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3.586 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^{20}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{d+10 e}{9 x^9}-\frac{2 d+9 e}{2 x^{10}}-\frac{15 (3 d+8 e)}{11 x^{11}}-\frac{5 (4 d+7 e)}{2 x^{12}}-\frac{42 (5 d+6 e)}{13 x^{13}}-\frac{3 (6 d+5 e)}{x^{14}}-\frac{2 (7 d+4 e)}{x^{15}}-\frac{15 (8 d+3 e)}{16 x^{16}}-\frac{5 (9 d+2 e)}{17 x^{17}}-\frac{10 d+e}{18 x^{18}}-\frac{d}{19 x^{19}}-\frac{e}{8 x^8} \]

[Out]

-d/(19*x^19) - (10*d + e)/(18*x^18) - (5*(9*d + 2*e))/(17*x^17) - (15*(8*d + 3*e))/(16*x^16) - (2*(7*d + 4*e))
/x^15 - (3*(6*d + 5*e))/x^14 - (42*(5*d + 6*e))/(13*x^13) - (5*(4*d + 7*e))/(2*x^12) - (15*(3*d + 8*e))/(11*x^
11) - (2*d + 9*e)/(2*x^10) - (d + 10*e)/(9*x^9) - e/(8*x^8)

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Rubi [A]  time = 0.0664167, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {27, 76} \[ -\frac{d+10 e}{9 x^9}-\frac{2 d+9 e}{2 x^{10}}-\frac{15 (3 d+8 e)}{11 x^{11}}-\frac{5 (4 d+7 e)}{2 x^{12}}-\frac{42 (5 d+6 e)}{13 x^{13}}-\frac{3 (6 d+5 e)}{x^{14}}-\frac{2 (7 d+4 e)}{x^{15}}-\frac{15 (8 d+3 e)}{16 x^{16}}-\frac{5 (9 d+2 e)}{17 x^{17}}-\frac{10 d+e}{18 x^{18}}-\frac{d}{19 x^{19}}-\frac{e}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^20,x]

[Out]

-d/(19*x^19) - (10*d + e)/(18*x^18) - (5*(9*d + 2*e))/(17*x^17) - (15*(8*d + 3*e))/(16*x^16) - (2*(7*d + 4*e))
/x^15 - (3*(6*d + 5*e))/x^14 - (42*(5*d + 6*e))/(13*x^13) - (5*(4*d + 7*e))/(2*x^12) - (15*(3*d + 8*e))/(11*x^
11) - (2*d + 9*e)/(2*x^10) - (d + 10*e)/(9*x^9) - e/(8*x^8)

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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{(d+e x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^{20}} \, dx\\ &=\int \left (\frac{d}{x^{20}}+\frac{10 d+e}{x^{19}}+\frac{5 (9 d+2 e)}{x^{18}}+\frac{15 (8 d+3 e)}{x^{17}}+\frac{30 (7 d+4 e)}{x^{16}}+\frac{42 (6 d+5 e)}{x^{15}}+\frac{42 (5 d+6 e)}{x^{14}}+\frac{30 (4 d+7 e)}{x^{13}}+\frac{15 (3 d+8 e)}{x^{12}}+\frac{5 (2 d+9 e)}{x^{11}}+\frac{d+10 e}{x^{10}}+\frac{e}{x^9}\right ) \, dx\\ &=-\frac{d}{19 x^{19}}-\frac{10 d+e}{18 x^{18}}-\frac{5 (9 d+2 e)}{17 x^{17}}-\frac{15 (8 d+3 e)}{16 x^{16}}-\frac{2 (7 d+4 e)}{x^{15}}-\frac{3 (6 d+5 e)}{x^{14}}-\frac{42 (5 d+6 e)}{13 x^{13}}-\frac{5 (4 d+7 e)}{2 x^{12}}-\frac{15 (3 d+8 e)}{11 x^{11}}-\frac{2 d+9 e}{2 x^{10}}-\frac{d+10 e}{9 x^9}-\frac{e}{8 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0401039, size = 149, normalized size = 1. \[ -\frac{d+10 e}{9 x^9}-\frac{2 d+9 e}{2 x^{10}}-\frac{15 (3 d+8 e)}{11 x^{11}}-\frac{5 (4 d+7 e)}{2 x^{12}}-\frac{42 (5 d+6 e)}{13 x^{13}}-\frac{3 (6 d+5 e)}{x^{14}}-\frac{2 (7 d+4 e)}{x^{15}}-\frac{15 (8 d+3 e)}{16 x^{16}}-\frac{5 (9 d+2 e)}{17 x^{17}}-\frac{10 d+e}{18 x^{18}}-\frac{d}{19 x^{19}}-\frac{e}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^20,x]

[Out]

-d/(19*x^19) - (10*d + e)/(18*x^18) - (5*(9*d + 2*e))/(17*x^17) - (15*(8*d + 3*e))/(16*x^16) - (2*(7*d + 4*e))
/x^15 - (3*(6*d + 5*e))/x^14 - (42*(5*d + 6*e))/(13*x^13) - (5*(4*d + 7*e))/(2*x^12) - (15*(3*d + 8*e))/(11*x^
11) - (2*d + 9*e)/(2*x^10) - (d + 10*e)/(9*x^9) - e/(8*x^8)

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Maple [A]  time = 0.007, size = 130, normalized size = 0.9 \begin{align*} -{\frac{d+10\,e}{9\,{x}^{9}}}-{\frac{10\,d+e}{18\,{x}^{18}}}-{\frac{45\,d+10\,e}{17\,{x}^{17}}}-{\frac{d}{19\,{x}^{19}}}-{\frac{e}{8\,{x}^{8}}}-{\frac{120\,d+45\,e}{16\,{x}^{16}}}-{\frac{10\,d+45\,e}{10\,{x}^{10}}}-{\frac{210\,d+120\,e}{15\,{x}^{15}}}-{\frac{210\,d+252\,e}{13\,{x}^{13}}}-{\frac{45\,d+120\,e}{11\,{x}^{11}}}-{\frac{252\,d+210\,e}{14\,{x}^{14}}}-{\frac{120\,d+210\,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^20,x)

[Out]

-1/9*(d+10*e)/x^9-1/18*(10*d+e)/x^18-1/17*(45*d+10*e)/x^17-1/19*d/x^19-1/8*e/x^8-1/16*(120*d+45*e)/x^16-1/10*(
10*d+45*e)/x^10-1/15*(210*d+120*e)/x^15-1/13*(210*d+252*e)/x^13-1/11*(45*d+120*e)/x^11-1/14*(252*d+210*e)/x^14
-1/12*(120*d+210*e)/x^12

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Maxima [A]  time = 1.01144, size = 174, normalized size = 1.17 \begin{align*} -\frac{831402 \, e x^{11} + 739024 \,{\left (d + 10 \, e\right )} x^{10} + 3325608 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 9069840 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 16628040 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 21488544 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 19953648 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 13302432 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 6235515 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 1956240 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 369512 \,{\left (10 \, d + e\right )} x + 350064 \, d}{6651216 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^20,x, algorithm="maxima")

[Out]

-1/6651216*(831402*e*x^11 + 739024*(d + 10*e)*x^10 + 3325608*(2*d + 9*e)*x^9 + 9069840*(3*d + 8*e)*x^8 + 16628
040*(4*d + 7*e)*x^7 + 21488544*(5*d + 6*e)*x^6 + 19953648*(6*d + 5*e)*x^5 + 13302432*(7*d + 4*e)*x^4 + 6235515
*(8*d + 3*e)*x^3 + 1956240*(9*d + 2*e)*x^2 + 369512*(10*d + e)*x + 350064*d)/x^19

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Fricas [A]  time = 1.22833, size = 409, normalized size = 2.74 \begin{align*} -\frac{831402 \, e x^{11} + 739024 \,{\left (d + 10 \, e\right )} x^{10} + 3325608 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 9069840 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 16628040 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 21488544 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 19953648 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 13302432 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 6235515 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 1956240 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 369512 \,{\left (10 \, d + e\right )} x + 350064 \, d}{6651216 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^20,x, algorithm="fricas")

[Out]

-1/6651216*(831402*e*x^11 + 739024*(d + 10*e)*x^10 + 3325608*(2*d + 9*e)*x^9 + 9069840*(3*d + 8*e)*x^8 + 16628
040*(4*d + 7*e)*x^7 + 21488544*(5*d + 6*e)*x^6 + 19953648*(6*d + 5*e)*x^5 + 13302432*(7*d + 4*e)*x^4 + 6235515
*(8*d + 3*e)*x^3 + 1956240*(9*d + 2*e)*x^2 + 369512*(10*d + e)*x + 350064*d)/x^19

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Sympy [A]  time = 28.0435, size = 116, normalized size = 0.78 \begin{align*} - \frac{350064 d + 831402 e x^{11} + x^{10} \left (739024 d + 7390240 e\right ) + x^{9} \left (6651216 d + 29930472 e\right ) + x^{8} \left (27209520 d + 72558720 e\right ) + x^{7} \left (66512160 d + 116396280 e\right ) + x^{6} \left (107442720 d + 128931264 e\right ) + x^{5} \left (119721888 d + 99768240 e\right ) + x^{4} \left (93117024 d + 53209728 e\right ) + x^{3} \left (49884120 d + 18706545 e\right ) + x^{2} \left (17606160 d + 3912480 e\right ) + x \left (3695120 d + 369512 e\right )}{6651216 x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**20,x)

[Out]

-(350064*d + 831402*e*x**11 + x**10*(739024*d + 7390240*e) + x**9*(6651216*d + 29930472*e) + x**8*(27209520*d
+ 72558720*e) + x**7*(66512160*d + 116396280*e) + x**6*(107442720*d + 128931264*e) + x**5*(119721888*d + 99768
240*e) + x**4*(93117024*d + 53209728*e) + x**3*(49884120*d + 18706545*e) + x**2*(17606160*d + 3912480*e) + x*(
3695120*d + 369512*e))/(6651216*x**19)

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Giac [A]  time = 1.16663, size = 192, normalized size = 1.29 \begin{align*} -\frac{831402 \, x^{11} e + 739024 \, d x^{10} + 7390240 \, x^{10} e + 6651216 \, d x^{9} + 29930472 \, x^{9} e + 27209520 \, d x^{8} + 72558720 \, x^{8} e + 66512160 \, d x^{7} + 116396280 \, x^{7} e + 107442720 \, d x^{6} + 128931264 \, x^{6} e + 119721888 \, d x^{5} + 99768240 \, x^{5} e + 93117024 \, d x^{4} + 53209728 \, x^{4} e + 49884120 \, d x^{3} + 18706545 \, x^{3} e + 17606160 \, d x^{2} + 3912480 \, x^{2} e + 3695120 \, d x + 369512 \, x e + 350064 \, d}{6651216 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^20,x, algorithm="giac")

[Out]

-1/6651216*(831402*x^11*e + 739024*d*x^10 + 7390240*x^10*e + 6651216*d*x^9 + 29930472*x^9*e + 27209520*d*x^8 +
 72558720*x^8*e + 66512160*d*x^7 + 116396280*x^7*e + 107442720*d*x^6 + 128931264*x^6*e + 119721888*d*x^5 + 997
68240*x^5*e + 93117024*d*x^4 + 53209728*x^4*e + 49884120*d*x^3 + 18706545*x^3*e + 17606160*d*x^2 + 3912480*x^2
*e + 3695120*d*x + 369512*x*e + 350064*d)/x^19

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