Integrand size = 19, antiderivative size = 87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x} \, dx=10 d x+\frac {45 d x^2}{2}+40 d x^3+\frac {105 d x^4}{2}+\frac {252 d x^5}{5}+35 d x^6+\frac {120 d x^7}{7}+\frac {45 d x^8}{8}+\frac {10 d x^9}{9}+\frac {d x^{10}}{10}+\frac {1}{11} e (1+x)^{11}+d \log (x) \] Output:
10*d*x+45/2*d*x^2+40*d*x^3+105/2*d*x^4+252/5*d*x^5+35*d*x^6+120/7*d*x^7+45 /8*d*x^8+10/9*d*x^9+1/10*d*x^10+1/11*e*(1+x)^11+d*ln(x)
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x} \, dx=\frac {1}{11} e (1+x)^{11}+d \left (\frac {7381}{2520}+10 x+\frac {45 x^2}{2}+40 x^3+\frac {105 x^4}{2}+\frac {252 x^5}{5}+35 x^6+\frac {120 x^7}{7}+\frac {45 x^8}{8}+\frac {10 x^9}{9}+\frac {x^{10}}{10}\right )+d \log (-x) \] Input:
Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x,x]
Output:
(e*(1 + x)^11)/11 + d*(7381/2520 + 10*x + (45*x^2)/2 + 40*x^3 + (105*x^4)/ 2 + (252*x^5)/5 + 35*x^6 + (120*x^7)/7 + (45*x^8)/8 + (10*x^9)/9 + x^10/10 ) + d*Log[-x]
Time = 0.33 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1184, 90, 49, 2009}
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.
\(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x}dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle d \int \frac {(x+1)^{10}}{x}dx+\frac {1}{11} e (x+1)^{11}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle d \int \left (x^9+10 x^8+45 x^7+120 x^6+210 x^5+252 x^4+210 x^3+120 x^2+45 x+10+\frac {1}{x}\right )dx+\frac {1}{11} e (x+1)^{11}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle d \left (\frac {x^{10}}{10}+\frac {10 x^9}{9}+\frac {45 x^8}{8}+\frac {120 x^7}{7}+35 x^6+\frac {252 x^5}{5}+\frac {105 x^4}{2}+40 x^3+\frac {45 x^2}{2}+10 x+\log (x)\right )+\frac {1}{11} e (x+1)^{11}\) |
Input:
Int[((d + e*x)*(1 + 2*x + x^2)^5)/x,x]
Output:
(e*(1 + x)^11)/11 + d*(10*x + (45*x^2)/2 + 40*x^3 + (105*x^4)/2 + (252*x^5 )/5 + 35*x^6 + (120*x^7)/7 + (45*x^8)/8 + (10*x^9)/9 + x^10/10 + Log[x])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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]
Time = 0.90 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.33
method | result | size |
norman | \(\left (10 d +e \right ) x +\left (35 d +42 e \right ) x^{6}+\left (40 d +15 e \right ) x^{3}+\left (\frac {d}{10}+e \right ) x^{10}+\left (\frac {10 d}{9}+5 e \right ) x^{9}+\left (\frac {45 d}{2}+5 e \right ) x^{2}+\left (\frac {45 d}{8}+15 e \right ) x^{8}+\left (\frac {105 d}{2}+30 e \right ) x^{4}+\left (\frac {120 d}{7}+30 e \right ) x^{7}+\left (\frac {252 d}{5}+42 e \right ) x^{5}+\frac {e \,x^{11}}{11}+d \ln \left (x \right )\) | \(116\) |
default | \(\frac {e \,x^{11}}{11}+\frac {d \,x^{10}}{10}+e \,x^{10}+\frac {10 d \,x^{9}}{9}+5 e \,x^{9}+\frac {45 d \,x^{8}}{8}+15 e \,x^{8}+\frac {120 d \,x^{7}}{7}+30 e \,x^{7}+35 d \,x^{6}+42 e \,x^{6}+\frac {252 d \,x^{5}}{5}+42 x^{5} e +\frac {105 d \,x^{4}}{2}+30 x^{4} e +40 d \,x^{3}+15 x^{3} e +\frac {45 d \,x^{2}}{2}+5 e \,x^{2}+10 d x +e x +d \ln \left (x \right )\) | \(126\) |
risch | \(\frac {e \,x^{11}}{11}+\frac {d \,x^{10}}{10}+e \,x^{10}+\frac {10 d \,x^{9}}{9}+5 e \,x^{9}+\frac {45 d \,x^{8}}{8}+15 e \,x^{8}+\frac {120 d \,x^{7}}{7}+30 e \,x^{7}+35 d \,x^{6}+42 e \,x^{6}+\frac {252 d \,x^{5}}{5}+42 x^{5} e +\frac {105 d \,x^{4}}{2}+30 x^{4} e +40 d \,x^{3}+15 x^{3} e +\frac {45 d \,x^{2}}{2}+5 e \,x^{2}+10 d x +e x +d \ln \left (x \right )\) | \(126\) |
parallelrisch | \(\frac {e \,x^{11}}{11}+\frac {d \,x^{10}}{10}+e \,x^{10}+\frac {10 d \,x^{9}}{9}+5 e \,x^{9}+\frac {45 d \,x^{8}}{8}+15 e \,x^{8}+\frac {120 d \,x^{7}}{7}+30 e \,x^{7}+35 d \,x^{6}+42 e \,x^{6}+\frac {252 d \,x^{5}}{5}+42 x^{5} e +\frac {105 d \,x^{4}}{2}+30 x^{4} e +40 d \,x^{3}+15 x^{3} e +\frac {45 d \,x^{2}}{2}+5 e \,x^{2}+10 d x +e x +d \ln \left (x \right )\) | \(126\) |
Input:
int((e*x+d)*(x^2+2*x+1)^5/x,x,method=_RETURNVERBOSE)
Output:
(10*d+e)*x+(35*d+42*e)*x^6+(40*d+15*e)*x^3+(1/10*d+e)*x^10+(10/9*d+5*e)*x^ 9+(45/2*d+5*e)*x^2+(45/8*d+15*e)*x^8+(105/2*d+30*e)*x^4+(120/7*d+30*e)*x^7 +(252/5*d+42*e)*x^5+1/11*e*x^11+d*ln(x)
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.43 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x} \, dx=\frac {1}{11} \, e x^{11} + \frac {1}{10} \, {\left (d + 10 \, e\right )} x^{10} + \frac {5}{9} \, {\left (2 \, d + 9 \, e\right )} x^{9} + \frac {15}{8} \, {\left (3 \, d + 8 \, e\right )} x^{8} + \frac {30}{7} \, {\left (4 \, d + 7 \, e\right )} x^{7} + 7 \, {\left (5 \, d + 6 \, e\right )} x^{6} + \frac {42}{5} \, {\left (6 \, d + 5 \, e\right )} x^{5} + \frac {15}{2} \, {\left (7 \, d + 4 \, e\right )} x^{4} + 5 \, {\left (8 \, d + 3 \, e\right )} x^{3} + \frac {5}{2} \, {\left (9 \, d + 2 \, e\right )} x^{2} + {\left (10 \, d + e\right )} x + d \log \left (x\right ) \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x,x, algorithm="fricas")
Output:
1/11*e*x^11 + 1/10*(d + 10*e)*x^10 + 5/9*(2*d + 9*e)*x^9 + 15/8*(3*d + 8*e )*x^8 + 30/7*(4*d + 7*e)*x^7 + 7*(5*d + 6*e)*x^6 + 42/5*(6*d + 5*e)*x^5 + 15/2*(7*d + 4*e)*x^4 + 5*(8*d + 3*e)*x^3 + 5/2*(9*d + 2*e)*x^2 + (10*d + e )*x + d*log(x)
Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.34 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x} \, dx=d \log {\left (x \right )} + \frac {e x^{11}}{11} + x^{10} \left (\frac {d}{10} + e\right ) + x^{9} \cdot \left (\frac {10 d}{9} + 5 e\right ) + x^{8} \cdot \left (\frac {45 d}{8} + 15 e\right ) + x^{7} \cdot \left (\frac {120 d}{7} + 30 e\right ) + x^{6} \cdot \left (35 d + 42 e\right ) + x^{5} \cdot \left (\frac {252 d}{5} + 42 e\right ) + x^{4} \cdot \left (\frac {105 d}{2} + 30 e\right ) + x^{3} \cdot \left (40 d + 15 e\right ) + x^{2} \cdot \left (\frac {45 d}{2} + 5 e\right ) + x \left (10 d + e\right ) \] Input:
integrate((e*x+d)*(x**2+2*x+1)**5/x,x)
Output:
d*log(x) + e*x**11/11 + x**10*(d/10 + e) + x**9*(10*d/9 + 5*e) + x**8*(45* d/8 + 15*e) + x**7*(120*d/7 + 30*e) + x**6*(35*d + 42*e) + x**5*(252*d/5 + 42*e) + x**4*(105*d/2 + 30*e) + x**3*(40*d + 15*e) + x**2*(45*d/2 + 5*e) + x*(10*d + e)
Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.43 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x} \, dx=\frac {1}{11} \, e x^{11} + \frac {1}{10} \, {\left (d + 10 \, e\right )} x^{10} + \frac {5}{9} \, {\left (2 \, d + 9 \, e\right )} x^{9} + \frac {15}{8} \, {\left (3 \, d + 8 \, e\right )} x^{8} + \frac {30}{7} \, {\left (4 \, d + 7 \, e\right )} x^{7} + 7 \, {\left (5 \, d + 6 \, e\right )} x^{6} + \frac {42}{5} \, {\left (6 \, d + 5 \, e\right )} x^{5} + \frac {15}{2} \, {\left (7 \, d + 4 \, e\right )} x^{4} + 5 \, {\left (8 \, d + 3 \, e\right )} x^{3} + \frac {5}{2} \, {\left (9 \, d + 2 \, e\right )} x^{2} + {\left (10 \, d + e\right )} x + d \log \left (x\right ) \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x,x, algorithm="maxima")
Output:
1/11*e*x^11 + 1/10*(d + 10*e)*x^10 + 5/9*(2*d + 9*e)*x^9 + 15/8*(3*d + 8*e )*x^8 + 30/7*(4*d + 7*e)*x^7 + 7*(5*d + 6*e)*x^6 + 42/5*(6*d + 5*e)*x^5 + 15/2*(7*d + 4*e)*x^4 + 5*(8*d + 3*e)*x^3 + 5/2*(9*d + 2*e)*x^2 + (10*d + e )*x + d*log(x)
Time = 0.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.45 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x} \, dx=\frac {1}{11} \, e x^{11} + \frac {1}{10} \, d x^{10} + e x^{10} + \frac {10}{9} \, d x^{9} + 5 \, e x^{9} + \frac {45}{8} \, d x^{8} + 15 \, e x^{8} + \frac {120}{7} \, d x^{7} + 30 \, e x^{7} + 35 \, d x^{6} + 42 \, e x^{6} + \frac {252}{5} \, d x^{5} + 42 \, e x^{5} + \frac {105}{2} \, d x^{4} + 30 \, e x^{4} + 40 \, d x^{3} + 15 \, e x^{3} + \frac {45}{2} \, d x^{2} + 5 \, e x^{2} + 10 \, d x + e x + d \log \left ({\left | x \right |}\right ) \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x,x, algorithm="giac")
Output:
1/11*e*x^11 + 1/10*d*x^10 + e*x^10 + 10/9*d*x^9 + 5*e*x^9 + 45/8*d*x^8 + 1 5*e*x^8 + 120/7*d*x^7 + 30*e*x^7 + 35*d*x^6 + 42*e*x^6 + 252/5*d*x^5 + 42* e*x^5 + 105/2*d*x^4 + 30*e*x^4 + 40*d*x^3 + 15*e*x^3 + 45/2*d*x^2 + 5*e*x^ 2 + 10*d*x + e*x + d*log(abs(x))
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x} \, dx=x^9\,\left (\frac {10\,d}{9}+5\,e\right )+x^2\,\left (\frac {45\,d}{2}+5\,e\right )+x^3\,\left (40\,d+15\,e\right )+x^8\,\left (\frac {45\,d}{8}+15\,e\right )+x^6\,\left (35\,d+42\,e\right )+x^4\,\left (\frac {105\,d}{2}+30\,e\right )+x^7\,\left (\frac {120\,d}{7}+30\,e\right )+x^5\,\left (\frac {252\,d}{5}+42\,e\right )+x\,\left (10\,d+e\right )+\frac {e\,x^{11}}{11}+d\,\ln \left (x\right )+x^{10}\,\left (\frac {d}{10}+e\right ) \] Input:
int(((d + e*x)*(2*x + x^2 + 1)^5)/x,x)
Output:
x^9*((10*d)/9 + 5*e) + x^2*((45*d)/2 + 5*e) + x^3*(40*d + 15*e) + x^8*((45 *d)/8 + 15*e) + x^6*(35*d + 42*e) + x^4*((105*d)/2 + 30*e) + x^7*((120*d)/ 7 + 30*e) + x^5*((252*d)/5 + 42*e) + x*(10*d + e) + (e*x^11)/11 + d*log(x) + x^10*(d/10 + e)
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.44 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x} \, dx=\mathrm {log}\left (x \right ) d +\frac {d \,x^{10}}{10}+\frac {10 d \,x^{9}}{9}+\frac {45 d \,x^{8}}{8}+\frac {120 d \,x^{7}}{7}+35 d \,x^{6}+\frac {252 d \,x^{5}}{5}+\frac {105 d \,x^{4}}{2}+40 d \,x^{3}+\frac {45 d \,x^{2}}{2}+10 d x +\frac {e \,x^{11}}{11}+e \,x^{10}+5 e \,x^{9}+15 e \,x^{8}+30 e \,x^{7}+42 e \,x^{6}+42 e \,x^{5}+30 e \,x^{4}+15 e \,x^{3}+5 e \,x^{2}+e x \] Input:
int((e*x+d)*(x^2+2*x+1)^5/x,x)
Output:
(27720*log(x)*d + 2772*d*x**10 + 30800*d*x**9 + 155925*d*x**8 + 475200*d*x **7 + 970200*d*x**6 + 1397088*d*x**5 + 1455300*d*x**4 + 1108800*d*x**3 + 6 23700*d*x**2 + 277200*d*x + 2520*e*x**11 + 27720*e*x**10 + 138600*e*x**9 + 415800*e*x**8 + 831600*e*x**7 + 1164240*e*x**6 + 1164240*e*x**5 + 831600* e*x**4 + 415800*e*x**3 + 138600*e*x**2 + 27720*e*x)/27720