d}{14 x^{14}}-\frac{e}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-d/(14*x^14) - (10*d + e)/(13*x^13) - (5*(9*d + 2*e))/(12*x^12) - (15*(8*d + 3*e))/(11*x^11) - (3*(7*d + 4*e))
/x^10 - (14*(6*d + 5*e))/(3*x^9) - (21*(5*d + 6*e))/(4*x^8) - (30*(4*d + 7*e))/(7*x^7) - (5*(3*d + 8*e))/(2*x^
6) - (2*d + 9*e)/x^5 - (d + 10*e)/(4*x^4) - e/(3*x^3)

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Maple [B]  time = 0.005, size = 130, normalized size = 1.8 \begin{align*} -{\frac{252\,d+210\,e}{9\,{x}^{9}}}-{\frac{45\,d+120\,e}{6\,{x}^{6}}}-{\frac{10\,d+e}{13\,{x}^{13}}}-{\frac{e}{3\,{x}^{3}}}-{\frac{210\,d+252\,e}{8\,{x}^{8}}}-{\frac{120\,d+210\,e}{7\,{x}^{7}}}-{\frac{210\,d+120\,e}{10\,{x}^{10}}}-{\frac{d+10\,e}{4\,{x}^{4}}}-{\frac{120\,d+45\,e}{11\,{x}^{11}}}-{\frac{d}{14\,{x}^{14}}}-{\frac{10\,d+45\,e}{5\,{x}^{5}}}-{\frac{45\,d+10\,e}{12\,{x}^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^15,x)

[Out]

-1/9*(252*d+210*e)/x^9-1/6*(45*d+120*e)/x^6-1/13*(10*d+e)/x^13-1/3*e/x^3-1/8*(210*d+252*e)/x^8-1/7*(120*d+210*
e)/x^7-1/10*(210*d+120*e)/x^10-1/4*(d+10*e)/x^4-1/11*(120*d+45*e)/x^11-1/14*d/x^14-1/5*(10*d+45*e)/x^5-1/12*(4
5*d+10*e)/x^12

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Maxima [B]  time = 1.0282, size = 174, normalized size = 2.45 \begin{align*} -\frac{4004 \, e x^{11} + 3003 \,{\left (d + 10 \, e\right )} x^{10} + 12012 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 30030 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 51480 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 63063 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 56056 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 36036 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 16380 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 5005 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 924 \,{\left (10 \, d + e\right )} x + 858 \, d}{12012 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12012*(4004*e*x^11 + 3003*(d + 10*e)*x^10 + 12012*(2*d + 9*e)*x^9 + 30030*(3*d + 8*e)*x^8 + 51480*(4*d + 7*
e)*x^7 + 63063*(5*d + 6*e)*x^6 + 56056*(6*d + 5*e)*x^5 + 36036*(7*d + 4*e)*x^4 + 16380*(8*d + 3*e)*x^3 + 5005*
(9*d + 2*e)*x^2 + 924*(10*d + e)*x + 858*d)/x^14

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Fricas [B]  time = 1.17564, size = 365, normalized size = 5.14 \begin{align*} -\frac{4004 \, e x^{11} + 3003 \,{\left (d + 10 \, e\right )} x^{10} + 12012 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 30030 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 51480 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 63063 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 56056 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 36036 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 16380 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 5005 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 924 \,{\left (10 \, d + e\right )} x + 858 \, d}{12012 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12012*(4004*e*x^11 + 3003*(d + 10*e)*x^10 + 12012*(2*d + 9*e)*x^9 + 30030*(3*d + 8*e)*x^8 + 51480*(4*d + 7*
e)*x^7 + 63063*(5*d + 6*e)*x^6 + 56056*(6*d + 5*e)*x^5 + 36036*(7*d + 4*e)*x^4 + 16380*(8*d + 3*e)*x^3 + 5005*
(9*d + 2*e)*x^2 + 924*(10*d + e)*x + 858*d)/x^14

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Sympy [A]  time = 18.6148, size = 116, normalized size = 1.63 \begin{align*} - \frac{858 d + 4004 e x^{11} + x^{10} \left (3003 d + 30030 e\right ) + x^{9} \left (24024 d + 108108 e\right ) + x^{8} \left (90090 d + 240240 e\right ) + x^{7} \left (205920 d + 360360 e\right ) + x^{6} \left (315315 d + 378378 e\right ) + x^{5} \left (336336 d + 280280 e\right ) + x^{4} \left (252252 d + 144144 e\right ) + x^{3} \left (131040 d + 49140 e\right ) + x^{2} \left (45045 d + 10010 e\right ) + x \left (9240 d + 924 e\right )}{12012 x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(858*d + 4004*e*x**11 + x**10*(3003*d + 30030*e) + x**9*(24024*d + 108108*e) + x**8*(90090*d + 240240*e) + x*
*7*(205920*d + 360360*e) + x**6*(315315*d + 378378*e) + x**5*(336336*d + 280280*e) + x**4*(252252*d + 144144*e
) + x**3*(131040*d + 49140*e) + x**2*(45045*d + 10010*e) + x*(9240*d + 924*e))/(12012*x**14)

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Giac [B]  time = 1.18837, size = 192, normalized size = 2.7 \begin{align*} -\frac{4004 \, x^{11} e + 3003 \, d x^{10} + 30030 \, x^{10} e + 24024 \, d x^{9} + 108108 \, x^{9} e + 90090 \, d x^{8} + 240240 \, x^{8} e + 205920 \, d x^{7} + 360360 \, x^{7} e + 315315 \, d x^{6} + 378378 \, x^{6} e + 336336 \, d x^{5} + 280280 \, x^{5} e + 252252 \, d x^{4} + 144144 \, x^{4} e + 131040 \, d x^{3} + 49140 \, x^{3} e + 45045 \, d x^{2} + 10010 \, x^{2} e + 9240 \, d x + 924 \, x e + 858 \, d}{12012 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12012*(4004*x^11*e + 3003*d*x^10 + 30030*x^10*e + 24024*d*x^9 + 108108*x^9*e + 90090*d*x^8 + 240240*x^8*e +
205920*d*x^7 + 360360*x^7*e + 315315*d*x^6 + 378378*x^6*e + 336336*d*x^5 + 280280*x^5*e + 252252*d*x^4 + 1441
44*x^4*e + 131040*d*x^3 + 49140*x^3*e + 45045*d*x^2 + 10010*x^2*e + 9240*d*x + 924*x*e + 858*d)/x^14