Integrand size = 19, antiderivative size = 137 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^5} \, dx=-\frac {d}{4 x^4}-\frac {10 d+e}{3 x^3}-\frac {5 (9 d+2 e)}{2 x^2}-\frac {15 (8 d+3 e)}{x}+42 (6 d+5 e) x+21 (5 d+6 e) x^2+10 (4 d+7 e) x^3+\frac {15}{4} (3 d+8 e) x^4+(2 d+9 e) x^5+\frac {1}{6} (d+10 e) x^6+\frac {e x^7}{7}+30 (7 d+4 e) \log (x) \]
-1/4*d/x^4+1/3*(-10*d-e)/x^3-5/2*(9*d+2*e)/x^2-15*(8*d+3*e)/x+42*(6*d+5*e) *x+21*(5*d+6*e)*x^2+10*(4*d+7*e)*x^3+15/4*(3*d+8*e)*x^4+(2*d+9*e)*x^5+1/6* (d+10*e)*x^6+1/7*e*x^7+30*(7*d+4*e)*ln(x)
Time = 0.03 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^5} \, dx=-\frac {d}{4 x^4}+\frac {-10 d-e}{3 x^3}-\frac {5 (9 d+2 e)}{2 x^2}-\frac {15 (8 d+3 e)}{x}+42 (6 d+5 e) x+21 (5 d+6 e) x^2+10 (4 d+7 e) x^3+\frac {15}{4} (3 d+8 e) x^4+(2 d+9 e) x^5+\frac {1}{6} (d+10 e) x^6+\frac {e x^7}{7}+30 (7 d+4 e) \log (x) \]
-1/4*d/x^4 + (-10*d - e)/(3*x^3) - (5*(9*d + 2*e))/(2*x^2) - (15*(8*d + 3* e))/x + 42*(6*d + 5*e)*x + 21*(5*d + 6*e)*x^2 + 10*(4*d + 7*e)*x^3 + (15*( 3*d + 8*e)*x^4)/4 + (2*d + 9*e)*x^5 + ((d + 10*e)*x^6)/6 + (e*x^7)/7 + 30* (7*d + 4*e)*Log[x]
Time = 0.30 (sec) , antiderivative size = 137, 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.
\(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x^5} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^5}dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (x^5 (d+10 e)+5 x^4 (2 d+9 e)+\frac {10 d+e}{x^4}+15 x^3 (3 d+8 e)+\frac {5 (9 d+2 e)}{x^3}+30 x^2 (4 d+7 e)+\frac {15 (8 d+3 e)}{x^2}+42 x (5 d+6 e)+\frac {30 (7 d+4 e)}{x}+42 (6 d+5 e)+\frac {d}{x^5}+e x^6\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} x^6 (d+10 e)+x^5 (2 d+9 e)+\frac {15}{4} x^4 (3 d+8 e)+10 x^3 (4 d+7 e)-\frac {10 d+e}{3 x^3}+21 x^2 (5 d+6 e)-\frac {5 (9 d+2 e)}{2 x^2}+42 x (6 d+5 e)-\frac {15 (8 d+3 e)}{x}+30 (7 d+4 e) \log (x)-\frac {d}{4 x^4}+\frac {e x^7}{7}\) |
-1/4*d/x^4 - (10*d + e)/(3*x^3) - (5*(9*d + 2*e))/(2*x^2) - (15*(8*d + 3*e ))/x + 42*(6*d + 5*e)*x + 21*(5*d + 6*e)*x^2 + 10*(4*d + 7*e)*x^3 + (15*(3 *d + 8*e)*x^4)/4 + (2*d + 9*e)*x^5 + ((d + 10*e)*x^6)/6 + (e*x^7)/7 + 30*( 7*d + 4*e)*Log[x]
3.6.71.3.1 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 = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90
method | result | size |
norman | \(\frac {\left (-120 d -45 e \right ) x^{3}+\left (2 d +9 e \right ) x^{9}+\left (40 d +70 e \right ) x^{7}+\left (105 d +126 e \right ) x^{6}+\left (252 d +210 e \right ) x^{5}+\left (-\frac {45 d}{2}-5 e \right ) x^{2}+\left (-\frac {10 d}{3}-\frac {e}{3}\right ) x +\left (\frac {d}{6}+\frac {5 e}{3}\right ) x^{10}+\left (\frac {45 d}{4}+30 e \right ) x^{8}-\frac {d}{4}+\frac {e \,x^{11}}{7}}{x^{4}}+\left (210 d +120 e \right ) \ln \left (x \right )\) | \(123\) |
risch | \(\frac {e \,x^{7}}{7}+\frac {d \,x^{6}}{6}+\frac {5 e \,x^{6}}{3}+2 d \,x^{5}+9 e \,x^{5}+\frac {45 d \,x^{4}}{4}+30 e \,x^{4}+40 d \,x^{3}+70 e \,x^{3}+105 d \,x^{2}+126 e \,x^{2}+252 d x +210 e x +\frac {\left (-120 d -45 e \right ) x^{3}+\left (-\frac {45 d}{2}-5 e \right ) x^{2}+\left (-\frac {10 d}{3}-\frac {e}{3}\right ) x -\frac {d}{4}}{x^{4}}+210 d \ln \left (x \right )+120 e \ln \left (x \right )\) | \(125\) |
default | \(\frac {e \,x^{7}}{7}+\frac {d \,x^{6}}{6}+\frac {5 e \,x^{6}}{3}+2 d \,x^{5}+9 e \,x^{5}+\frac {45 d \,x^{4}}{4}+30 e \,x^{4}+40 d \,x^{3}+70 e \,x^{3}+105 d \,x^{2}+126 e \,x^{2}+252 d x +210 e x -\frac {d}{4 x^{4}}+\left (210 d +120 e \right ) \ln \left (x \right )-\frac {45 d +10 e}{2 x^{2}}-\frac {120 d +45 e}{x}-\frac {10 d +e}{3 x^{3}}\) | \(126\) |
parallelrisch | \(\frac {12 e \,x^{11}+14 d \,x^{10}+140 e \,x^{10}+168 d \,x^{9}+756 e \,x^{9}+945 d \,x^{8}+2520 e \,x^{8}+3360 d \,x^{7}+5880 e \,x^{7}+8820 d \,x^{6}+10584 e \,x^{6}+17640 \ln \left (x \right ) x^{4} d +10080 \ln \left (x \right ) x^{4} e +21168 d \,x^{5}+17640 e \,x^{5}-10080 d \,x^{3}-3780 e \,x^{3}-1890 d \,x^{2}-420 e \,x^{2}-280 d x -28 e x -21 d}{84 x^{4}}\) | \(136\) |
((-120*d-45*e)*x^3+(2*d+9*e)*x^9+(40*d+70*e)*x^7+(105*d+126*e)*x^6+(252*d+ 210*e)*x^5+(-45/2*d-5*e)*x^2+(-10/3*d-1/3*e)*x+(1/6*d+5/3*e)*x^10+(45/4*d+ 30*e)*x^8-1/4*d+1/7*e*x^11)/x^4+(210*d+120*e)*ln(x)
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^5} \, dx=\frac {12 \, e x^{11} + 14 \, {\left (d + 10 \, e\right )} x^{10} + 84 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 315 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 840 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 1764 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 3528 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 2520 \, {\left (7 \, d + 4 \, e\right )} x^{4} \log \left (x\right ) - 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 210 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 28 \, {\left (10 \, d + e\right )} x - 21 \, d}{84 \, x^{4}} \]
1/84*(12*e*x^11 + 14*(d + 10*e)*x^10 + 84*(2*d + 9*e)*x^9 + 315*(3*d + 8*e )*x^8 + 840*(4*d + 7*e)*x^7 + 1764*(5*d + 6*e)*x^6 + 3528*(6*d + 5*e)*x^5 + 2520*(7*d + 4*e)*x^4*log(x) - 1260*(8*d + 3*e)*x^3 - 210*(9*d + 2*e)*x^2 - 28*(10*d + e)*x - 21*d)/x^4
Time = 0.55 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.89 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^5} \, dx=\frac {e x^{7}}{7} + x^{6} \left (\frac {d}{6} + \frac {5 e}{3}\right ) + x^{5} \cdot \left (2 d + 9 e\right ) + x^{4} \cdot \left (\frac {45 d}{4} + 30 e\right ) + x^{3} \cdot \left (40 d + 70 e\right ) + x^{2} \cdot \left (105 d + 126 e\right ) + x \left (252 d + 210 e\right ) + 30 \cdot \left (7 d + 4 e\right ) \log {\left (x \right )} + \frac {- 3 d + x^{3} \left (- 1440 d - 540 e\right ) + x^{2} \left (- 270 d - 60 e\right ) + x \left (- 40 d - 4 e\right )}{12 x^{4}} \]
e*x**7/7 + x**6*(d/6 + 5*e/3) + x**5*(2*d + 9*e) + x**4*(45*d/4 + 30*e) + x**3*(40*d + 70*e) + x**2*(105*d + 126*e) + x*(252*d + 210*e) + 30*(7*d + 4*e)*log(x) + (-3*d + x**3*(-1440*d - 540*e) + x**2*(-270*d - 60*e) + x*(- 40*d - 4*e))/(12*x**4)
Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^5} \, dx=\frac {1}{7} \, e x^{7} + \frac {1}{6} \, {\left (d + 10 \, e\right )} x^{6} + {\left (2 \, d + 9 \, e\right )} x^{5} + \frac {15}{4} \, {\left (3 \, d + 8 \, e\right )} x^{4} + 10 \, {\left (4 \, d + 7 \, e\right )} x^{3} + 21 \, {\left (5 \, d + 6 \, e\right )} x^{2} + 42 \, {\left (6 \, d + 5 \, e\right )} x + 30 \, {\left (7 \, d + 4 \, e\right )} \log \left (x\right ) - \frac {180 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 30 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 4 \, {\left (10 \, d + e\right )} x + 3 \, d}{12 \, x^{4}} \]
1/7*e*x^7 + 1/6*(d + 10*e)*x^6 + (2*d + 9*e)*x^5 + 15/4*(3*d + 8*e)*x^4 + 10*(4*d + 7*e)*x^3 + 21*(5*d + 6*e)*x^2 + 42*(6*d + 5*e)*x + 30*(7*d + 4*e )*log(x) - 1/12*(180*(8*d + 3*e)*x^3 + 30*(9*d + 2*e)*x^2 + 4*(10*d + e)*x + 3*d)/x^4
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^5} \, dx=\frac {1}{7} \, e x^{7} + \frac {1}{6} \, d x^{6} + \frac {5}{3} \, e x^{6} + 2 \, d x^{5} + 9 \, e x^{5} + \frac {45}{4} \, d x^{4} + 30 \, e x^{4} + 40 \, d x^{3} + 70 \, e x^{3} + 105 \, d x^{2} + 126 \, e x^{2} + 252 \, d x + 210 \, e x + 30 \, {\left (7 \, d + 4 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {180 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 30 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 4 \, {\left (10 \, d + e\right )} x + 3 \, d}{12 \, x^{4}} \]
1/7*e*x^7 + 1/6*d*x^6 + 5/3*e*x^6 + 2*d*x^5 + 9*e*x^5 + 45/4*d*x^4 + 30*e* x^4 + 40*d*x^3 + 70*e*x^3 + 105*d*x^2 + 126*e*x^2 + 252*d*x + 210*e*x + 30 *(7*d + 4*e)*log(abs(x)) - 1/12*(180*(8*d + 3*e)*x^3 + 30*(9*d + 2*e)*x^2 + 4*(10*d + e)*x + 3*d)/x^4
Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^5} \, dx=x^5\,\left (2\,d+9\,e\right )+x^6\,\left (\frac {d}{6}+\frac {5\,e}{3}\right )+x^4\,\left (\frac {45\,d}{4}+30\,e\right )+x^3\,\left (40\,d+70\,e\right )+x^2\,\left (105\,d+126\,e\right )+\ln \left (x\right )\,\left (210\,d+120\,e\right )-\frac {\left (120\,d+45\,e\right )\,x^3+\left (\frac {45\,d}{2}+5\,e\right )\,x^2+\left (\frac {10\,d}{3}+\frac {e}{3}\right )\,x+\frac {d}{4}}{x^4}+\frac {e\,x^7}{7}+x\,\left (252\,d+210\,e\right ) \]