\(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^{16}} \, dx\) [211]

Optimal result

Integrand size = 19, antiderivative size = 90 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {d (1+x)^{11}}{15 x^{15}}+\frac {(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac {(4 d-15 e) (1+x)^{11}}{910 x^{13}}+\frac {(4 d-15 e) (1+x)^{11}}{5460 x^{12}}-\frac {(4 d-15 e) (1+x)^{11}}{60060 x^{11}} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.70 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {d}{15 x^{15}}-\frac {10 d+e}{14 x^{14}}-\frac {5 (9 d+2 e)}{13 x^{13}}-\frac {5 (8 d+3 e)}{4 x^{12}}-\frac {30 (7 d+4 e)}{11 x^{11}}-\frac {21 (6 d+5 e)}{5 x^{10}}-\frac {14 (5 d+6 e)}{3 x^9}-\frac {15 (4 d+7 e)}{4 x^8}-\frac {15 (3 d+8 e)}{7 x^7}-\frac {5 (2 d+9 e)}{6 x^6}-\frac {d+10 e}{5 x^5}-\frac {e}{4 x^4} \] Input:

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

Output:

-1/15*d/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)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1184, 87, 55, 55, 55, 48}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1184

\(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^{16}}dx\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {1}{15} (4 d-15 e) \int \frac {(x+1)^{10}}{x^{15}}dx-\frac {d (x+1)^{11}}{15 x^{15}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {1}{15} (4 d-15 e) \left (-\frac {3}{14} \int \frac {(x+1)^{10}}{x^{14}}dx-\frac {(x+1)^{11}}{14 x^{14}}\right )-\frac {d (x+1)^{11}}{15 x^{15}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {1}{15} (4 d-15 e) \left (-\frac {3}{14} \left (-\frac {2}{13} \int \frac {(x+1)^{10}}{x^{13}}dx-\frac {(x+1)^{11}}{13 x^{13}}\right )-\frac {(x+1)^{11}}{14 x^{14}}\right )-\frac {d (x+1)^{11}}{15 x^{15}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {1}{15} (4 d-15 e) \left (-\frac {3}{14} \left (-\frac {2}{13} \left (-\frac {1}{12} \int \frac {(x+1)^{10}}{x^{12}}dx-\frac {(x+1)^{11}}{12 x^{12}}\right )-\frac {(x+1)^{11}}{13 x^{13}}\right )-\frac {(x+1)^{11}}{14 x^{14}}\right )-\frac {d (x+1)^{11}}{15 x^{15}}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {1}{15} \left (-\frac {(x+1)^{11}}{14 x^{14}}-\frac {3}{14} \left (-\frac {(x+1)^{11}}{13 x^{13}}-\frac {2}{13} \left (\frac {(x+1)^{11}}{132 x^{11}}-\frac {(x+1)^{11}}{12 x^{12}}\right )\right )\right ) (4 d-15 e)-\frac {d (x+1)^{11}}{15 x^{15}}\)

Input:

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

Output:

-1/15*(d*(1 + x)^11)/x^15 - ((4*d - 15*e)*(-1/14*(1 + x)^11/x^14 - (3*(-1/ 
13*(1 + x)^11/x^13 - (2*(-1/12*(1 + x)^11/x^12 + (1 + x)^11/(132*x^11)))/1 
3))/14))/15
 

Defintions of rubi rules used

rule 48

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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

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] - Simp[d*(S 
implify[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] && 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] ||  !SumSimp 
lerQ[n, 1])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(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] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.37

method	result	size
norman	\(\frac {-\frac {d}{15}+\left (-\frac {5 d}{7}-\frac {e}{14}\right ) x +\left (-\frac {45 d}{13}-\frac {10 e}{13}\right ) x^{2}+\left (-10 d -\frac {15 e}{4}\right ) x^{3}+\left (-\frac {210 d}{11}-\frac {120 e}{11}\right ) x^{4}+\left (-\frac {126 d}{5}-21 e \right ) x^{5}+\left (-\frac {70 d}{3}-28 e \right ) x^{6}+\left (-15 d -\frac {105 e}{4}\right ) x^{7}+\left (-\frac {45 d}{7}-\frac {120 e}{7}\right ) x^{8}+\left (-\frac {5 d}{3}-\frac {15 e}{2}\right ) x^{9}+\left (-\frac {d}{5}-2 e \right ) x^{10}-\frac {e \,x^{11}}{4}}{x^{15}}\)	\(123\)
risch	\(\frac {-\frac {d}{15}+\left (-\frac {5 d}{7}-\frac {e}{14}\right ) x +\left (-\frac {45 d}{13}-\frac {10 e}{13}\right ) x^{2}+\left (-10 d -\frac {15 e}{4}\right ) x^{3}+\left (-\frac {210 d}{11}-\frac {120 e}{11}\right ) x^{4}+\left (-\frac {126 d}{5}-21 e \right ) x^{5}+\left (-\frac {70 d}{3}-28 e \right ) x^{6}+\left (-15 d -\frac {105 e}{4}\right ) x^{7}+\left (-\frac {45 d}{7}-\frac {120 e}{7}\right ) x^{8}+\left (-\frac {5 d}{3}-\frac {15 e}{2}\right ) x^{9}+\left (-\frac {d}{5}-2 e \right ) x^{10}-\frac {e \,x^{11}}{4}}{x^{15}}\)	\(123\)
default	\(-\frac {d +10 e}{5 x^{5}}-\frac {10 d +45 e}{6 x^{6}}-\frac {45 d +10 e}{13 x^{13}}-\frac {120 d +210 e}{8 x^{8}}-\frac {e}{4 x^{4}}-\frac {10 d +e}{14 x^{14}}-\frac {210 d +252 e}{9 x^{9}}-\frac {45 d +120 e}{7 x^{7}}-\frac {252 d +210 e}{10 x^{10}}-\frac {d}{15 x^{15}}-\frac {210 d +120 e}{11 x^{11}}-\frac {120 d +45 e}{12 x^{12}}\)	\(130\)
gosper	\(-\frac {15015 e \,x^{11}+12012 d \,x^{10}+120120 e \,x^{10}+100100 d \,x^{9}+450450 e \,x^{9}+386100 d \,x^{8}+1029600 e \,x^{8}+900900 d \,x^{7}+1576575 e \,x^{7}+1401400 d \,x^{6}+1681680 e \,x^{6}+1513512 d \,x^{5}+1261260 x^{5} e +1146600 d \,x^{4}+655200 x^{4} e +600600 d \,x^{3}+225225 x^{3} e +207900 d \,x^{2}+46200 e \,x^{2}+42900 d x +4290 e x +4004 d}{60060 x^{15}}\)	\(132\)
parallelrisch	\(\frac {-15015 e \,x^{11}-12012 d \,x^{10}-120120 e \,x^{10}-100100 d \,x^{9}-450450 e \,x^{9}-386100 d \,x^{8}-1029600 e \,x^{8}-900900 d \,x^{7}-1576575 e \,x^{7}-1401400 d \,x^{6}-1681680 e \,x^{6}-1513512 d \,x^{5}-1261260 x^{5} e -1146600 d \,x^{4}-655200 x^{4} e -600600 d \,x^{3}-225225 x^{3} e -207900 d \,x^{2}-46200 e \,x^{2}-42900 d x -4290 e x -4004 d}{60060 x^{15}}\)	\(132\)
orering	\(-\frac {\left (15015 e \,x^{11}+12012 d \,x^{10}+120120 e \,x^{10}+100100 d \,x^{9}+450450 e \,x^{9}+386100 d \,x^{8}+1029600 e \,x^{8}+900900 d \,x^{7}+1576575 e \,x^{7}+1401400 d \,x^{6}+1681680 e \,x^{6}+1513512 d \,x^{5}+1261260 x^{5} e +1146600 d \,x^{4}+655200 x^{4} e +600600 d \,x^{3}+225225 x^{3} e +207900 d \,x^{2}+46200 e \,x^{2}+42900 d x +4290 e x +4004 d \right ) \left (x^{2}+2 x +1\right )^{5}}{60060 x^{15} \left (x +1\right )^{10}}\)	\(147\)

Input:

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

Output:

(-1/15*d+(-5/7*d-1/14*e)*x+(-45/13*d-10/13*e)*x^2+(-10*d-15/4*e)*x^3+(-210 
/11*d-120/11*e)*x^4+(-126/5*d-21*e)*x^5+(-70/3*d-28*e)*x^6+(-15*d-105/4*e) 
*x^7+(-45/7*d-120/7*e)*x^8+(-5/3*d-15/2*e)*x^9+(-1/5*d-2*e)*x^10-1/4*e*x^1 
1)/x^15
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.43 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {15015 \, e x^{11} + 12012 \, {\left (d + 10 \, e\right )} x^{10} + 50050 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 128700 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 225225 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 280280 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 252252 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 163800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 75075 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 23100 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 4290 \, {\left (10 \, d + e\right )} x + 4004 \, d}{60060 \, x^{15}} \] Input:

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

Output:

-1/60060*(15015*e*x^11 + 12012*(d + 10*e)*x^10 + 50050*(2*d + 9*e)*x^9 + 1 
28700*(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
 

Sympy [A] (verification not implemented)

Time = 10.52 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=\frac {- 4004 d - 15015 e x^{11} + x^{10} \left (- 12012 d - 120120 e\right ) + x^{9} \left (- 100100 d - 450450 e\right ) + x^{8} \left (- 386100 d - 1029600 e\right ) + x^{7} \left (- 900900 d - 1576575 e\right ) + x^{6} \left (- 1401400 d - 1681680 e\right ) + x^{5} \left (- 1513512 d - 1261260 e\right ) + x^{4} \left (- 1146600 d - 655200 e\right ) + x^{3} \left (- 600600 d - 225225 e\right ) + x^{2} \left (- 207900 d - 46200 e\right ) + x \left (- 42900 d - 4290 e\right )}{60060 x^{15}} \] Input:

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

Output:

(-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*(-11 
46600*d - 655200*e) + x**3*(-600600*d - 225225*e) + x**2*(-207900*d - 4620 
0*e) + x*(-42900*d - 4290*e))/(60060*x**15)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.43 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {15015 \, e x^{11} + 12012 \, {\left (d + 10 \, e\right )} x^{10} + 50050 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 128700 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 225225 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 280280 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 252252 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 163800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 75075 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 23100 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 4290 \, {\left (10 \, d + e\right )} x + 4004 \, d}{60060 \, x^{15}} \] Input:

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

Output:

-1/60060*(15015*e*x^11 + 12012*(d + 10*e)*x^10 + 50050*(2*d + 9*e)*x^9 + 1 
28700*(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
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {15015 \, e x^{11} + 12012 \, d x^{10} + 120120 \, e x^{10} + 100100 \, d x^{9} + 450450 \, e x^{9} + 386100 \, d x^{8} + 1029600 \, e x^{8} + 900900 \, d x^{7} + 1576575 \, e x^{7} + 1401400 \, d x^{6} + 1681680 \, e x^{6} + 1513512 \, d x^{5} + 1261260 \, e x^{5} + 1146600 \, d x^{4} + 655200 \, e x^{4} + 600600 \, d x^{3} + 225225 \, e x^{3} + 207900 \, d x^{2} + 46200 \, e x^{2} + 42900 \, d x + 4290 \, e x + 4004 \, d}{60060 \, x^{15}} \] Input:

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

Output:

-1/60060*(15015*e*x^11 + 12012*d*x^10 + 120120*e*x^10 + 100100*d*x^9 + 450 
450*e*x^9 + 386100*d*x^8 + 1029600*e*x^8 + 900900*d*x^7 + 1576575*e*x^7 + 
1401400*d*x^6 + 1681680*e*x^6 + 1513512*d*x^5 + 1261260*e*x^5 + 1146600*d* 
x^4 + 655200*e*x^4 + 600600*d*x^3 + 225225*e*x^3 + 207900*d*x^2 + 46200*e* 
x^2 + 42900*d*x + 4290*e*x + 4004*d)/x^15
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.37 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {\frac {e\,x^{11}}{4}+\left (\frac {d}{5}+2\,e\right )\,x^{10}+\left (\frac {5\,d}{3}+\frac {15\,e}{2}\right )\,x^9+\left (\frac {45\,d}{7}+\frac {120\,e}{7}\right )\,x^8+\left (15\,d+\frac {105\,e}{4}\right )\,x^7+\left (\frac {70\,d}{3}+28\,e\right )\,x^6+\left (\frac {126\,d}{5}+21\,e\right )\,x^5+\left (\frac {210\,d}{11}+\frac {120\,e}{11}\right )\,x^4+\left (10\,d+\frac {15\,e}{4}\right )\,x^3+\left (\frac {45\,d}{13}+\frac {10\,e}{13}\right )\,x^2+\left (\frac {5\,d}{7}+\frac {e}{14}\right )\,x+\frac {d}{15}}{x^{15}} \] Input:

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

Output:

-(d/15 + x^10*(d/5 + 2*e) + x^3*(10*d + (15*e)/4) + x^9*((5*d)/3 + (15*e)/ 
2) + x^2*((45*d)/13 + (10*e)/13) + x^6*((70*d)/3 + 28*e) + x^7*(15*d + (10 
5*e)/4) + x^5*((126*d)/5 + 21*e) + x^8*((45*d)/7 + (120*e)/7) + x^4*((210* 
d)/11 + (120*e)/11) + (e*x^11)/4 + x*((5*d)/7 + e/14))/x^15
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=\frac {-15015 e \,x^{11}-12012 d \,x^{10}-120120 e \,x^{10}-100100 d \,x^{9}-450450 e \,x^{9}-386100 d \,x^{8}-1029600 e \,x^{8}-900900 d \,x^{7}-1576575 e \,x^{7}-1401400 d \,x^{6}-1681680 e \,x^{6}-1513512 d \,x^{5}-1261260 e \,x^{5}-1146600 d \,x^{4}-655200 e \,x^{4}-600600 d \,x^{3}-225225 e \,x^{3}-207900 d \,x^{2}-46200 e \,x^{2}-42900 d x -4290 e x -4004 d}{60060 x^{15}} \] Input:

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

Output:

( - 12012*d*x**10 - 100100*d*x**9 - 386100*d*x**8 - 900900*d*x**7 - 140140 
0*d*x**6 - 1513512*d*x**5 - 1146600*d*x**4 - 600600*d*x**3 - 207900*d*x**2 
 - 42900*d*x - 4004*d - 15015*e*x**11 - 120120*e*x**10 - 450450*e*x**9 - 1 
029600*e*x**8 - 1576575*e*x**7 - 1681680*e*x**6 - 1261260*e*x**5 - 655200* 
e*x**4 - 225225*e*x**3 - 46200*e*x**2 - 4290*e*x)/(60060*x**15)