Integrand size = 19, antiderivative size = 137 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{10}} \, dx=-\frac {d}{9 x^9}-\frac {10 d+e}{8 x^8}-\frac {5 (9 d+2 e)}{7 x^7}-\frac {5 (8 d+3 e)}{2 x^6}-\frac {6 (7 d+4 e)}{x^5}-\frac {21 (6 d+5 e)}{2 x^4}-\frac {14 (5 d+6 e)}{x^3}-\frac {15 (4 d+7 e)}{x^2}-\frac {15 (3 d+8 e)}{x}+(d+10 e) x+\frac {e x^2}{2}+5 (2 d+9 e) \log (x) \]
-1/9*d/x^9+1/8*(-10*d-e)/x^8-5/7*(9*d+2*e)/x^7-5/2*(8*d+3*e)/x^6-6*(7*d+4* e)/x^5-21/2*(6*d+5*e)/x^4-14*(5*d+6*e)/x^3-15*(4*d+7*e)/x^2-15*(3*d+8*e)/x +(d+10*e)*x+1/2*e*x^2+5*(2*d+9*e)*ln(x)
Time = 0.03 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{10}} \, dx=-\frac {d}{9 x^9}+\frac {-10 d-e}{8 x^8}-\frac {5 (9 d+2 e)}{7 x^7}-\frac {5 (8 d+3 e)}{2 x^6}-\frac {6 (7 d+4 e)}{x^5}-\frac {21 (6 d+5 e)}{2 x^4}-\frac {14 (5 d+6 e)}{x^3}-\frac {15 (4 d+7 e)}{x^2}-\frac {15 (3 d+8 e)}{x}+(d+10 e) x+\frac {e x^2}{2}+5 (2 d+9 e) \log (x) \]
-1/9*d/x^9 + (-10*d - e)/(8*x^8) - (5*(9*d + 2*e))/(7*x^7) - (5*(8*d + 3*e ))/(2*x^6) - (6*(7*d + 4*e))/x^5 - (21*(6*d + 5*e))/(2*x^4) - (14*(5*d + 6 *e))/x^3 - (15*(4*d + 7*e))/x^2 - (15*(3*d + 8*e))/x + (d + 10*e)*x + (e*x ^2)/2 + 5*(2*d + 9*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.
\(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x^{10}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^{10}}dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (\frac {10 d+e}{x^9}+\frac {5 (9 d+2 e)}{x^8}+\frac {15 (8 d+3 e)}{x^7}+\frac {30 (7 d+4 e)}{x^6}+\frac {42 (6 d+5 e)}{x^5}+\frac {42 (5 d+6 e)}{x^4}+\frac {30 (4 d+7 e)}{x^3}+\frac {15 (3 d+8 e)}{x^2}+\frac {5 (2 d+9 e)}{x}+d \left (\frac {10 e}{d}+1\right )+\frac {d}{x^{10}}+e x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {10 d+e}{8 x^8}-\frac {5 (9 d+2 e)}{7 x^7}-\frac {5 (8 d+3 e)}{2 x^6}-\frac {6 (7 d+4 e)}{x^5}-\frac {21 (6 d+5 e)}{2 x^4}-\frac {14 (5 d+6 e)}{x^3}-\frac {15 (4 d+7 e)}{x^2}+x (d+10 e)-\frac {15 (3 d+8 e)}{x}+5 (2 d+9 e) \log (x)-\frac {d}{9 x^9}+\frac {e x^2}{2}\) |
-1/9*d/x^9 - (10*d + e)/(8*x^8) - (5*(9*d + 2*e))/(7*x^7) - (5*(8*d + 3*e) )/(2*x^6) - (6*(7*d + 4*e))/x^5 - (21*(6*d + 5*e))/(2*x^4) - (14*(5*d + 6* e))/x^3 - (15*(4*d + 7*e))/x^2 - (15*(3*d + 8*e))/x + (d + 10*e)*x + (e*x^ 2)/2 + 5*(2*d + 9*e)*Log[x]
3.6.76.3.1 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 = 119, normalized size of antiderivative = 0.87
method | result | size |
risch | \(\frac {e \,x^{2}}{2}+d x +10 e x +\frac {\left (-45 d -120 e \right ) x^{8}+\left (-60 d -105 e \right ) x^{7}+\left (-70 d -84 e \right ) x^{6}+\left (-63 d -\frac {105 e}{2}\right ) x^{5}+\left (-42 d -24 e \right ) x^{4}+\left (-20 d -\frac {15 e}{2}\right ) x^{3}+\left (-\frac {45 d}{7}-\frac {10 e}{7}\right ) x^{2}+\left (-\frac {5 d}{4}-\frac {e}{8}\right ) x -\frac {d}{9}}{x^{9}}+10 d \ln \left (x \right )+45 e \ln \left (x \right )\) | \(119\) |
norman | \(\frac {\left (-70 d -84 e \right ) x^{6}+\left (-63 d -\frac {105 e}{2}\right ) x^{5}+\left (-60 d -105 e \right ) x^{7}+\left (-45 d -120 e \right ) x^{8}+\left (-42 d -24 e \right ) x^{4}+\left (-20 d -\frac {15 e}{2}\right ) x^{3}+\left (d +10 e \right ) x^{10}+\left (-\frac {45 d}{7}-\frac {10 e}{7}\right ) x^{2}+\left (-\frac {5 d}{4}-\frac {e}{8}\right ) x -\frac {d}{9}+\frac {e \,x^{11}}{2}}{x^{9}}+\left (10 d +45 e \right ) \ln \left (x \right )\) | \(121\) |
default | \(\frac {e \,x^{2}}{2}+d x +10 e x -\frac {120 d +45 e}{6 x^{6}}-\frac {252 d +210 e}{4 x^{4}}-\frac {210 d +120 e}{5 x^{5}}+\left (10 d +45 e \right ) \ln \left (x \right )-\frac {120 d +210 e}{2 x^{2}}-\frac {d}{9 x^{9}}-\frac {45 d +10 e}{7 x^{7}}-\frac {45 d +120 e}{x}-\frac {210 d +252 e}{3 x^{3}}-\frac {10 d +e}{8 x^{8}}\) | \(125\) |
parallelrisch | \(\frac {252 e \,x^{11}+5040 \ln \left (x \right ) x^{9} d +22680 \ln \left (x \right ) x^{9} e +504 d \,x^{10}+5040 e \,x^{10}-22680 d \,x^{8}-60480 e \,x^{8}-30240 d \,x^{7}-52920 e \,x^{7}-35280 d \,x^{6}-42336 e \,x^{6}-31752 d \,x^{5}-26460 e \,x^{5}-21168 d \,x^{4}-12096 e \,x^{4}-10080 d \,x^{3}-3780 e \,x^{3}-3240 d \,x^{2}-720 e \,x^{2}-630 d x -63 e x -56 d}{504 x^{9}}\) | \(136\) |
1/2*e*x^2+d*x+10*e*x+((-45*d-120*e)*x^8+(-60*d-105*e)*x^7+(-70*d-84*e)*x^6 +(-63*d-105/2*e)*x^5+(-42*d-24*e)*x^4+(-20*d-15/2*e)*x^3+(-45/7*d-10/7*e)* x^2+(-5/4*d-1/8*e)*x-1/9*d)/x^9+10*d*ln(x)+45*e*ln(x)
Time = 0.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{10}} \, dx=\frac {252 \, e x^{11} + 504 \, {\left (d + 10 \, e\right )} x^{10} + 2520 \, {\left (2 \, d + 9 \, e\right )} x^{9} \log \left (x\right ) - 7560 \, {\left (3 \, d + 8 \, e\right )} x^{8} - 7560 \, {\left (4 \, d + 7 \, e\right )} x^{7} - 7056 \, {\left (5 \, d + 6 \, e\right )} x^{6} - 5292 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 3024 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 360 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 63 \, {\left (10 \, d + e\right )} x - 56 \, d}{504 \, x^{9}} \]
1/504*(252*e*x^11 + 504*(d + 10*e)*x^10 + 2520*(2*d + 9*e)*x^9*log(x) - 75 60*(3*d + 8*e)*x^8 - 7560*(4*d + 7*e)*x^7 - 7056*(5*d + 6*e)*x^6 - 5292*(6 *d + 5*e)*x^5 - 3024*(7*d + 4*e)*x^4 - 1260*(8*d + 3*e)*x^3 - 360*(9*d + 2 *e)*x^2 - 63*(10*d + e)*x - 56*d)/x^9
Time = 3.49 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{10}} \, dx=\frac {e x^{2}}{2} + x \left (d + 10 e\right ) + 5 \cdot \left (2 d + 9 e\right ) \log {\left (x \right )} + \frac {- 56 d + x^{8} \left (- 22680 d - 60480 e\right ) + x^{7} \left (- 30240 d - 52920 e\right ) + x^{6} \left (- 35280 d - 42336 e\right ) + x^{5} \left (- 31752 d - 26460 e\right ) + x^{4} \left (- 21168 d - 12096 e\right ) + x^{3} \left (- 10080 d - 3780 e\right ) + x^{2} \left (- 3240 d - 720 e\right ) + x \left (- 630 d - 63 e\right )}{504 x^{9}} \]
e*x**2/2 + x*(d + 10*e) + 5*(2*d + 9*e)*log(x) + (-56*d + x**8*(-22680*d - 60480*e) + x**7*(-30240*d - 52920*e) + x**6*(-35280*d - 42336*e) + x**5*( -31752*d - 26460*e) + x**4*(-21168*d - 12096*e) + x**3*(-10080*d - 3780*e) + x**2*(-3240*d - 720*e) + x*(-630*d - 63*e))/(504*x**9)
Time = 0.19 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{10}} \, dx=\frac {1}{2} \, e x^{2} + {\left (d + 10 \, e\right )} x + 5 \, {\left (2 \, d + 9 \, e\right )} \log \left (x\right ) - \frac {7560 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 7560 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 7056 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 5292 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 3024 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 360 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 63 \, {\left (10 \, d + e\right )} x + 56 \, d}{504 \, x^{9}} \]
1/2*e*x^2 + (d + 10*e)*x + 5*(2*d + 9*e)*log(x) - 1/504*(7560*(3*d + 8*e)* x^8 + 7560*(4*d + 7*e)*x^7 + 7056*(5*d + 6*e)*x^6 + 5292*(6*d + 5*e)*x^5 + 3024*(7*d + 4*e)*x^4 + 1260*(8*d + 3*e)*x^3 + 360*(9*d + 2*e)*x^2 + 63*(1 0*d + e)*x + 56*d)/x^9
Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{10}} \, dx=\frac {1}{2} \, e x^{2} + d x + 10 \, e x + 5 \, {\left (2 \, d + 9 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac {7560 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 7560 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 7056 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 5292 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 3024 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 1260 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 360 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 63 \, {\left (10 \, d + e\right )} x + 56 \, d}{504 \, x^{9}} \]
1/2*e*x^2 + d*x + 10*e*x + 5*(2*d + 9*e)*log(abs(x)) - 1/504*(7560*(3*d + 8*e)*x^8 + 7560*(4*d + 7*e)*x^7 + 7056*(5*d + 6*e)*x^6 + 5292*(6*d + 5*e)* x^5 + 3024*(7*d + 4*e)*x^4 + 1260*(8*d + 3*e)*x^3 + 360*(9*d + 2*e)*x^2 + 63*(10*d + e)*x + 56*d)/x^9
Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{10}} \, dx=\ln \left (x\right )\,\left (10\,d+45\,e\right )+x\,\left (d+10\,e\right )+\frac {e\,x^2}{2}-\frac {\left (45\,d+120\,e\right )\,x^8+\left (60\,d+105\,e\right )\,x^7+\left (70\,d+84\,e\right )\,x^6+\left (63\,d+\frac {105\,e}{2}\right )\,x^5+\left (42\,d+24\,e\right )\,x^4+\left (20\,d+\frac {15\,e}{2}\right )\,x^3+\left (\frac {45\,d}{7}+\frac {10\,e}{7}\right )\,x^2+\left (\frac {5\,d}{4}+\frac {e}{8}\right )\,x+\frac {d}{9}}{x^9} \]