3.6.11 \(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^{18}} \, dx\)

Optimal. Leaf size=128 \[ \frac {(x+1)^{11} (6 d-17 e)}{272 x^{16}}-\frac {(x+1)^{11} (6 d-17 e)}{816 x^{15}}+\frac {(x+1)^{11} (6 d-17 e)}{2856 x^{14}}-\frac {(x+1)^{11} (6 d-17 e)}{12376 x^{13}}+\frac {(x+1)^{11} (6 d-17 e)}{74256 x^{12}}-\frac {(x+1)^{11} (6 d-17 e)}{816816 x^{11}}-\frac {d (x+1)^{11}}{17 x^{17}} \]

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Rubi [A]  time = 0.03, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {27, 78, 45, 37} \begin {gather*} -\frac {(x+1)^{11} (6 d-17 e)}{816816 x^{11}}+\frac {(x+1)^{11} (6 d-17 e)}{74256 x^{12}}-\frac {(x+1)^{11} (6 d-17 e)}{12376 x^{13}}+\frac {(x+1)^{11} (6 d-17 e)}{2856 x^{14}}-\frac {(x+1)^{11} (6 d-17 e)}{816 x^{15}}+\frac {(x+1)^{11} (6 d-17 e)}{272 x^{16}}-\frac {d (x+1)^{11}}{17 x^{17}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^18,x]

[Out]

-(d*(1 + x)^11)/(17*x^17) + ((6*d - 17*e)*(1 + x)^11)/(272*x^16) - ((6*d - 17*e)*(1 + x)^11)/(816*x^15) + ((6*
d - 17*e)*(1 + x)^11)/(2856*x^14) - ((6*d - 17*e)*(1 + x)^11)/(12376*x^13) + ((6*d - 17*e)*(1 + x)^11)/(74256*
x^12) - ((6*d - 17*e)*(1 + x)^11)/(816816*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 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]

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 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]))))

Rubi steps

\begin {align*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{18}} \, dx\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}-\frac {1}{17} (6 d-17 e) \int \frac {(1+x)^{10}}{x^{17}} \, dx\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}+\frac {1}{272} (5 (6 d-17 e)) \int \frac {(1+x)^{10}}{x^{16}} \, dx\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {1}{204} (-6 d+17 e) \int \frac {(1+x)^{10}}{x^{15}} \, dx\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {(6 d-17 e) (1+x)^{11}}{2856 x^{14}}+\frac {1}{952} (6 d-17 e) \int \frac {(1+x)^{10}}{x^{14}} \, dx\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {(6 d-17 e) (1+x)^{11}}{2856 x^{14}}-\frac {(6 d-17 e) (1+x)^{11}}{12376 x^{13}}+\frac {(-6 d+17 e) \int \frac {(1+x)^{10}}{x^{13}} \, dx}{6188}\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {(6 d-17 e) (1+x)^{11}}{2856 x^{14}}-\frac {(6 d-17 e) (1+x)^{11}}{12376 x^{13}}+\frac {(6 d-17 e) (1+x)^{11}}{74256 x^{12}}+\frac {(6 d-17 e) \int \frac {(1+x)^{10}}{x^{12}} \, dx}{74256}\\ &=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {(6 d-17 e) (1+x)^{11}}{2856 x^{14}}-\frac {(6 d-17 e) (1+x)^{11}}{12376 x^{13}}+\frac {(6 d-17 e) (1+x)^{11}}{74256 x^{12}}-\frac {(6 d-17 e) (1+x)^{11}}{816816 x^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 151, normalized size = 1.18 \begin {gather*} -\frac {10 d+e}{16 x^{16}}-\frac {9 d+2 e}{3 x^{15}}-\frac {15 (8 d+3 e)}{14 x^{14}}-\frac {30 (7 d+4 e)}{13 x^{13}}-\frac {7 (6 d+5 e)}{2 x^{12}}-\frac {42 (5 d+6 e)}{11 x^{11}}-\frac {3 (4 d+7 e)}{x^{10}}-\frac {5 (3 d+8 e)}{3 x^9}-\frac {5 (2 d+9 e)}{8 x^8}-\frac {d+10 e}{7 x^7}-\frac {d}{17 x^{17}}-\frac {e}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^18,x]

[Out]

-1/17*d/x^17 - (10*d + e)/(16*x^16) - (9*d + 2*e)/(3*x^15) - (15*(8*d + 3*e))/(14*x^14) - (30*(7*d + 4*e))/(13
*x^13) - (7*(6*d + 5*e))/(2*x^12) - (42*(5*d + 6*e))/(11*x^11) - (3*(4*d + 7*e))/x^10 - (5*(3*d + 8*e))/(3*x^9
) - (5*(2*d + 9*e))/(8*x^8) - (d + 10*e)/(7*x^7) - e/(6*x^6)

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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^{18}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^18,x]

[Out]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^18, x]

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fricas [A]  time = 0.40, size = 129, normalized size = 1.01 \begin {gather*} -\frac {136136 \, e x^{11} + 116688 \, {\left (d + 10 \, e\right )} x^{10} + 510510 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 1361360 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 2450448 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3118752 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2858856 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1884960 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 875160 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 272272 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 51051 \, {\left (10 \, d + e\right )} x + 48048 \, d}{816816 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^18,x, algorithm="fricas")

[Out]

-1/816816*(136136*e*x^11 + 116688*(d + 10*e)*x^10 + 510510*(2*d + 9*e)*x^9 + 1361360*(3*d + 8*e)*x^8 + 2450448
*(4*d + 7*e)*x^7 + 3118752*(5*d + 6*e)*x^6 + 2858856*(6*d + 5*e)*x^5 + 1884960*(7*d + 4*e)*x^4 + 875160*(8*d +
3*e)*x^3 + 272272*(9*d + 2*e)*x^2 + 51051*(10*d + e)*x + 48048*d)/x^17

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giac [A]  time = 0.16, size = 142, normalized size = 1.11 \begin {gather*} -\frac {136136 \, x^{11} e + 116688 \, d x^{10} + 1166880 \, x^{10} e + 1021020 \, d x^{9} + 4594590 \, x^{9} e + 4084080 \, d x^{8} + 10890880 \, x^{8} e + 9801792 \, d x^{7} + 17153136 \, x^{7} e + 15593760 \, d x^{6} + 18712512 \, x^{6} e + 17153136 \, d x^{5} + 14294280 \, x^{5} e + 13194720 \, d x^{4} + 7539840 \, x^{4} e + 7001280 \, d x^{3} + 2625480 \, x^{3} e + 2450448 \, d x^{2} + 544544 \, x^{2} e + 510510 \, d x + 51051 \, x e + 48048 \, d}{816816 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^18,x, algorithm="giac")

[Out]

-1/816816*(136136*x^11*e + 116688*d*x^10 + 1166880*x^10*e + 1021020*d*x^9 + 4594590*x^9*e + 4084080*d*x^8 + 10
890880*x^8*e + 9801792*d*x^7 + 17153136*x^7*e + 15593760*d*x^6 + 18712512*x^6*e + 17153136*d*x^5 + 14294280*x^
5*e + 13194720*d*x^4 + 7539840*x^4*e + 7001280*d*x^3 + 2625480*x^3*e + 2450448*d*x^2 + 544544*x^2*e + 510510*d
*x + 51051*x*e + 48048*d)/x^17

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maple [A]  time = 0.05, size = 130, normalized size = 1.02 \begin {gather*} -\frac {e}{6 x^{6}}-\frac {d +10 e}{7 x^{7}}-\frac {10 d +45 e}{8 x^{8}}-\frac {45 d +120 e}{9 x^{9}}-\frac {120 d +210 e}{10 x^{10}}-\frac {210 d +252 e}{11 x^{11}}-\frac {252 d +210 e}{12 x^{12}}-\frac {210 d +120 e}{13 x^{13}}-\frac {120 d +45 e}{14 x^{14}}-\frac {45 d +10 e}{15 x^{15}}-\frac {d}{17 x^{17}}-\frac {10 d +e}{16 x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^18,x)

[Out]

-1/14*(120*d+45*e)/x^14-1/13*(210*d+120*e)/x^13-1/16*(10*d+e)/x^16-1/8*(10*d+45*e)/x^8-1/10*(120*d+210*e)/x^10
-1/15*(45*d+10*e)/x^15-1/9*(45*d+120*e)/x^9-1/17*d/x^17-1/12*(252*d+210*e)/x^12-1/7*(d+10*e)/x^7-1/6*e/x^6-1/1
1*(210*d+252*e)/x^11

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maxima [A]  time = 0.51, size = 129, normalized size = 1.01 \begin {gather*} -\frac {136136 \, e x^{11} + 116688 \, {\left (d + 10 \, e\right )} x^{10} + 510510 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 1361360 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 2450448 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3118752 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2858856 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1884960 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 875160 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 272272 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 51051 \, {\left (10 \, d + e\right )} x + 48048 \, d}{816816 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^18,x, algorithm="maxima")

[Out]

-1/816816*(136136*e*x^11 + 116688*(d + 10*e)*x^10 + 510510*(2*d + 9*e)*x^9 + 1361360*(3*d + 8*e)*x^8 + 2450448
*(4*d + 7*e)*x^7 + 3118752*(5*d + 6*e)*x^6 + 2858856*(6*d + 5*e)*x^5 + 1884960*(7*d + 4*e)*x^4 + 875160*(8*d +
3*e)*x^3 + 272272*(9*d + 2*e)*x^2 + 51051*(10*d + e)*x + 48048*d)/x^17

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mupad [B]  time = 1.13, size = 123, normalized size = 0.96 \begin {gather*} -\frac {\frac {e\,x^{11}}{6}+\left (\frac {d}{7}+\frac {10\,e}{7}\right )\,x^{10}+\left (\frac {5\,d}{4}+\frac {45\,e}{8}\right )\,x^9+\left (5\,d+\frac {40\,e}{3}\right )\,x^8+\left (12\,d+21\,e\right )\,x^7+\left (\frac {210\,d}{11}+\frac {252\,e}{11}\right )\,x^6+\left (21\,d+\frac {35\,e}{2}\right )\,x^5+\left (\frac {210\,d}{13}+\frac {120\,e}{13}\right )\,x^4+\left (\frac {60\,d}{7}+\frac {45\,e}{14}\right )\,x^3+\left (3\,d+\frac {2\,e}{3}\right )\,x^2+\left (\frac {5\,d}{8}+\frac {e}{16}\right )\,x+\frac {d}{17}}{x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x)*(2*x + x^2 + 1)^5)/x^18,x)

[Out]

-(d/17 + x^2*(3*d + (2*e)/3) + x^10*(d/7 + (10*e)/7) + x^7*(12*d + 21*e) + x^8*(5*d + (40*e)/3) + x^5*(21*d +
(35*e)/2) + x^9*((5*d)/4 + (45*e)/8) + x^3*((60*d)/7 + (45*e)/14) + x^4*((210*d)/13 + (120*e)/13) + x^6*((210*
d)/11 + (252*e)/11) + (e*x^11)/6 + x*((5*d)/8 + e/16))/x^17

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sympy [A]  time = 24.87, size = 131, normalized size = 1.02 \begin {gather*} \frac {- 48048 d - 136136 e x^{11} + x^{10} \left (- 116688 d - 1166880 e\right ) + x^{9} \left (- 1021020 d - 4594590 e\right ) + x^{8} \left (- 4084080 d - 10890880 e\right ) + x^{7} \left (- 9801792 d - 17153136 e\right ) + x^{6} \left (- 15593760 d - 18712512 e\right ) + x^{5} \left (- 17153136 d - 14294280 e\right ) + x^{4} \left (- 13194720 d - 7539840 e\right ) + x^{3} \left (- 7001280 d - 2625480 e\right ) + x^{2} \left (- 2450448 d - 544544 e\right ) + x \left (- 510510 d - 51051 e\right )}{816816 x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**18,x)

[Out]

(-48048*d - 136136*e*x**11 + x**10*(-116688*d - 1166880*e) + x**9*(-1021020*d - 4594590*e) + x**8*(-4084080*d
- 10890880*e) + x**7*(-9801792*d - 17153136*e) + x**6*(-15593760*d - 18712512*e) + x**5*(-17153136*d - 1429428
0*e) + x**4*(-13194720*d - 7539840*e) + x**3*(-7001280*d - 2625480*e) + x**2*(-2450448*d - 544544*e) + x*(-510
510*d - 51051*e))/(816816*x**17)

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