3.582 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^{16}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{(x+1)^{11} (4 d-15 e)}{60060 x^{11}}+\frac{(x+1)^{11} (4 d-15 e)}{5460 x^{12}}-\frac{(x+1)^{11} (4 d-15 e)}{910 x^{13}}+\frac{(x+1)^{11} (4 d-15 e)}{210 x^{14}}-\frac{d (x+1)^{11}}{15 x^{15}} \]

[Out]

-(d*(1 + x)^11)/(15*x^15) + ((4*d - 15*e)*(1 + x)^11)/(210*x^14) - ((4*d - 15*e)*(1 + x)^11)/(910*x^13) + ((4*
d - 15*e)*(1 + x)^11)/(5460*x^12) - ((4*d - 15*e)*(1 + x)^11)/(60060*x^11)

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Rubi [A]  time = 0.0223859, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {27, 78, 45, 37} \[ -\frac{(x+1)^{11} (4 d-15 e)}{60060 x^{11}}+\frac{(x+1)^{11} (4 d-15 e)}{5460 x^{12}}-\frac{(x+1)^{11} (4 d-15 e)}{910 x^{13}}+\frac{(x+1)^{11} (4 d-15 e)}{210 x^{14}}-\frac{d (x+1)^{11}}{15 x^{15}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^16,x]

[Out]

-(d*(1 + x)^11)/(15*x^15) + ((4*d - 15*e)*(1 + x)^11)/(210*x^14) - ((4*d - 15*e)*(1 + x)^11)/(910*x^13) + ((4*
d - 15*e)*(1 + x)^11)/(5460*x^12) - ((4*d - 15*e)*(1 + x)^11)/(60060*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 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 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^{16}} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^{16}} \, dx\\ &=-\frac{d (1+x)^{11}}{15 x^{15}}-\frac{1}{15} (4 d-15 e) \int \frac{(1+x)^{10}}{x^{15}} \, dx\\ &=-\frac{d (1+x)^{11}}{15 x^{15}}+\frac{(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac{1}{70} (-4 d+15 e) \int \frac{(1+x)^{10}}{x^{14}} \, dx\\ &=-\frac{d (1+x)^{11}}{15 x^{15}}+\frac{(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac{(4 d-15 e) (1+x)^{11}}{910 x^{13}}-\frac{1}{455} (4 d-15 e) \int \frac{(1+x)^{10}}{x^{13}} \, dx\\ &=-\frac{d (1+x)^{11}}{15 x^{15}}+\frac{(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac{(4 d-15 e) (1+x)^{11}}{910 x^{13}}+\frac{(4 d-15 e) (1+x)^{11}}{5460 x^{12}}-\frac{(-4 d+15 e) \int \frac{(1+x)^{10}}{x^{12}} \, dx}{5460}\\ &=-\frac{d (1+x)^{11}}{15 x^{15}}+\frac{(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac{(4 d-15 e) (1+x)^{11}}{910 x^{13}}+\frac{(4 d-15 e) (1+x)^{11}}{5460 x^{12}}-\frac{(4 d-15 e) (1+x)^{11}}{60060 x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.0362069, size = 153, normalized size = 1.7 \[ -\frac{d+10 e}{5 x^5}-\frac{5 (2 d+9 e)}{6 x^6}-\frac{15 (3 d+8 e)}{7 x^7}-\frac{15 (4 d+7 e)}{4 x^8}-\frac{14 (5 d+6 e)}{3 x^9}-\frac{21 (6 d+5 e)}{5 x^{10}}-\frac{30 (7 d+4 e)}{11 x^{11}}-\frac{5 (8 d+3 e)}{4 x^{12}}-\frac{5 (9 d+2 e)}{13 x^{13}}-\frac{10 d+e}{14 x^{14}}-\frac{d}{15 x^{15}}-\frac{e}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^16,x]

[Out]

-d/(15*x^15) - (10*d + e)/(14*x^14) - (5*(9*d + 2*e))/(13*x^13) - (5*(8*d + 3*e))/(4*x^12) - (30*(7*d + 4*e))/
(11*x^11) - (21*(6*d + 5*e))/(5*x^10) - (14*(5*d + 6*e))/(3*x^9) - (15*(4*d + 7*e))/(4*x^8) - (15*(3*d + 8*e))
/(7*x^7) - (5*(2*d + 9*e))/(6*x^6) - (d + 10*e)/(5*x^5) - e/(4*x^4)

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Maple [A]  time = 0.007, size = 130, normalized size = 1.4 \begin{align*} -{\frac{210\,d+252\,e}{9\,{x}^{9}}}-{\frac{d}{15\,{x}^{15}}}-{\frac{10\,d+45\,e}{6\,{x}^{6}}}-{\frac{210\,d+120\,e}{11\,{x}^{11}}}-{\frac{10\,d+e}{14\,{x}^{14}}}-{\frac{45\,d+120\,e}{7\,{x}^{7}}}-{\frac{120\,d+210\,e}{8\,{x}^{8}}}-{\frac{45\,d+10\,e}{13\,{x}^{13}}}-{\frac{e}{4\,{x}^{4}}}-{\frac{120\,d+45\,e}{12\,{x}^{12}}}-{\frac{d+10\,e}{5\,{x}^{5}}}-{\frac{252\,d+210\,e}{10\,{x}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^16,x)

[Out]

-1/9*(210*d+252*e)/x^9-1/15*d/x^15-1/6*(10*d+45*e)/x^6-1/11*(210*d+120*e)/x^11-1/14*(10*d+e)/x^14-1/7*(45*d+12
0*e)/x^7-1/8*(120*d+210*e)/x^8-1/13*(45*d+10*e)/x^13-1/4*e/x^4-1/12*(120*d+45*e)/x^12-1/5*(d+10*e)/x^5-1/10*(2
52*d+210*e)/x^10

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Maxima [A]  time = 1.0258, size = 174, normalized size = 1.93 \begin{align*} -\frac{15015 \, e x^{11} + 12012 \,{\left (d + 10 \, e\right )} x^{10} + 50050 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 128700 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 225225 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 280280 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 252252 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 163800 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 75075 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 23100 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 4290 \,{\left (10 \, d + e\right )} x + 4004 \, d}{60060 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^16,x, algorithm="maxima")

[Out]

-1/60060*(15015*e*x^11 + 12012*(d + 10*e)*x^10 + 50050*(2*d + 9*e)*x^9 + 128700*(3*d + 8*e)*x^8 + 225225*(4*d
+ 7*e)*x^7 + 280280*(5*d + 6*e)*x^6 + 252252*(6*d + 5*e)*x^5 + 163800*(7*d + 4*e)*x^4 + 75075*(8*d + 3*e)*x^3
+ 23100*(9*d + 2*e)*x^2 + 4290*(10*d + e)*x + 4004*d)/x^15

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Fricas [A]  time = 1.13441, size = 378, normalized size = 4.2 \begin{align*} -\frac{15015 \, e x^{11} + 12012 \,{\left (d + 10 \, e\right )} x^{10} + 50050 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 128700 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 225225 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 280280 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 252252 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 163800 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 75075 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 23100 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 4290 \,{\left (10 \, d + e\right )} x + 4004 \, d}{60060 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^16,x, algorithm="fricas")

[Out]

-1/60060*(15015*e*x^11 + 12012*(d + 10*e)*x^10 + 50050*(2*d + 9*e)*x^9 + 128700*(3*d + 8*e)*x^8 + 225225*(4*d
+ 7*e)*x^7 + 280280*(5*d + 6*e)*x^6 + 252252*(6*d + 5*e)*x^5 + 163800*(7*d + 4*e)*x^4 + 75075*(8*d + 3*e)*x^3
+ 23100*(9*d + 2*e)*x^2 + 4290*(10*d + e)*x + 4004*d)/x^15

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Sympy [A]  time = 17.8235, size = 116, normalized size = 1.29 \begin{align*} - \frac{4004 d + 15015 e x^{11} + x^{10} \left (12012 d + 120120 e\right ) + x^{9} \left (100100 d + 450450 e\right ) + x^{8} \left (386100 d + 1029600 e\right ) + x^{7} \left (900900 d + 1576575 e\right ) + x^{6} \left (1401400 d + 1681680 e\right ) + x^{5} \left (1513512 d + 1261260 e\right ) + x^{4} \left (1146600 d + 655200 e\right ) + x^{3} \left (600600 d + 225225 e\right ) + x^{2} \left (207900 d + 46200 e\right ) + x \left (42900 d + 4290 e\right )}{60060 x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**16,x)

[Out]

-(4004*d + 15015*e*x**11 + x**10*(12012*d + 120120*e) + x**9*(100100*d + 450450*e) + x**8*(386100*d + 1029600*
e) + x**7*(900900*d + 1576575*e) + x**6*(1401400*d + 1681680*e) + x**5*(1513512*d + 1261260*e) + x**4*(1146600
*d + 655200*e) + x**3*(600600*d + 225225*e) + x**2*(207900*d + 46200*e) + x*(42900*d + 4290*e))/(60060*x**15)

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Giac [A]  time = 1.15089, size = 192, normalized size = 2.13 \begin{align*} -\frac{15015 \, x^{11} e + 12012 \, d x^{10} + 120120 \, x^{10} e + 100100 \, d x^{9} + 450450 \, x^{9} e + 386100 \, d x^{8} + 1029600 \, x^{8} e + 900900 \, d x^{7} + 1576575 \, x^{7} e + 1401400 \, d x^{6} + 1681680 \, x^{6} e + 1513512 \, d x^{5} + 1261260 \, x^{5} e + 1146600 \, d x^{4} + 655200 \, x^{4} e + 600600 \, d x^{3} + 225225 \, x^{3} e + 207900 \, d x^{2} + 46200 \, x^{2} e + 42900 \, d x + 4290 \, x e + 4004 \, d}{60060 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^16,x, algorithm="giac")

[Out]

-1/60060*(15015*x^11*e + 12012*d*x^10 + 120120*x^10*e + 100100*d*x^9 + 450450*x^9*e + 386100*d*x^8 + 1029600*x
^8*e + 900900*d*x^7 + 1576575*x^7*e + 1401400*d*x^6 + 1681680*x^6*e + 1513512*d*x^5 + 1261260*x^5*e + 1146600*
d*x^4 + 655200*x^4*e + 600600*d*x^3 + 225225*x^3*e + 207900*d*x^2 + 46200*x^2*e + 42900*d*x + 4290*x*e + 4004*
d)/x^15