3.579 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^{13}} \, dx\)

Optimal. Leaf size=31 \[ \frac{(x+1)^{11} (d-12 e)}{132 x^{11}}-\frac{d (x+1)^{11}}{12 x^{12}} \]

[Out]

-(d*(1 + x)^11)/(12*x^12) + ((d - 12*e)*(1 + x)^11)/(132*x^11)

________________________________________________________________________________________

Rubi [A]  time = 0.0064312, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {27, 78, 37} \[ \frac{(x+1)^{11} (d-12 e)}{132 x^{11}}-\frac{d (x+1)^{11}}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^13,x]

[Out]

-(d*(1 + x)^11)/(12*x^12) + ((d - 12*e)*(1 + x)^11)/(132*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 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^{13}} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^{13}} \, dx\\ &=-\frac{d (1+x)^{11}}{12 x^{12}}-\frac{1}{12} (d-12 e) \int \frac{(1+x)^{10}}{x^{12}} \, dx\\ &=-\frac{d (1+x)^{11}}{12 x^{12}}+\frac{(d-12 e) (1+x)^{11}}{132 x^{11}}\\ \end{align*}

Mathematica [B]  time = 0.0239248, size = 114, normalized size = 3.68 \[ -\frac{d \left (66 x^{10}+440 x^9+1485 x^8+3168 x^7+4620 x^6+4752 x^5+3465 x^4+1760 x^3+594 x^2+120 x+11\right )+12 e x \left (11 x^{10}+55 x^9+165 x^8+330 x^7+462 x^6+462 x^5+330 x^4+165 x^3+55 x^2+11 x+1\right )}{132 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^13,x]

[Out]

-(12*e*x*(1 + 11*x + 55*x^2 + 165*x^3 + 330*x^4 + 462*x^5 + 462*x^6 + 330*x^7 + 165*x^8 + 55*x^9 + 11*x^10) +
d*(11 + 120*x + 594*x^2 + 1760*x^3 + 3465*x^4 + 4752*x^5 + 4620*x^6 + 3168*x^7 + 1485*x^8 + 440*x^9 + 66*x^10)
)/(132*x^12)

________________________________________________________________________________________

Maple [B]  time = 0.005, size = 130, normalized size = 4.2 \begin{align*} -{\frac{120\,d+45\,e}{9\,{x}^{9}}}-{\frac{45\,d+10\,e}{10\,{x}^{10}}}-{\frac{10\,d+45\,e}{3\,{x}^{3}}}-{\frac{210\,d+120\,e}{8\,{x}^{8}}}-{\frac{d+10\,e}{2\,{x}^{2}}}-{\frac{e}{x}}-{\frac{252\,d+210\,e}{7\,{x}^{7}}}-{\frac{210\,d+252\,e}{6\,{x}^{6}}}-{\frac{120\,d+210\,e}{5\,{x}^{5}}}-{\frac{45\,d+120\,e}{4\,{x}^{4}}}-{\frac{d}{12\,{x}^{12}}}-{\frac{10\,d+e}{11\,{x}^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^13,x)

[Out]

-1/9*(120*d+45*e)/x^9-1/10*(45*d+10*e)/x^10-1/3*(10*d+45*e)/x^3-1/8*(210*d+120*e)/x^8-1/2*(d+10*e)/x^2-e/x-1/7
*(252*d+210*e)/x^7-1/6*(210*d+252*e)/x^6-1/5*(120*d+210*e)/x^5-1/4*(45*d+120*e)/x^4-1/12*d/x^12-1/11*(10*d+e)/
x^11

________________________________________________________________________________________

Maxima [B]  time = 0.977487, size = 174, normalized size = 5.61 \begin{align*} -\frac{132 \, e x^{11} + 66 \,{\left (d + 10 \, e\right )} x^{10} + 220 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 495 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 792 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 924 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 792 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 495 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 220 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 66 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 12 \,{\left (10 \, d + e\right )} x + 11 \, d}{132 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^13,x, algorithm="maxima")

[Out]

-1/132*(132*e*x^11 + 66*(d + 10*e)*x^10 + 220*(2*d + 9*e)*x^9 + 495*(3*d + 8*e)*x^8 + 792*(4*d + 7*e)*x^7 + 92
4*(5*d + 6*e)*x^6 + 792*(6*d + 5*e)*x^5 + 495*(7*d + 4*e)*x^4 + 220*(8*d + 3*e)*x^3 + 66*(9*d + 2*e)*x^2 + 12*
(10*d + e)*x + 11*d)/x^12

________________________________________________________________________________________

Fricas [B]  time = 1.25642, size = 333, normalized size = 10.74 \begin{align*} -\frac{132 \, e x^{11} + 66 \,{\left (d + 10 \, e\right )} x^{10} + 220 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 495 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 792 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 924 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 792 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 495 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 220 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 66 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 12 \,{\left (10 \, d + e\right )} x + 11 \, d}{132 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^13,x, algorithm="fricas")

[Out]

-1/132*(132*e*x^11 + 66*(d + 10*e)*x^10 + 220*(2*d + 9*e)*x^9 + 495*(3*d + 8*e)*x^8 + 792*(4*d + 7*e)*x^7 + 92
4*(5*d + 6*e)*x^6 + 792*(6*d + 5*e)*x^5 + 495*(7*d + 4*e)*x^4 + 220*(8*d + 3*e)*x^3 + 66*(9*d + 2*e)*x^2 + 12*
(10*d + e)*x + 11*d)/x^12

________________________________________________________________________________________

Sympy [B]  time = 18.6296, size = 116, normalized size = 3.74 \begin{align*} - \frac{11 d + 132 e x^{11} + x^{10} \left (66 d + 660 e\right ) + x^{9} \left (440 d + 1980 e\right ) + x^{8} \left (1485 d + 3960 e\right ) + x^{7} \left (3168 d + 5544 e\right ) + x^{6} \left (4620 d + 5544 e\right ) + x^{5} \left (4752 d + 3960 e\right ) + x^{4} \left (3465 d + 1980 e\right ) + x^{3} \left (1760 d + 660 e\right ) + x^{2} \left (594 d + 132 e\right ) + x \left (120 d + 12 e\right )}{132 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**13,x)

[Out]

-(11*d + 132*e*x**11 + x**10*(66*d + 660*e) + x**9*(440*d + 1980*e) + x**8*(1485*d + 3960*e) + x**7*(3168*d +
5544*e) + x**6*(4620*d + 5544*e) + x**5*(4752*d + 3960*e) + x**4*(3465*d + 1980*e) + x**3*(1760*d + 660*e) + x
**2*(594*d + 132*e) + x*(120*d + 12*e))/(132*x**12)

________________________________________________________________________________________

Giac [B]  time = 1.15334, size = 192, normalized size = 6.19 \begin{align*} -\frac{132 \, x^{11} e + 66 \, d x^{10} + 660 \, x^{10} e + 440 \, d x^{9} + 1980 \, x^{9} e + 1485 \, d x^{8} + 3960 \, x^{8} e + 3168 \, d x^{7} + 5544 \, x^{7} e + 4620 \, d x^{6} + 5544 \, x^{6} e + 4752 \, d x^{5} + 3960 \, x^{5} e + 3465 \, d x^{4} + 1980 \, x^{4} e + 1760 \, d x^{3} + 660 \, x^{3} e + 594 \, d x^{2} + 132 \, x^{2} e + 120 \, d x + 12 \, x e + 11 \, d}{132 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^13,x, algorithm="giac")

[Out]

-1/132*(132*x^11*e + 66*d*x^10 + 660*x^10*e + 440*d*x^9 + 1980*x^9*e + 1485*d*x^8 + 3960*x^8*e + 3168*d*x^7 +
5544*x^7*e + 4620*d*x^6 + 5544*x^6*e + 4752*d*x^5 + 3960*x^5*e + 3465*d*x^4 + 1980*x^4*e + 1760*d*x^3 + 660*x^
3*e + 594*d*x^2 + 132*x^2*e + 120*d*x + 12*x*e + 11*d)/x^12