Integrand size = 19, antiderivative size = 138 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^9} \, dx=-\frac {d}{8 x^8}-\frac {10 d+e}{7 x^7}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {3 (8 d+3 e)}{x^5}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {14 (6 d+5 e)}{x^3}-\frac {21 (5 d+6 e)}{x^2}-\frac {30 (4 d+7 e)}{x}+5 (2 d+9 e) x+\frac {1}{2} (d+10 e) x^2+\frac {e x^3}{3}+15 (3 d+8 e) \log (x) \]
-1/8*d/x^8+1/7*(-10*d-e)/x^7-5/6*(9*d+2*e)/x^6-3*(8*d+3*e)/x^5-15/2*(7*d+4 *e)/x^4-14*(6*d+5*e)/x^3-21*(5*d+6*e)/x^2-30*(4*d+7*e)/x+5*(2*d+9*e)*x+1/2 *(d+10*e)*x^2+1/3*e*x^3+15*(3*d+8*e)*ln(x)
Time = 0.03 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^9} \, dx=-\frac {d}{8 x^8}+\frac {-10 d-e}{7 x^7}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {3 (8 d+3 e)}{x^5}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {14 (6 d+5 e)}{x^3}-\frac {21 (5 d+6 e)}{x^2}-\frac {30 (4 d+7 e)}{x}+5 (2 d+9 e) x+\frac {1}{2} (d+10 e) x^2+\frac {e x^3}{3}+15 (3 d+8 e) \log (x) \]
-1/8*d/x^8 + (-10*d - e)/(7*x^7) - (5*(9*d + 2*e))/(6*x^6) - (3*(8*d + 3*e ))/x^5 - (15*(7*d + 4*e))/(2*x^4) - (14*(6*d + 5*e))/x^3 - (21*(5*d + 6*e) )/x^2 - (30*(4*d + 7*e))/x + 5*(2*d + 9*e)*x + ((d + 10*e)*x^2)/2 + (e*x^3 )/3 + 15*(3*d + 8*e)*Log[x]
Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1184, 85, 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.
\(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x^9} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^9}dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (\frac {10 d+e}{x^8}+\frac {5 (9 d+2 e)}{x^7}+\frac {15 (8 d+3 e)}{x^6}+\frac {30 (7 d+4 e)}{x^5}+\frac {42 (6 d+5 e)}{x^4}+\frac {42 (5 d+6 e)}{x^3}+\frac {30 (4 d+7 e)}{x^2}+x (d+10 e)+\frac {15 (3 d+8 e)}{x}+5 (2 d+9 e)+\frac {d}{x^9}+e x^2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {10 d+e}{7 x^7}-\frac {5 (9 d+2 e)}{6 x^6}-\frac {3 (8 d+3 e)}{x^5}-\frac {15 (7 d+4 e)}{2 x^4}-\frac {14 (6 d+5 e)}{x^3}+\frac {1}{2} x^2 (d+10 e)-\frac {21 (5 d+6 e)}{x^2}+5 x (2 d+9 e)-\frac {30 (4 d+7 e)}{x}+15 (3 d+8 e) \log (x)-\frac {d}{8 x^8}+\frac {e x^3}{3}\) |
-1/8*d/x^8 - (10*d + e)/(7*x^7) - (5*(9*d + 2*e))/(6*x^6) - (3*(8*d + 3*e) )/x^5 - (15*(7*d + 4*e))/(2*x^4) - (14*(6*d + 5*e))/x^3 - (21*(5*d + 6*e)) /x^2 - (30*(4*d + 7*e))/x + 5*(2*d + 9*e)*x + ((d + 10*e)*x^2)/2 + (e*x^3) /3 + 15*(3*d + 8*e)*Log[x]
3.6.75.3.1 Defintions of rubi rules used
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > 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]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
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 = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {e \,x^{3}}{3}+\frac {d \,x^{2}}{2}+5 e \,x^{2}+10 d x +45 e x +\frac {\left (-120 d -210 e \right ) x^{7}+\left (-105 d -126 e \right ) x^{6}+\left (-84 d -70 e \right ) x^{5}+\left (-\frac {105 d}{2}-30 e \right ) x^{4}+\left (-24 d -9 e \right ) x^{3}+\left (-\frac {15 d}{2}-\frac {5 e}{3}\right ) x^{2}+\left (-\frac {10 d}{7}-\frac {e}{7}\right ) x -\frac {d}{8}}{x^{8}}+45 d \ln \left (x \right )+120 e \ln \left (x \right )\) | \(121\) |
norman | \(\frac {\left (-120 d -210 e \right ) x^{7}+\left (-105 d -126 e \right ) x^{6}+\left (-84 d -70 e \right ) x^{5}+\left (-24 d -9 e \right ) x^{3}+\left (10 d +45 e \right ) x^{9}+\left (-\frac {105 d}{2}-30 e \right ) x^{4}+\left (-\frac {15 d}{2}-\frac {5 e}{3}\right ) x^{2}+\left (-\frac {10 d}{7}-\frac {e}{7}\right ) x +\left (\frac {d}{2}+5 e \right ) x^{10}-\frac {d}{8}+\frac {e \,x^{11}}{3}}{x^{8}}+\left (45 d +120 e \right ) \ln \left (x \right )\) | \(123\) |
default | \(\frac {e \,x^{3}}{3}+\frac {d \,x^{2}}{2}+5 e \,x^{2}+10 d x +45 e x -\frac {45 d +10 e}{6 x^{6}}-\frac {210 d +120 e}{4 x^{4}}-\frac {120 d +45 e}{5 x^{5}}+\left (45 d +120 e \right ) \ln \left (x \right )-\frac {210 d +252 e}{2 x^{2}}-\frac {10 d +e}{7 x^{7}}-\frac {120 d +210 e}{x}-\frac {252 d +210 e}{3 x^{3}}-\frac {d}{8 x^{8}}\) | \(126\) |
parallelrisch | \(\frac {56 e \,x^{11}+84 d \,x^{10}+840 e \,x^{10}+7560 \ln \left (x \right ) x^{8} d +20160 \ln \left (x \right ) x^{8} e +1680 d \,x^{9}+7560 e \,x^{9}-20160 d \,x^{7}-35280 e \,x^{7}-17640 d \,x^{6}-21168 e \,x^{6}-14112 d \,x^{5}-11760 e \,x^{5}-8820 d \,x^{4}-5040 e \,x^{4}-4032 d \,x^{3}-1512 e \,x^{3}-1260 d \,x^{2}-280 e \,x^{2}-240 d x -24 e x -21 d}{168 x^{8}}\) | \(136\) |
1/3*e*x^3+1/2*d*x^2+5*e*x^2+10*d*x+45*e*x+((-120*d-210*e)*x^7+(-105*d-126* e)*x^6+(-84*d-70*e)*x^5+(-105/2*d-30*e)*x^4+(-24*d-9*e)*x^3+(-15/2*d-5/3*e )*x^2+(-10/7*d-1/7*e)*x-1/8*d)/x^8+45*d*ln(x)+120*e*ln(x)
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^9} \, dx=\frac {56 \, e x^{11} + 84 \, {\left (d + 10 \, e\right )} x^{10} + 840 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 2520 \, {\left (3 \, d + 8 \, e\right )} x^{8} \log \left (x\right ) - 5040 \, {\left (4 \, d + 7 \, e\right )} x^{7} - 3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} - 2352 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 1260 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 504 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 140 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 24 \, {\left (10 \, d + e\right )} x - 21 \, d}{168 \, x^{8}} \]
1/168*(56*e*x^11 + 84*(d + 10*e)*x^10 + 840*(2*d + 9*e)*x^9 + 2520*(3*d + 8*e)*x^8*log(x) - 5040*(4*d + 7*e)*x^7 - 3528*(5*d + 6*e)*x^6 - 2352*(6*d + 5*e)*x^5 - 1260*(7*d + 4*e)*x^4 - 504*(8*d + 3*e)*x^3 - 140*(9*d + 2*e)* x^2 - 24*(10*d + e)*x - 21*d)/x^8
Time = 2.68 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^9} \, dx=\frac {e x^{3}}{3} + x^{2} \left (\frac {d}{2} + 5 e\right ) + x \left (10 d + 45 e\right ) + 15 \cdot \left (3 d + 8 e\right ) \log {\left (x \right )} + \frac {- 21 d + x^{7} \left (- 20160 d - 35280 e\right ) + x^{6} \left (- 17640 d - 21168 e\right ) + x^{5} \left (- 14112 d - 11760 e\right ) + x^{4} \left (- 8820 d - 5040 e\right ) + x^{3} \left (- 4032 d - 1512 e\right ) + x^{2} \left (- 1260 d - 280 e\right ) + x \left (- 240 d - 24 e\right )}{168 x^{8}} \]
e*x**3/3 + x**2*(d/2 + 5*e) + x*(10*d + 45*e) + 15*(3*d + 8*e)*log(x) + (- 21*d + x**7*(-20160*d - 35280*e) + x**6*(-17640*d - 21168*e) + x**5*(-1411 2*d - 11760*e) + x**4*(-8820*d - 5040*e) + x**3*(-4032*d - 1512*e) + x**2* (-1260*d - 280*e) + x*(-240*d - 24*e))/(168*x**8)
Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^9} \, dx=\frac {1}{3} \, e x^{3} + \frac {1}{2} \, {\left (d + 10 \, e\right )} x^{2} + 5 \, {\left (2 \, d + 9 \, e\right )} x + 15 \, {\left (3 \, d + 8 \, e\right )} \log \left (x\right ) - \frac {5040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2352 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1260 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 504 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 140 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 24 \, {\left (10 \, d + e\right )} x + 21 \, d}{168 \, x^{8}} \]
1/3*e*x^3 + 1/2*(d + 10*e)*x^2 + 5*(2*d + 9*e)*x + 15*(3*d + 8*e)*log(x) - 1/168*(5040*(4*d + 7*e)*x^7 + 3528*(5*d + 6*e)*x^6 + 2352*(6*d + 5*e)*x^5 + 1260*(7*d + 4*e)*x^4 + 504*(8*d + 3*e)*x^3 + 140*(9*d + 2*e)*x^2 + 24*( 10*d + e)*x + 21*d)/x^8
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^9} \, dx=\frac {1}{3} \, e x^{3} + \frac {1}{2} \, d x^{2} + 5 \, e x^{2} + 10 \, d x + 45 \, e x + 15 \, {\left (3 \, d + 8 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {5040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3528 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2352 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1260 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 504 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 140 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 24 \, {\left (10 \, d + e\right )} x + 21 \, d}{168 \, x^{8}} \]
1/3*e*x^3 + 1/2*d*x^2 + 5*e*x^2 + 10*d*x + 45*e*x + 15*(3*d + 8*e)*log(abs (x)) - 1/168*(5040*(4*d + 7*e)*x^7 + 3528*(5*d + 6*e)*x^6 + 2352*(6*d + 5* e)*x^5 + 1260*(7*d + 4*e)*x^4 + 504*(8*d + 3*e)*x^3 + 140*(9*d + 2*e)*x^2 + 24*(10*d + e)*x + 21*d)/x^8
Time = 9.99 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^9} \, dx=x^2\,\left (\frac {d}{2}+5\,e\right )+\ln \left (x\right )\,\left (45\,d+120\,e\right )+\frac {e\,x^3}{3}+x\,\left (10\,d+45\,e\right )-\frac {\left (120\,d+210\,e\right )\,x^7+\left (105\,d+126\,e\right )\,x^6+\left (84\,d+70\,e\right )\,x^5+\left (\frac {105\,d}{2}+30\,e\right )\,x^4+\left (24\,d+9\,e\right )\,x^3+\left (\frac {15\,d}{2}+\frac {5\,e}{3}\right )\,x^2+\left (\frac {10\,d}{7}+\frac {e}{7}\right )\,x+\frac {d}{8}}{x^8} \]