Integrand size = 16, antiderivative size = 216 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^4} \, dx=-\frac {a^{10} A}{3 x^3}-\frac {a^9 (10 A b+a B)}{2 x^2}-\frac {5 a^8 b (9 A b+2 a B)}{x}+30 a^6 b^3 (7 A b+4 a B) x+21 a^5 b^4 (6 A b+5 a B) x^2+14 a^4 b^5 (5 A b+6 a B) x^3+\frac {15}{2} a^3 b^6 (4 A b+7 a B) x^4+3 a^2 b^7 (3 A b+8 a B) x^5+\frac {5}{6} a b^8 (2 A b+9 a B) x^6+\frac {1}{7} b^9 (A b+10 a B) x^7+\frac {1}{8} b^{10} B x^8+15 a^7 b^2 (8 A b+3 a B) \log (x) \]
-1/3*a^10*A/x^3-1/2*a^9*(10*A*b+B*a)/x^2-5*a^8*b*(9*A*b+2*B*a)/x+30*a^6*b^ 3*(7*A*b+4*B*a)*x+21*a^5*b^4*(6*A*b+5*B*a)*x^2+14*a^4*b^5*(5*A*b+6*B*a)*x^ 3+15/2*a^3*b^6*(4*A*b+7*B*a)*x^4+3*a^2*b^7*(3*A*b+8*B*a)*x^5+5/6*a*b^8*(2* A*b+9*B*a)*x^6+1/7*b^9*(A*b+10*B*a)*x^7+1/8*b^10*B*x^8+15*a^7*b^2*(8*A*b+3 *B*a)*ln(x)
Time = 0.06 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^4} \, dx=-\frac {45 a^8 A b^2}{x}+120 a^7 b^3 B x+105 a^6 b^4 x (2 A+B x)-\frac {5 a^9 b (A+2 B x)}{x^2}+42 a^5 b^5 x^2 (3 A+2 B x)-\frac {a^{10} (2 A+3 B x)}{6 x^3}+\frac {35}{2} a^4 b^6 x^3 (4 A+3 B x)+6 a^3 b^7 x^4 (5 A+4 B x)+\frac {3}{2} a^2 b^8 x^5 (6 A+5 B x)+\frac {5}{21} a b^9 x^6 (7 A+6 B x)+\frac {1}{56} b^{10} x^7 (8 A+7 B x)+15 a^7 b^2 (8 A b+3 a B) \log (x) \]
(-45*a^8*A*b^2)/x + 120*a^7*b^3*B*x + 105*a^6*b^4*x*(2*A + B*x) - (5*a^9*b *(A + 2*B*x))/x^2 + 42*a^5*b^5*x^2*(3*A + 2*B*x) - (a^10*(2*A + 3*B*x))/(6 *x^3) + (35*a^4*b^6*x^3*(4*A + 3*B*x))/2 + 6*a^3*b^7*x^4*(5*A + 4*B*x) + ( 3*a^2*b^8*x^5*(6*A + 5*B*x))/2 + (5*a*b^9*x^6*(7*A + 6*B*x))/21 + (b^10*x^ 7*(8*A + 7*B*x))/56 + 15*a^7*b^2*(8*A*b + 3*a*B)*Log[x]
Time = 0.38 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{x^4} \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {a^{10} A}{x^4}+\frac {a^9 (a B+10 A b)}{x^3}+\frac {5 a^8 b (2 a B+9 A b)}{x^2}+\frac {15 a^7 b^2 (3 a B+8 A b)}{x}+30 a^6 b^3 (4 a B+7 A b)+42 a^5 b^4 x (5 a B+6 A b)+42 a^4 b^5 x^2 (6 a B+5 A b)+30 a^3 b^6 x^3 (7 a B+4 A b)+15 a^2 b^7 x^4 (8 a B+3 A b)+b^9 x^6 (10 a B+A b)+5 a b^8 x^5 (9 a B+2 A b)+b^{10} B x^7\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{10} A}{3 x^3}-\frac {a^9 (a B+10 A b)}{2 x^2}-\frac {5 a^8 b (2 a B+9 A b)}{x}+15 a^7 b^2 \log (x) (3 a B+8 A b)+30 a^6 b^3 x (4 a B+7 A b)+21 a^5 b^4 x^2 (5 a B+6 A b)+14 a^4 b^5 x^3 (6 a B+5 A b)+\frac {15}{2} a^3 b^6 x^4 (7 a B+4 A b)+3 a^2 b^7 x^5 (8 a B+3 A b)+\frac {1}{7} b^9 x^7 (10 a B+A b)+\frac {5}{6} a b^8 x^6 (9 a B+2 A b)+\frac {1}{8} b^{10} B x^8\) |
-1/3*(a^10*A)/x^3 - (a^9*(10*A*b + a*B))/(2*x^2) - (5*a^8*b*(9*A*b + 2*a*B ))/x + 30*a^6*b^3*(7*A*b + 4*a*B)*x + 21*a^5*b^4*(6*A*b + 5*a*B)*x^2 + 14* a^4*b^5*(5*A*b + 6*a*B)*x^3 + (15*a^3*b^6*(4*A*b + 7*a*B)*x^4)/2 + 3*a^2*b ^7*(3*A*b + 8*a*B)*x^5 + (5*a*b^8*(2*A*b + 9*a*B)*x^6)/6 + (b^9*(A*b + 10* a*B)*x^7)/7 + (b^10*B*x^8)/8 + 15*a^7*b^2*(8*A*b + 3*a*B)*Log[x]
3.2.51.3.1 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])
Time = 0.41 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(\frac {b^{10} B \,x^{8}}{8}+\frac {A \,b^{10} x^{7}}{7}+\frac {10 B a \,b^{9} x^{7}}{7}+\frac {5 A a \,b^{9} x^{6}}{3}+\frac {15 B \,a^{2} b^{8} x^{6}}{2}+9 A \,a^{2} b^{8} x^{5}+24 B \,a^{3} b^{7} x^{5}+30 A \,a^{3} b^{7} x^{4}+\frac {105 B \,a^{4} b^{6} x^{4}}{2}+70 A \,a^{4} b^{6} x^{3}+84 B \,a^{5} b^{5} x^{3}+126 A \,a^{5} b^{5} x^{2}+105 B \,a^{6} b^{4} x^{2}+210 A \,a^{6} b^{4} x +120 B \,a^{7} b^{3} x +15 a^{7} b^{2} \left (8 A b +3 B a \right ) \ln \left (x \right )-\frac {a^{10} A}{3 x^{3}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{x}-\frac {a^{9} \left (10 A b +B a \right )}{2 x^{2}}\) | \(230\) |
| norman | \(\frac {\left (\frac {1}{7} b^{10} A +\frac {10}{7} a \,b^{9} B \right ) x^{10}+\left (\frac {5}{3} a \,b^{9} A +\frac {15}{2} a^{2} b^{8} B \right ) x^{9}+\left (30 a^{3} b^{7} A +\frac {105}{2} a^{4} b^{6} B \right ) x^{7}+\left (-5 a^{9} b A -\frac {1}{2} a^{10} B \right ) x +\left (9 a^{2} b^{8} A +24 a^{3} b^{7} B \right ) x^{8}+\left (70 a^{4} b^{6} A +84 a^{5} b^{5} B \right ) x^{6}+\left (126 a^{5} b^{5} A +105 a^{6} b^{4} B \right ) x^{5}+\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{4}+\left (-45 a^{8} b^{2} A -10 a^{9} b B \right ) x^{2}-\frac {a^{10} A}{3}+\frac {b^{10} B \,x^{11}}{8}}{x^{3}}+\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) \ln \left (x \right )\) | \(235\) |
| risch | \(\frac {b^{10} B \,x^{8}}{8}+\frac {A \,b^{10} x^{7}}{7}+\frac {10 B a \,b^{9} x^{7}}{7}+\frac {5 A a \,b^{9} x^{6}}{3}+\frac {15 B \,a^{2} b^{8} x^{6}}{2}+9 A \,a^{2} b^{8} x^{5}+24 B \,a^{3} b^{7} x^{5}+30 A \,a^{3} b^{7} x^{4}+\frac {105 B \,a^{4} b^{6} x^{4}}{2}+70 A \,a^{4} b^{6} x^{3}+84 B \,a^{5} b^{5} x^{3}+126 A \,a^{5} b^{5} x^{2}+105 B \,a^{6} b^{4} x^{2}+210 A \,a^{6} b^{4} x +120 B \,a^{7} b^{3} x +\frac {\left (-45 a^{8} b^{2} A -10 a^{9} b B \right ) x^{2}+\left (-5 a^{9} b A -\frac {1}{2} a^{10} B \right ) x -\frac {a^{10} A}{3}}{x^{3}}+120 A \ln \left (x \right ) a^{7} b^{3}+45 B \ln \left (x \right ) a^{8} b^{2}\) | \(238\) |
| parallelrisch | \(\frac {21 b^{10} B \,x^{11}+24 A \,b^{10} x^{10}+240 B a \,b^{9} x^{10}+280 a A \,b^{9} x^{9}+1260 B \,a^{2} b^{8} x^{9}+1512 a^{2} A \,b^{8} x^{8}+4032 B \,a^{3} b^{7} x^{8}+5040 a^{3} A \,b^{7} x^{7}+8820 B \,a^{4} b^{6} x^{7}+11760 a^{4} A \,b^{6} x^{6}+14112 B \,a^{5} b^{5} x^{6}+21168 a^{5} A \,b^{5} x^{5}+17640 B \,a^{6} b^{4} x^{5}+20160 A \ln \left (x \right ) x^{3} a^{7} b^{3}+35280 a^{6} A \,b^{4} x^{4}+7560 B \ln \left (x \right ) x^{3} a^{8} b^{2}+20160 B \,a^{7} b^{3} x^{4}-7560 a^{8} A \,b^{2} x^{2}-1680 B \,a^{9} b \,x^{2}-840 a^{9} A b x -84 a^{10} B x -56 a^{10} A}{168 x^{3}}\) | \(248\) |
1/8*b^10*B*x^8+1/7*A*b^10*x^7+10/7*B*a*b^9*x^7+5/3*A*a*b^9*x^6+15/2*B*a^2* b^8*x^6+9*A*a^2*b^8*x^5+24*B*a^3*b^7*x^5+30*A*a^3*b^7*x^4+105/2*B*a^4*b^6* x^4+70*A*a^4*b^6*x^3+84*B*a^5*b^5*x^3+126*A*a^5*b^5*x^2+105*B*a^6*b^4*x^2+ 210*A*a^6*b^4*x+120*B*a^7*b^3*x+15*a^7*b^2*(8*A*b+3*B*a)*ln(x)-1/3*a^10*A/ x^3-5*a^8*b*(9*A*b+2*B*a)/x-1/2*a^9*(10*A*b+B*a)/x^2
Time = 0.22 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^4} \, dx=\frac {21 \, B b^{10} x^{11} - 56 \, A a^{10} + 24 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 140 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 504 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 1260 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 2352 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 3528 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 5040 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 2520 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} \log \left (x\right ) - 840 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 84 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{168 \, x^{3}} \]
1/168*(21*B*b^10*x^11 - 56*A*a^10 + 24*(10*B*a*b^9 + A*b^10)*x^10 + 140*(9 *B*a^2*b^8 + 2*A*a*b^9)*x^9 + 504*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 1260*( 7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 2352*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 35 28*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 5040*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 2520*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3*log(x) - 840*(2*B*a^9*b + 9*A*a^8*b ^2)*x^2 - 84*(B*a^10 + 10*A*a^9*b)*x)/x^3
Time = 0.51 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^4} \, dx=\frac {B b^{10} x^{8}}{8} + 15 a^{7} b^{2} \cdot \left (8 A b + 3 B a\right ) \log {\left (x \right )} + x^{7} \left (\frac {A b^{10}}{7} + \frac {10 B a b^{9}}{7}\right ) + x^{6} \cdot \left (\frac {5 A a b^{9}}{3} + \frac {15 B a^{2} b^{8}}{2}\right ) + x^{5} \cdot \left (9 A a^{2} b^{8} + 24 B a^{3} b^{7}\right ) + x^{4} \cdot \left (30 A a^{3} b^{7} + \frac {105 B a^{4} b^{6}}{2}\right ) + x^{3} \cdot \left (70 A a^{4} b^{6} + 84 B a^{5} b^{5}\right ) + x^{2} \cdot \left (126 A a^{5} b^{5} + 105 B a^{6} b^{4}\right ) + x \left (210 A a^{6} b^{4} + 120 B a^{7} b^{3}\right ) + \frac {- 2 A a^{10} + x^{2} \left (- 270 A a^{8} b^{2} - 60 B a^{9} b\right ) + x \left (- 30 A a^{9} b - 3 B a^{10}\right )}{6 x^{3}} \]
B*b**10*x**8/8 + 15*a**7*b**2*(8*A*b + 3*B*a)*log(x) + x**7*(A*b**10/7 + 1 0*B*a*b**9/7) + x**6*(5*A*a*b**9/3 + 15*B*a**2*b**8/2) + x**5*(9*A*a**2*b* *8 + 24*B*a**3*b**7) + x**4*(30*A*a**3*b**7 + 105*B*a**4*b**6/2) + x**3*(7 0*A*a**4*b**6 + 84*B*a**5*b**5) + x**2*(126*A*a**5*b**5 + 105*B*a**6*b**4) + x*(210*A*a**6*b**4 + 120*B*a**7*b**3) + (-2*A*a**10 + x**2*(-270*A*a**8 *b**2 - 60*B*a**9*b) + x*(-30*A*a**9*b - 3*B*a**10))/(6*x**3)
Time = 0.20 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^4} \, dx=\frac {1}{8} \, B b^{10} x^{8} + \frac {1}{7} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{7} + \frac {5}{6} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{6} + 3 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{5} + \frac {15}{2} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{4} + 14 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{3} + 21 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{2} + 30 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x + 15 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} \log \left (x\right ) - \frac {2 \, A a^{10} + 30 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 3 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6 \, x^{3}} \]
1/8*B*b^10*x^8 + 1/7*(10*B*a*b^9 + A*b^10)*x^7 + 5/6*(9*B*a^2*b^8 + 2*A*a* b^9)*x^6 + 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^5 + 15/2*(7*B*a^4*b^6 + 4*A*a^3 *b^7)*x^4 + 14*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^3 + 21*(5*B*a^6*b^4 + 6*A*a^5 *b^5)*x^2 + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x + 15*(3*B*a^8*b^2 + 8*A*a^7*b ^3)*log(x) - 1/6*(2*A*a^10 + 30*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 3*(B*a^10 + 10*A*a^9*b)*x)/x^3
Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^4} \, dx=\frac {1}{8} \, B b^{10} x^{8} + \frac {10}{7} \, B a b^{9} x^{7} + \frac {1}{7} \, A b^{10} x^{7} + \frac {15}{2} \, B a^{2} b^{8} x^{6} + \frac {5}{3} \, A a b^{9} x^{6} + 24 \, B a^{3} b^{7} x^{5} + 9 \, A a^{2} b^{8} x^{5} + \frac {105}{2} \, B a^{4} b^{6} x^{4} + 30 \, A a^{3} b^{7} x^{4} + 84 \, B a^{5} b^{5} x^{3} + 70 \, A a^{4} b^{6} x^{3} + 105 \, B a^{6} b^{4} x^{2} + 126 \, A a^{5} b^{5} x^{2} + 120 \, B a^{7} b^{3} x + 210 \, A a^{6} b^{4} x + 15 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, A a^{10} + 30 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 3 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{6 \, x^{3}} \]
1/8*B*b^10*x^8 + 10/7*B*a*b^9*x^7 + 1/7*A*b^10*x^7 + 15/2*B*a^2*b^8*x^6 + 5/3*A*a*b^9*x^6 + 24*B*a^3*b^7*x^5 + 9*A*a^2*b^8*x^5 + 105/2*B*a^4*b^6*x^4 + 30*A*a^3*b^7*x^4 + 84*B*a^5*b^5*x^3 + 70*A*a^4*b^6*x^3 + 105*B*a^6*b^4* x^2 + 126*A*a^5*b^5*x^2 + 120*B*a^7*b^3*x + 210*A*a^6*b^4*x + 15*(3*B*a^8* b^2 + 8*A*a^7*b^3)*log(abs(x)) - 1/6*(2*A*a^10 + 30*(2*B*a^9*b + 9*A*a^8*b ^2)*x^2 + 3*(B*a^10 + 10*A*a^9*b)*x)/x^3
Time = 0.36 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^4} \, dx=x^7\,\left (\frac {A\,b^{10}}{7}+\frac {10\,B\,a\,b^9}{7}\right )-\frac {x\,\left (\frac {B\,a^{10}}{2}+5\,A\,b\,a^9\right )+\frac {A\,a^{10}}{3}+x^2\,\left (10\,B\,a^9\,b+45\,A\,a^8\,b^2\right )}{x^3}+\ln \left (x\right )\,\left (45\,B\,a^8\,b^2+120\,A\,a^7\,b^3\right )+\frac {B\,b^{10}\,x^8}{8}+21\,a^5\,b^4\,x^2\,\left (6\,A\,b+5\,B\,a\right )+14\,a^4\,b^5\,x^3\,\left (5\,A\,b+6\,B\,a\right )+\frac {15\,a^3\,b^6\,x^4\,\left (4\,A\,b+7\,B\,a\right )}{2}+3\,a^2\,b^7\,x^5\,\left (3\,A\,b+8\,B\,a\right )+30\,a^6\,b^3\,x\,\left (7\,A\,b+4\,B\,a\right )+\frac {5\,a\,b^8\,x^6\,\left (2\,A\,b+9\,B\,a\right )}{6} \]
x^7*((A*b^10)/7 + (10*B*a*b^9)/7) - (x*((B*a^10)/2 + 5*A*a^9*b) + (A*a^10) /3 + x^2*(45*A*a^8*b^2 + 10*B*a^9*b))/x^3 + log(x)*(120*A*a^7*b^3 + 45*B*a ^8*b^2) + (B*b^10*x^8)/8 + 21*a^5*b^4*x^2*(6*A*b + 5*B*a) + 14*a^4*b^5*x^3 *(5*A*b + 6*B*a) + (15*a^3*b^6*x^4*(4*A*b + 7*B*a))/2 + 3*a^2*b^7*x^5*(3*A *b + 8*B*a) + 30*a^6*b^3*x*(7*A*b + 4*B*a) + (5*a*b^8*x^6*(2*A*b + 9*B*a)) /6