3.587 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^{21}} \, dx\)

Optimal. Leaf size=151 \[ -\frac{d+10 e}{10 x^{10}}-\frac{5 (2 d+9 e)}{11 x^{11}}-\frac{5 (3 d+8 e)}{4 x^{12}}-\frac{30 (4 d+7 e)}{13 x^{13}}-\frac{3 (5 d+6 e)}{x^{14}}-\frac{14 (6 d+5 e)}{5 x^{15}}-\frac{15 (7 d+4 e)}{8 x^{16}}-\frac{15 (8 d+3 e)}{17 x^{17}}-\frac{5 (9 d+2 e)}{18 x^{18}}-\frac{10 d+e}{19 x^{19}}-\frac{d}{20 x^{20}}-\frac{e}{9 x^9} \]

[Out]

-d/(20*x^20) - (10*d + e)/(19*x^19) - (5*(9*d + 2*e))/(18*x^18) - (15*(8*d + 3*e))/(17*x^17) - (15*(7*d + 4*e)
)/(8*x^16) - (14*(6*d + 5*e))/(5*x^15) - (3*(5*d + 6*e))/x^14 - (30*(4*d + 7*e))/(13*x^13) - (5*(3*d + 8*e))/(
4*x^12) - (5*(2*d + 9*e))/(11*x^11) - (d + 10*e)/(10*x^10) - e/(9*x^9)

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Rubi [A]  time = 0.0642711, antiderivative size = 151, 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} \[ -\frac{d+10 e}{10 x^{10}}-\frac{5 (2 d+9 e)}{11 x^{11}}-\frac{5 (3 d+8 e)}{4 x^{12}}-\frac{30 (4 d+7 e)}{13 x^{13}}-\frac{3 (5 d+6 e)}{x^{14}}-\frac{14 (6 d+5 e)}{5 x^{15}}-\frac{15 (7 d+4 e)}{8 x^{16}}-\frac{15 (8 d+3 e)}{17 x^{17}}-\frac{5 (9 d+2 e)}{18 x^{18}}-\frac{10 d+e}{19 x^{19}}-\frac{d}{20 x^{20}}-\frac{e}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^21,x]

[Out]

-d/(20*x^20) - (10*d + e)/(19*x^19) - (5*(9*d + 2*e))/(18*x^18) - (15*(8*d + 3*e))/(17*x^17) - (15*(7*d + 4*e)
)/(8*x^16) - (14*(6*d + 5*e))/(5*x^15) - (3*(5*d + 6*e))/x^14 - (30*(4*d + 7*e))/(13*x^13) - (5*(3*d + 8*e))/(
4*x^12) - (5*(2*d + 9*e))/(11*x^11) - (d + 10*e)/(10*x^10) - e/(9*x^9)

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^{21}} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^{21}} \, dx\\ &=\int \left (\frac{d}{x^{21}}+\frac{10 d+e}{x^{20}}+\frac{5 (9 d+2 e)}{x^{19}}+\frac{15 (8 d+3 e)}{x^{18}}+\frac{30 (7 d+4 e)}{x^{17}}+\frac{42 (6 d+5 e)}{x^{16}}+\frac{42 (5 d+6 e)}{x^{15}}+\frac{30 (4 d+7 e)}{x^{14}}+\frac{15 (3 d+8 e)}{x^{13}}+\frac{5 (2 d+9 e)}{x^{12}}+\frac{d+10 e}{x^{11}}+\frac{e}{x^{10}}\right ) \, dx\\ &=-\frac{d}{20 x^{20}}-\frac{10 d+e}{19 x^{19}}-\frac{5 (9 d+2 e)}{18 x^{18}}-\frac{15 (8 d+3 e)}{17 x^{17}}-\frac{15 (7 d+4 e)}{8 x^{16}}-\frac{14 (6 d+5 e)}{5 x^{15}}-\frac{3 (5 d+6 e)}{x^{14}}-\frac{30 (4 d+7 e)}{13 x^{13}}-\frac{5 (3 d+8 e)}{4 x^{12}}-\frac{5 (2 d+9 e)}{11 x^{11}}-\frac{d+10 e}{10 x^{10}}-\frac{e}{9 x^9}\\ \end{align*}

Mathematica [A]  time = 0.0472311, size = 151, normalized size = 1. \[ -\frac{d+10 e}{10 x^{10}}-\frac{5 (2 d+9 e)}{11 x^{11}}-\frac{5 (3 d+8 e)}{4 x^{12}}-\frac{30 (4 d+7 e)}{13 x^{13}}-\frac{3 (5 d+6 e)}{x^{14}}-\frac{14 (6 d+5 e)}{5 x^{15}}-\frac{15 (7 d+4 e)}{8 x^{16}}-\frac{15 (8 d+3 e)}{17 x^{17}}-\frac{5 (9 d+2 e)}{18 x^{18}}-\frac{10 d+e}{19 x^{19}}-\frac{d}{20 x^{20}}-\frac{e}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^21,x]

[Out]

-d/(20*x^20) - (10*d + e)/(19*x^19) - (5*(9*d + 2*e))/(18*x^18) - (15*(8*d + 3*e))/(17*x^17) - (15*(7*d + 4*e)
)/(8*x^16) - (14*(6*d + 5*e))/(5*x^15) - (3*(5*d + 6*e))/x^14 - (30*(4*d + 7*e))/(13*x^13) - (5*(3*d + 8*e))/(
4*x^12) - (5*(2*d + 9*e))/(11*x^11) - (d + 10*e)/(10*x^10) - e/(9*x^9)

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Maple [A]  time = 0.006, size = 130, normalized size = 0.9 \begin{align*} -{\frac{e}{9\,{x}^{9}}}-{\frac{120\,d+210\,e}{13\,{x}^{13}}}-{\frac{d+10\,e}{10\,{x}^{10}}}-{\frac{210\,d+120\,e}{16\,{x}^{16}}}-{\frac{120\,d+45\,e}{17\,{x}^{17}}}-{\frac{252\,d+210\,e}{15\,{x}^{15}}}-{\frac{d}{20\,{x}^{20}}}-{\frac{210\,d+252\,e}{14\,{x}^{14}}}-{\frac{45\,d+10\,e}{18\,{x}^{18}}}-{\frac{45\,d+120\,e}{12\,{x}^{12}}}-{\frac{10\,d+e}{19\,{x}^{19}}}-{\frac{10\,d+45\,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^21,x)

[Out]

-1/9*e/x^9-1/13*(120*d+210*e)/x^13-1/10*(d+10*e)/x^10-1/16*(210*d+120*e)/x^16-1/17*(120*d+45*e)/x^17-1/15*(252
*d+210*e)/x^15-1/20*d/x^20-1/14*(210*d+252*e)/x^14-1/18*(45*d+10*e)/x^18-1/12*(45*d+120*e)/x^12-1/19*(10*d+e)/
x^19-1/11*(10*d+45*e)/x^11

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Maxima [A]  time = 0.967287, size = 174, normalized size = 1.15 \begin{align*} -\frac{1847560 \, e x^{11} + 1662804 \,{\left (d + 10 \, e\right )} x^{10} + 7558200 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 20785050 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 38372400 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 49884120 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 46558512 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 31177575 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 14671800 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 4618900 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 875160 \,{\left (10 \, d + e\right )} x + 831402 \, d}{16628040 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^21,x, algorithm="maxima")

[Out]

-1/16628040*(1847560*e*x^11 + 1662804*(d + 10*e)*x^10 + 7558200*(2*d + 9*e)*x^9 + 20785050*(3*d + 8*e)*x^8 + 3
8372400*(4*d + 7*e)*x^7 + 49884120*(5*d + 6*e)*x^6 + 46558512*(6*d + 5*e)*x^5 + 31177575*(7*d + 4*e)*x^4 + 146
71800*(8*d + 3*e)*x^3 + 4618900*(9*d + 2*e)*x^2 + 875160*(10*d + e)*x + 831402*d)/x^20

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Fricas [A]  time = 1.18992, size = 416, normalized size = 2.75 \begin{align*} -\frac{1847560 \, e x^{11} + 1662804 \,{\left (d + 10 \, e\right )} x^{10} + 7558200 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 20785050 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 38372400 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 49884120 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 46558512 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 31177575 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 14671800 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 4618900 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 875160 \,{\left (10 \, d + e\right )} x + 831402 \, d}{16628040 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^21,x, algorithm="fricas")

[Out]

-1/16628040*(1847560*e*x^11 + 1662804*(d + 10*e)*x^10 + 7558200*(2*d + 9*e)*x^9 + 20785050*(3*d + 8*e)*x^8 + 3
8372400*(4*d + 7*e)*x^7 + 49884120*(5*d + 6*e)*x^6 + 46558512*(6*d + 5*e)*x^5 + 31177575*(7*d + 4*e)*x^4 + 146
71800*(8*d + 3*e)*x^3 + 4618900*(9*d + 2*e)*x^2 + 875160*(10*d + e)*x + 831402*d)/x^20

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Sympy [A]  time = 32.0442, size = 116, normalized size = 0.77 \begin{align*} - \frac{831402 d + 1847560 e x^{11} + x^{10} \left (1662804 d + 16628040 e\right ) + x^{9} \left (15116400 d + 68023800 e\right ) + x^{8} \left (62355150 d + 166280400 e\right ) + x^{7} \left (153489600 d + 268606800 e\right ) + x^{6} \left (249420600 d + 299304720 e\right ) + x^{5} \left (279351072 d + 232792560 e\right ) + x^{4} \left (218243025 d + 124710300 e\right ) + x^{3} \left (117374400 d + 44015400 e\right ) + x^{2} \left (41570100 d + 9237800 e\right ) + x \left (8751600 d + 875160 e\right )}{16628040 x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**21,x)

[Out]

-(831402*d + 1847560*e*x**11 + x**10*(1662804*d + 16628040*e) + x**9*(15116400*d + 68023800*e) + x**8*(6235515
0*d + 166280400*e) + x**7*(153489600*d + 268606800*e) + x**6*(249420600*d + 299304720*e) + x**5*(279351072*d +
232792560*e) + x**4*(218243025*d + 124710300*e) + x**3*(117374400*d + 44015400*e) + x**2*(41570100*d + 923780
0*e) + x*(8751600*d + 875160*e))/(16628040*x**20)

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Giac [A]  time = 1.13512, size = 192, normalized size = 1.27 \begin{align*} -\frac{1847560 \, x^{11} e + 1662804 \, d x^{10} + 16628040 \, x^{10} e + 15116400 \, d x^{9} + 68023800 \, x^{9} e + 62355150 \, d x^{8} + 166280400 \, x^{8} e + 153489600 \, d x^{7} + 268606800 \, x^{7} e + 249420600 \, d x^{6} + 299304720 \, x^{6} e + 279351072 \, d x^{5} + 232792560 \, x^{5} e + 218243025 \, d x^{4} + 124710300 \, x^{4} e + 117374400 \, d x^{3} + 44015400 \, x^{3} e + 41570100 \, d x^{2} + 9237800 \, x^{2} e + 8751600 \, d x + 875160 \, x e + 831402 \, d}{16628040 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^21,x, algorithm="giac")

[Out]

-1/16628040*(1847560*x^11*e + 1662804*d*x^10 + 16628040*x^10*e + 15116400*d*x^9 + 68023800*x^9*e + 62355150*d*
x^8 + 166280400*x^8*e + 153489600*d*x^7 + 268606800*x^7*e + 249420600*d*x^6 + 299304720*x^6*e + 279351072*d*x^
5 + 232792560*x^5*e + 218243025*d*x^4 + 124710300*x^4*e + 117374400*d*x^3 + 44015400*x^3*e + 41570100*d*x^2 +
9237800*x^2*e + 8751600*d*x + 875160*x*e + 831402*d)/x^20