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

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

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Rubi [A]  time = 0.07, antiderivative size = 149, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 149, normalized size = 1.00 \begin {gather*} -\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 {d}{19 x^{19}}-\frac {e}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/19*d/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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 129, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.15, size = 142, normalized size = 0.95 \begin {gather*} -\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 {gather*}

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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maple [A]  time = 0.05, size = 130, normalized size = 0.87 \begin {gather*} -\frac {e}{8 x^{8}}-\frac {d +10 e}{9 x^{9}}-\frac {10 d +45 e}{10 x^{10}}-\frac {45 d +120 e}{11 x^{11}}-\frac {120 d +210 e}{12 x^{12}}-\frac {210 d +252 e}{13 x^{13}}-\frac {252 d +210 e}{14 x^{14}}-\frac {210 d +120 e}{15 x^{15}}-\frac {120 d +45 e}{16 x^{16}}-\frac {45 d +10 e}{17 x^{17}}-\frac {d}{19 x^{19}}-\frac {10 d +e}{18 x^{18}} \end {gather*}

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]

-1/13*(210*d+252*e)/x^13-1/14*(252*d+210*e)/x^14-1/8*e/x^8-1/10*(10*d+45*e)/x^10-1/17*(45*d+10*e)/x^17-1/15*(2
10*d+120*e)/x^15-1/12*(120*d+210*e)/x^12-1/9*(d+10*e)/x^9-1/18*(10*d+e)/x^18-1/19*d/x^19-1/11*(45*d+120*e)/x^1
1-1/16*(120*d+45*e)/x^16

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maxima [A]  time = 0.59, size = 129, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.14, size = 121, normalized size = 0.81 \begin {gather*} -\frac {\frac {e\,x^{11}}{8}+\left (\frac {d}{9}+\frac {10\,e}{9}\right )\,x^{10}+\left (d+\frac {9\,e}{2}\right )\,x^9+\left (\frac {45\,d}{11}+\frac {120\,e}{11}\right )\,x^8+\left (10\,d+\frac {35\,e}{2}\right )\,x^7+\left (\frac {210\,d}{13}+\frac {252\,e}{13}\right )\,x^6+\left (18\,d+15\,e\right )\,x^5+\left (14\,d+8\,e\right )\,x^4+\left (\frac {15\,d}{2}+\frac {45\,e}{16}\right )\,x^3+\left (\frac {45\,d}{17}+\frac {10\,e}{17}\right )\,x^2+\left (\frac {5\,d}{9}+\frac {e}{18}\right )\,x+\frac {d}{19}}{x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d/19 + x^4*(14*d + 8*e) + x^5*(18*d + 15*e) + x^10*(d/9 + (10*e)/9) + x^7*(10*d + (35*e)/2) + x^3*((15*d)/2
+ (45*e)/16) + x^2*((45*d)/17 + (10*e)/17) + x^8*((45*d)/11 + (120*e)/11) + x^6*((210*d)/13 + (252*e)/13) + (e
*x^11)/8 + x*((5*d)/9 + e/18) + x^9*(d + (9*e)/2))/x^19

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sympy [A]  time = 31.79, size = 131, normalized size = 0.88 \begin {gather*} \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 {gather*}

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 -
99768240*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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