Integrand size = 16, antiderivative size = 153 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=-\frac {a^{10} B}{10 x^{10}}-\frac {10 a^9 b B}{9 x^9}-\frac {45 a^8 b^2 B}{8 x^8}-\frac {120 a^7 b^3 B}{7 x^7}-\frac {35 a^6 b^4 B}{x^6}-\frac {252 a^5 b^5 B}{5 x^5}-\frac {105 a^4 b^6 B}{2 x^4}-\frac {40 a^3 b^7 B}{x^3}-\frac {45 a^2 b^8 B}{2 x^2}-\frac {10 a b^9 B}{x}-\frac {A (a+b x)^{11}}{11 a x^{11}}+b^{10} B \log (x) \] Output:
-1/10*a^10*B/x^10-10/9*a^9*b*B/x^9-45/8*a^8*b^2*B/x^8-120/7*a^7*b^3*B/x^7- 35*a^6*b^4*B/x^6-252/5*a^5*b^5*B/x^5-105/2*a^4*b^6*B/x^4-40*a^3*b^7*B/x^3- 45/2*a^2*b^8*B/x^2-10*a*b^9*B/x-1/11*A*(b*x+a)^11/a/x^11+b^10*B*ln(x)
Time = 0.06 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=-\frac {A b^{10}}{x}-\frac {5 a b^9 (A+2 B x)}{x^2}-\frac {15 a^2 b^8 (2 A+3 B x)}{2 x^3}-\frac {10 a^3 b^7 (3 A+4 B x)}{x^4}-\frac {21 a^4 b^6 (4 A+5 B x)}{2 x^5}-\frac {42 a^5 b^5 (5 A+6 B x)}{5 x^6}-\frac {5 a^6 b^4 (6 A+7 B x)}{x^7}-\frac {15 a^7 b^3 (7 A+8 B x)}{7 x^8}-\frac {5 a^8 b^2 (8 A+9 B x)}{8 x^9}-\frac {a^9 b (9 A+10 B x)}{9 x^{10}}-\frac {a^{10} (10 A+11 B x)}{110 x^{11}}+b^{10} B \log (x) \] Input:
Integrate[((a + b*x)^10*(A + B*x))/x^12,x]
Output:
-((A*b^10)/x) - (5*a*b^9*(A + 2*B*x))/x^2 - (15*a^2*b^8*(2*A + 3*B*x))/(2* x^3) - (10*a^3*b^7*(3*A + 4*B*x))/x^4 - (21*a^4*b^6*(4*A + 5*B*x))/(2*x^5) - (42*a^5*b^5*(5*A + 6*B*x))/(5*x^6) - (5*a^6*b^4*(6*A + 7*B*x))/x^7 - (1 5*a^7*b^3*(7*A + 8*B*x))/(7*x^8) - (5*a^8*b^2*(8*A + 9*B*x))/(8*x^9) - (a^ 9*b*(9*A + 10*B*x))/(9*x^10) - (a^10*(10*A + 11*B*x))/(110*x^11) + b^10*B* Log[x]
Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {87, 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.
\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle B \int \frac {(a+b x)^{10}}{x^{11}}dx-\frac {A (a+b x)^{11}}{11 a x^{11}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle B \int \left (\frac {a^{10}}{x^{11}}+\frac {10 b a^9}{x^{10}}+\frac {45 b^2 a^8}{x^9}+\frac {120 b^3 a^7}{x^8}+\frac {210 b^4 a^6}{x^7}+\frac {252 b^5 a^5}{x^6}+\frac {210 b^6 a^4}{x^5}+\frac {120 b^7 a^3}{x^4}+\frac {45 b^8 a^2}{x^3}+\frac {10 b^9 a}{x^2}+\frac {b^{10}}{x}\right )dx-\frac {A (a+b x)^{11}}{11 a x^{11}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle B \left (-\frac {a^{10}}{10 x^{10}}-\frac {10 a^9 b}{9 x^9}-\frac {45 a^8 b^2}{8 x^8}-\frac {120 a^7 b^3}{7 x^7}-\frac {35 a^6 b^4}{x^6}-\frac {252 a^5 b^5}{5 x^5}-\frac {105 a^4 b^6}{2 x^4}-\frac {40 a^3 b^7}{x^3}-\frac {45 a^2 b^8}{2 x^2}-\frac {10 a b^9}{x}+b^{10} \log (x)\right )-\frac {A (a+b x)^{11}}{11 a x^{11}}\) |
Input:
Int[((a + b*x)^10*(A + B*x))/x^12,x]
Output:
-1/11*(A*(a + b*x)^11)/(a*x^11) + B*(-1/10*a^10/x^10 - (10*a^9*b)/(9*x^9) - (45*a^8*b^2)/(8*x^8) - (120*a^7*b^3)/(7*x^7) - (35*a^6*b^4)/x^6 - (252*a ^5*b^5)/(5*x^5) - (105*a^4*b^6)/(2*x^4) - (40*a^3*b^7)/x^3 - (45*a^2*b^8)/ (2*x^2) - (10*a*b^9)/x + b^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*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]))))
Time = 0.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.35
method | result | size |
default | \(-\frac {5 a^{2} b^{7} \left (3 A b +8 B a \right )}{x^{3}}-\frac {42 a^{4} b^{5} \left (5 A b +6 B a \right )}{5 x^{5}}-\frac {a^{10} A}{11 x^{11}}-\frac {5 a \,b^{8} \left (2 A b +9 B a \right )}{2 x^{2}}-\frac {30 a^{6} b^{3} \left (7 A b +4 B a \right )}{7 x^{7}}-\frac {15 a^{3} b^{6} \left (4 A b +7 B a \right )}{2 x^{4}}-\frac {15 a^{7} b^{2} \left (8 A b +3 B a \right )}{8 x^{8}}+b^{10} B \ln \left (x \right )-\frac {a^{9} \left (10 A b +B a \right )}{10 x^{10}}-\frac {b^{9} \left (A b +10 B a \right )}{x}-\frac {7 a^{5} b^{4} \left (6 A b +5 B a \right )}{x^{6}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{9 x^{9}}\) | \(206\) |
norman | \(\frac {\left (-5 a \,b^{9} A -\frac {45}{2} a^{2} b^{8} B \right ) x^{9}+\left (-30 a^{3} b^{7} A -\frac {105}{2} a^{4} b^{6} B \right ) x^{7}+\left (-42 a^{4} b^{6} A -\frac {252}{5} a^{5} b^{5} B \right ) x^{6}+\left (-30 a^{6} b^{4} A -\frac {120}{7} a^{7} b^{3} B \right ) x^{4}+\left (-15 a^{7} b^{3} A -\frac {45}{8} a^{8} b^{2} B \right ) x^{3}+\left (-5 a^{8} b^{2} A -\frac {10}{9} a^{9} b B \right ) x^{2}+\left (-a^{9} b A -\frac {1}{10} a^{10} B \right ) x +\left (-b^{10} A -10 a \,b^{9} B \right ) x^{10}+\left (-15 a^{2} b^{8} A -40 a^{3} b^{7} B \right ) x^{8}+\left (-42 a^{5} b^{5} A -35 a^{6} b^{4} B \right ) x^{5}-\frac {a^{10} A}{11}}{x^{11}}+b^{10} B \ln \left (x \right )\) | \(234\) |
risch | \(\frac {\left (-5 a \,b^{9} A -\frac {45}{2} a^{2} b^{8} B \right ) x^{9}+\left (-30 a^{3} b^{7} A -\frac {105}{2} a^{4} b^{6} B \right ) x^{7}+\left (-42 a^{4} b^{6} A -\frac {252}{5} a^{5} b^{5} B \right ) x^{6}+\left (-30 a^{6} b^{4} A -\frac {120}{7} a^{7} b^{3} B \right ) x^{4}+\left (-15 a^{7} b^{3} A -\frac {45}{8} a^{8} b^{2} B \right ) x^{3}+\left (-5 a^{8} b^{2} A -\frac {10}{9} a^{9} b B \right ) x^{2}+\left (-a^{9} b A -\frac {1}{10} a^{10} B \right ) x +\left (-b^{10} A -10 a \,b^{9} B \right ) x^{10}+\left (-15 a^{2} b^{8} A -40 a^{3} b^{7} B \right ) x^{8}+\left (-42 a^{5} b^{5} A -35 a^{6} b^{4} B \right ) x^{5}-\frac {a^{10} A}{11}}{x^{11}}+b^{10} B \ln \left (x \right )\) | \(234\) |
parallelrisch | \(-\frac {-27720 b^{10} B \ln \left (x \right ) x^{11}+27720 A \,b^{10} x^{10}+277200 B a \,b^{9} x^{10}+138600 a A \,b^{9} x^{9}+623700 B \,a^{2} b^{8} x^{9}+415800 a^{2} A \,b^{8} x^{8}+1108800 B \,a^{3} b^{7} x^{8}+831600 a^{3} A \,b^{7} x^{7}+1455300 B \,a^{4} b^{6} x^{7}+1164240 a^{4} A \,b^{6} x^{6}+1397088 B \,a^{5} b^{5} x^{6}+1164240 a^{5} A \,b^{5} x^{5}+970200 B \,a^{6} b^{4} x^{5}+831600 a^{6} A \,b^{4} x^{4}+475200 B \,a^{7} b^{3} x^{4}+415800 a^{7} A \,b^{3} x^{3}+155925 B \,a^{8} b^{2} x^{3}+138600 a^{8} A \,b^{2} x^{2}+30800 B \,a^{9} b \,x^{2}+27720 a^{9} A b x +2772 B \,a^{10} x +2520 a^{10} A}{27720 x^{11}}\) | \(246\) |
Input:
int((b*x+a)^10*(B*x+A)/x^12,x,method=_RETURNVERBOSE)
Output:
-5*a^2*b^7*(3*A*b+8*B*a)/x^3-42/5*a^4*b^5*(5*A*b+6*B*a)/x^5-1/11*a^10*A/x^ 11-5/2*a*b^8*(2*A*b+9*B*a)/x^2-30/7*a^6*b^3*(7*A*b+4*B*a)/x^7-15/2*a^3*b^6 *(4*A*b+7*B*a)/x^4-15/8*a^7*b^2*(8*A*b+3*B*a)/x^8+b^10*B*ln(x)-1/10*a^9*(1 0*A*b+B*a)/x^10-b^9*(A*b+10*B*a)/x-7*a^5*b^4*(6*A*b+5*B*a)/x^6-5/9*a^8*b*( 9*A*b+2*B*a)/x^9
Time = 0.07 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=\frac {27720 \, B b^{10} x^{11} \log \left (x\right ) - 2520 \, A a^{10} - 27720 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} - 69300 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} - 138600 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 207900 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 232848 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 194040 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 118800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 51975 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 15400 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 2772 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \] Input:
integrate((b*x+a)^10*(B*x+A)/x^12,x, algorithm="fricas")
Output:
1/27720*(27720*B*b^10*x^11*log(x) - 2520*A*a^10 - 27720*(10*B*a*b^9 + A*b^ 10)*x^10 - 69300*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 - 138600*(8*B*a^3*b^7 + 3*A *a^2*b^8)*x^8 - 207900*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 232848*(6*B*a^5*b ^5 + 5*A*a^4*b^6)*x^6 - 194040*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 118800*(4 *B*a^7*b^3 + 7*A*a^6*b^4)*x^4 - 51975*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 - 15 400*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 2772*(B*a^10 + 10*A*a^9*b)*x)/x^11
Time = 9.63 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=B b^{10} \log {\left (x \right )} + \frac {- 2520 A a^{10} + x^{10} \left (- 27720 A b^{10} - 277200 B a b^{9}\right ) + x^{9} \left (- 138600 A a b^{9} - 623700 B a^{2} b^{8}\right ) + x^{8} \left (- 415800 A a^{2} b^{8} - 1108800 B a^{3} b^{7}\right ) + x^{7} \left (- 831600 A a^{3} b^{7} - 1455300 B a^{4} b^{6}\right ) + x^{6} \left (- 1164240 A a^{4} b^{6} - 1397088 B a^{5} b^{5}\right ) + x^{5} \left (- 1164240 A a^{5} b^{5} - 970200 B a^{6} b^{4}\right ) + x^{4} \left (- 831600 A a^{6} b^{4} - 475200 B a^{7} b^{3}\right ) + x^{3} \left (- 415800 A a^{7} b^{3} - 155925 B a^{8} b^{2}\right ) + x^{2} \left (- 138600 A a^{8} b^{2} - 30800 B a^{9} b\right ) + x \left (- 27720 A a^{9} b - 2772 B a^{10}\right )}{27720 x^{11}} \] Input:
integrate((b*x+a)**10*(B*x+A)/x**12,x)
Output:
B*b**10*log(x) + (-2520*A*a**10 + x**10*(-27720*A*b**10 - 277200*B*a*b**9) + x**9*(-138600*A*a*b**9 - 623700*B*a**2*b**8) + x**8*(-415800*A*a**2*b** 8 - 1108800*B*a**3*b**7) + x**7*(-831600*A*a**3*b**7 - 1455300*B*a**4*b**6 ) + x**6*(-1164240*A*a**4*b**6 - 1397088*B*a**5*b**5) + x**5*(-1164240*A*a **5*b**5 - 970200*B*a**6*b**4) + x**4*(-831600*A*a**6*b**4 - 475200*B*a**7 *b**3) + x**3*(-415800*A*a**7*b**3 - 155925*B*a**8*b**2) + x**2*(-138600*A *a**8*b**2 - 30800*B*a**9*b) + x*(-27720*A*a**9*b - 2772*B*a**10))/(27720* x**11)
Time = 0.04 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.58 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=B b^{10} \log \left (x\right ) - \frac {2520 \, A a^{10} + 27720 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 69300 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 138600 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 207900 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 232848 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 194040 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 118800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 51975 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 15400 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 2772 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \] Input:
integrate((b*x+a)^10*(B*x+A)/x^12,x, algorithm="maxima")
Output:
B*b^10*log(x) - 1/27720*(2520*A*a^10 + 27720*(10*B*a*b^9 + A*b^10)*x^10 + 69300*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 138600*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x ^8 + 207900*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 232848*(6*B*a^5*b^5 + 5*A*a^ 4*b^6)*x^6 + 194040*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 118800*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 51975*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 15400*(2*B*a^ 9*b + 9*A*a^8*b^2)*x^2 + 2772*(B*a^10 + 10*A*a^9*b)*x)/x^11
Time = 0.12 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=B b^{10} \log \left ({\left | x \right |}\right ) - \frac {2520 \, A a^{10} + 27720 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 69300 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 138600 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 207900 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 232848 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 194040 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 118800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 51975 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 15400 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 2772 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{27720 \, x^{11}} \] Input:
integrate((b*x+a)^10*(B*x+A)/x^12,x, algorithm="giac")
Output:
B*b^10*log(abs(x)) - 1/27720*(2520*A*a^10 + 27720*(10*B*a*b^9 + A*b^10)*x^ 10 + 69300*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 138600*(8*B*a^3*b^7 + 3*A*a^2*b ^8)*x^8 + 207900*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 232848*(6*B*a^5*b^5 + 5 *A*a^4*b^6)*x^6 + 194040*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 118800*(4*B*a^7 *b^3 + 7*A*a^6*b^4)*x^4 + 51975*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 15400*(2 *B*a^9*b + 9*A*a^8*b^2)*x^2 + 2772*(B*a^10 + 10*A*a^9*b)*x)/x^11
Time = 0.24 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=B\,b^{10}\,\ln \left (x\right )-\frac {x\,\left (\frac {B\,a^{10}}{10}+A\,b\,a^9\right )+\frac {A\,a^{10}}{11}+x^2\,\left (\frac {10\,B\,a^9\,b}{9}+5\,A\,a^8\,b^2\right )+x^9\,\left (\frac {45\,B\,a^2\,b^8}{2}+5\,A\,a\,b^9\right )+x^{10}\,\left (A\,b^{10}+10\,B\,a\,b^9\right )+x^8\,\left (40\,B\,a^3\,b^7+15\,A\,a^2\,b^8\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{8}+15\,A\,a^7\,b^3\right )+x^5\,\left (35\,B\,a^6\,b^4+42\,A\,a^5\,b^5\right )+x^7\,\left (\frac {105\,B\,a^4\,b^6}{2}+30\,A\,a^3\,b^7\right )+x^4\,\left (\frac {120\,B\,a^7\,b^3}{7}+30\,A\,a^6\,b^4\right )+x^6\,\left (\frac {252\,B\,a^5\,b^5}{5}+42\,A\,a^4\,b^6\right )}{x^{11}} \] Input:
int(((A + B*x)*(a + b*x)^10)/x^12,x)
Output:
B*b^10*log(x) - (x*((B*a^10)/10 + A*a^9*b) + (A*a^10)/11 + x^2*(5*A*a^8*b^ 2 + (10*B*a^9*b)/9) + x^9*((45*B*a^2*b^8)/2 + 5*A*a*b^9) + x^10*(A*b^10 + 10*B*a*b^9) + x^8*(15*A*a^2*b^8 + 40*B*a^3*b^7) + x^3*(15*A*a^7*b^3 + (45* B*a^8*b^2)/8) + x^5*(42*A*a^5*b^5 + 35*B*a^6*b^4) + x^7*(30*A*a^3*b^7 + (1 05*B*a^4*b^6)/2) + x^4*(30*A*a^6*b^4 + (120*B*a^7*b^3)/7) + x^6*(42*A*a^4* b^6 + (252*B*a^5*b^5)/5))/x^11
Time = 0.16 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{12}} \, dx=\frac {27720 \,\mathrm {log}\left (x \right ) b^{11} x^{11}-2520 a^{11}-30492 a^{10} b x -169400 a^{9} b^{2} x^{2}-571725 a^{8} b^{3} x^{3}-1306800 a^{7} b^{4} x^{4}-2134440 a^{6} b^{5} x^{5}-2561328 a^{5} b^{6} x^{6}-2286900 a^{4} b^{7} x^{7}-1524600 a^{3} b^{8} x^{8}-762300 a^{2} b^{9} x^{9}-304920 a \,b^{10} x^{10}}{27720 x^{11}} \] Input:
int((b*x+a)^10*(B*x+A)/x^12,x)
Output:
(27720*log(x)*b**11*x**11 - 2520*a**11 - 30492*a**10*b*x - 169400*a**9*b** 2*x**2 - 571725*a**8*b**3*x**3 - 1306800*a**7*b**4*x**4 - 2134440*a**6*b** 5*x**5 - 2561328*a**5*b**6*x**6 - 2286900*a**4*b**7*x**7 - 1524600*a**3*b* *8*x**8 - 762300*a**2*b**9*x**9 - 304920*a*b**10*x**10)/(27720*x**11)