Integrand size = 16, antiderivative size = 217 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^2} \, dx=-\frac {a^{10} A}{x}+5 a^8 b (9 A b+2 a B) x+\frac {15}{2} a^7 b^2 (8 A b+3 a B) x^2+10 a^6 b^3 (7 A b+4 a B) x^3+\frac {21}{2} a^5 b^4 (6 A b+5 a B) x^4+\frac {42}{5} a^4 b^5 (5 A b+6 a B) x^5+5 a^3 b^6 (4 A b+7 a B) x^6+\frac {15}{7} a^2 b^7 (3 A b+8 a B) x^7+\frac {5}{8} a b^8 (2 A b+9 a B) x^8+\frac {1}{9} b^9 (A b+10 a B) x^9+\frac {1}{10} b^{10} B x^{10}+a^9 (10 A b+a B) \log (x) \]
-a^10*A/x+5*a^8*b*(9*A*b+2*B*a)*x+15/2*a^7*b^2*(8*A*b+3*B*a)*x^2+10*a^6*b^ 3*(7*A*b+4*B*a)*x^3+21/2*a^5*b^4*(6*A*b+5*B*a)*x^4+42/5*a^4*b^5*(5*A*b+6*B *a)*x^5+5*a^3*b^6*(4*A*b+7*B*a)*x^6+15/7*a^2*b^7*(3*A*b+8*B*a)*x^7+5/8*a*b ^8*(2*A*b+9*B*a)*x^8+1/9*b^9*(A*b+10*B*a)*x^9+1/10*b^10*B*x^10+a^9*(10*A*b +B*a)*ln(x)
Time = 0.05 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^2} \, dx=-\frac {a^{10} A}{x}+10 a^9 b B x+\frac {45}{2} a^8 b^2 x (2 A+B x)+20 a^7 b^3 x^2 (3 A+2 B x)+\frac {35}{2} a^6 b^4 x^3 (4 A+3 B x)+\frac {63}{5} a^5 b^5 x^4 (5 A+4 B x)+7 a^4 b^6 x^5 (6 A+5 B x)+\frac {20}{7} a^3 b^7 x^6 (7 A+6 B x)+\frac {45}{56} a^2 b^8 x^7 (8 A+7 B x)+\frac {5}{36} a b^9 x^8 (9 A+8 B x)+\frac {1}{90} b^{10} x^9 (10 A+9 B x)+a^9 (10 A b+a B) \log (x) \]
-((a^10*A)/x) + 10*a^9*b*B*x + (45*a^8*b^2*x*(2*A + B*x))/2 + 20*a^7*b^3*x ^2*(3*A + 2*B*x) + (35*a^6*b^4*x^3*(4*A + 3*B*x))/2 + (63*a^5*b^5*x^4*(5*A + 4*B*x))/5 + 7*a^4*b^6*x^5*(6*A + 5*B*x) + (20*a^3*b^7*x^6*(7*A + 6*B*x) )/7 + (45*a^2*b^8*x^7*(8*A + 7*B*x))/56 + (5*a*b^9*x^8*(9*A + 8*B*x))/36 + (b^10*x^9*(10*A + 9*B*x))/90 + a^9*(10*A*b + a*B)*Log[x]
Time = 0.38 (sec) , antiderivative size = 217, 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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{x^2} \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {a^{10} A}{x^2}+\frac {a^9 (a B+10 A b)}{x}+5 a^8 b (2 a B+9 A b)+15 a^7 b^2 x (3 a B+8 A b)+30 a^6 b^3 x^2 (4 a B+7 A b)+42 a^5 b^4 x^3 (5 a B+6 A b)+42 a^4 b^5 x^4 (6 a B+5 A b)+30 a^3 b^6 x^5 (7 a B+4 A b)+15 a^2 b^7 x^6 (8 a B+3 A b)+b^9 x^8 (10 a B+A b)+5 a b^8 x^7 (9 a B+2 A b)+b^{10} B x^9\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{10} A}{x}+a^9 \log (x) (a B+10 A b)+5 a^8 b x (2 a B+9 A b)+\frac {15}{2} a^7 b^2 x^2 (3 a B+8 A b)+10 a^6 b^3 x^3 (4 a B+7 A b)+\frac {21}{2} a^5 b^4 x^4 (5 a B+6 A b)+\frac {42}{5} a^4 b^5 x^5 (6 a B+5 A b)+5 a^3 b^6 x^6 (7 a B+4 A b)+\frac {15}{7} a^2 b^7 x^7 (8 a B+3 A b)+\frac {1}{9} b^9 x^9 (10 a B+A b)+\frac {5}{8} a b^8 x^8 (9 a B+2 A b)+\frac {1}{10} b^{10} B x^{10}\) |
-((a^10*A)/x) + 5*a^8*b*(9*A*b + 2*a*B)*x + (15*a^7*b^2*(8*A*b + 3*a*B)*x^ 2)/2 + 10*a^6*b^3*(7*A*b + 4*a*B)*x^3 + (21*a^5*b^4*(6*A*b + 5*a*B)*x^4)/2 + (42*a^4*b^5*(5*A*b + 6*a*B)*x^5)/5 + 5*a^3*b^6*(4*A*b + 7*a*B)*x^6 + (1 5*a^2*b^7*(3*A*b + 8*a*B)*x^7)/7 + (5*a*b^8*(2*A*b + 9*a*B)*x^8)/8 + (b^9* (A*b + 10*a*B)*x^9)/9 + (b^10*B*x^10)/10 + a^9*(10*A*b + a*B)*Log[x]
3.2.49.3.1 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 = 0.42 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.09
| method | result | size |
| norman | \(\frac {\left (\frac {1}{9} b^{10} A +\frac {10}{9} a \,b^{9} B \right ) x^{10}+\left (\frac {5}{4} a \,b^{9} A +\frac {45}{8} a^{2} b^{8} B \right ) x^{9}+\left (\frac {45}{7} a^{2} b^{8} A +\frac {120}{7} a^{3} b^{7} B \right ) x^{8}+\left (42 a^{4} b^{6} A +\frac {252}{5} a^{5} b^{5} B \right ) x^{6}+\left (63 a^{5} b^{5} A +\frac {105}{2} a^{6} b^{4} B \right ) x^{5}+\left (60 a^{7} b^{3} A +\frac {45}{2} a^{8} b^{2} B \right ) x^{3}+\left (20 a^{3} b^{7} A +35 a^{4} b^{6} B \right ) x^{7}+\left (70 a^{6} b^{4} A +40 a^{7} b^{3} B \right ) x^{4}+\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{2}-a^{10} A +\frac {b^{10} B \,x^{11}}{10}}{x}+\left (10 a^{9} b A +a^{10} B \right ) \ln \left (x \right )\) | \(236\) |
| default | \(\frac {b^{10} B \,x^{10}}{10}+\frac {A \,b^{10} x^{9}}{9}+\frac {10 B a \,b^{9} x^{9}}{9}+\frac {5 A a \,b^{9} x^{8}}{4}+\frac {45 B \,a^{2} b^{8} x^{8}}{8}+\frac {45 A \,a^{2} b^{8} x^{7}}{7}+\frac {120 B \,a^{3} b^{7} x^{7}}{7}+20 A \,a^{3} b^{7} x^{6}+35 B \,a^{4} b^{6} x^{6}+42 A \,a^{4} b^{6} x^{5}+\frac {252 B \,a^{5} b^{5} x^{5}}{5}+63 A \,a^{5} b^{5} x^{4}+\frac {105 B \,a^{6} b^{4} x^{4}}{2}+70 A \,a^{6} b^{4} x^{3}+40 B \,a^{7} b^{3} x^{3}+60 A \,a^{7} b^{3} x^{2}+\frac {45 B \,a^{8} b^{2} x^{2}}{2}+45 a^{8} b^{2} A x +10 a^{9} b B x +a^{9} \left (10 A b +B a \right ) \ln \left (x \right )-\frac {a^{10} A}{x}\) | \(237\) |
| risch | \(\frac {b^{10} B \,x^{10}}{10}+\frac {A \,b^{10} x^{9}}{9}+\frac {10 B a \,b^{9} x^{9}}{9}+\frac {5 A a \,b^{9} x^{8}}{4}+\frac {45 B \,a^{2} b^{8} x^{8}}{8}+\frac {45 A \,a^{2} b^{8} x^{7}}{7}+\frac {120 B \,a^{3} b^{7} x^{7}}{7}+20 A \,a^{3} b^{7} x^{6}+35 B \,a^{4} b^{6} x^{6}+42 A \,a^{4} b^{6} x^{5}+\frac {252 B \,a^{5} b^{5} x^{5}}{5}+63 A \,a^{5} b^{5} x^{4}+\frac {105 B \,a^{6} b^{4} x^{4}}{2}+70 A \,a^{6} b^{4} x^{3}+40 B \,a^{7} b^{3} x^{3}+60 A \,a^{7} b^{3} x^{2}+\frac {45 B \,a^{8} b^{2} x^{2}}{2}+45 a^{8} b^{2} A x +10 a^{9} b B x -\frac {a^{10} A}{x}+10 A \ln \left (x \right ) a^{9} b +B \ln \left (x \right ) a^{10}\) | \(239\) |
| parallelrisch | \(\frac {252 b^{10} B \,x^{11}+280 A \,b^{10} x^{10}+2800 B a \,b^{9} x^{10}+3150 a A \,b^{9} x^{9}+14175 B \,a^{2} b^{8} x^{9}+16200 a^{2} A \,b^{8} x^{8}+43200 B \,a^{3} b^{7} x^{8}+50400 a^{3} A \,b^{7} x^{7}+88200 B \,a^{4} b^{6} x^{7}+105840 a^{4} A \,b^{6} x^{6}+127008 B \,a^{5} b^{5} x^{6}+158760 a^{5} A \,b^{5} x^{5}+132300 B \,a^{6} b^{4} x^{5}+176400 a^{6} A \,b^{4} x^{4}+100800 B \,a^{7} b^{3} x^{4}+151200 a^{7} A \,b^{3} x^{3}+56700 B \,a^{8} b^{2} x^{3}+25200 A \ln \left (x \right ) x \,a^{9} b +113400 a^{8} A \,b^{2} x^{2}+2520 B \ln \left (x \right ) x \,a^{10}+25200 B \,a^{9} b \,x^{2}-2520 a^{10} A}{2520 x}\) | \(248\) |
((1/9*b^10*A+10/9*a*b^9*B)*x^10+(5/4*a*b^9*A+45/8*a^2*b^8*B)*x^9+(45/7*a^2 *b^8*A+120/7*a^3*b^7*B)*x^8+(42*a^4*b^6*A+252/5*a^5*b^5*B)*x^6+(63*a^5*b^5 *A+105/2*a^6*b^4*B)*x^5+(60*a^7*b^3*A+45/2*a^8*b^2*B)*x^3+(20*A*a^3*b^7+35 *B*a^4*b^6)*x^7+(70*A*a^6*b^4+40*B*a^7*b^3)*x^4+(45*A*a^8*b^2+10*B*a^9*b)* x^2-a^10*A+1/10*b^10*B*x^11)/x+(10*A*a^9*b+B*a^10)*ln(x)
Time = 0.22 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^2} \, dx=\frac {252 \, B b^{10} x^{11} - 2520 \, A a^{10} + 280 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 1575 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 5400 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 12600 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 21168 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 26460 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 25200 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 18900 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 12600 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 2520 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x \log \left (x\right )}{2520 \, x} \]
1/2520*(252*B*b^10*x^11 - 2520*A*a^10 + 280*(10*B*a*b^9 + A*b^10)*x^10 + 1 575*(9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 5400*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 12600*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 21168*(6*B*a^5*b^5 + 5*A*a^4*b^6) *x^6 + 26460*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 25200*(4*B*a^7*b^3 + 7*A*a^ 6*b^4)*x^4 + 18900*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 12600*(2*B*a^9*b + 9* A*a^8*b^2)*x^2 + 2520*(B*a^10 + 10*A*a^9*b)*x*log(x))/x
Time = 0.22 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^2} \, dx=- \frac {A a^{10}}{x} + \frac {B b^{10} x^{10}}{10} + a^{9} \cdot \left (10 A b + B a\right ) \log {\left (x \right )} + x^{9} \left (\frac {A b^{10}}{9} + \frac {10 B a b^{9}}{9}\right ) + x^{8} \cdot \left (\frac {5 A a b^{9}}{4} + \frac {45 B a^{2} b^{8}}{8}\right ) + x^{7} \cdot \left (\frac {45 A a^{2} b^{8}}{7} + \frac {120 B a^{3} b^{7}}{7}\right ) + x^{6} \cdot \left (20 A a^{3} b^{7} + 35 B a^{4} b^{6}\right ) + x^{5} \cdot \left (42 A a^{4} b^{6} + \frac {252 B a^{5} b^{5}}{5}\right ) + x^{4} \cdot \left (63 A a^{5} b^{5} + \frac {105 B a^{6} b^{4}}{2}\right ) + x^{3} \cdot \left (70 A a^{6} b^{4} + 40 B a^{7} b^{3}\right ) + x^{2} \cdot \left (60 A a^{7} b^{3} + \frac {45 B a^{8} b^{2}}{2}\right ) + x \left (45 A a^{8} b^{2} + 10 B a^{9} b\right ) \]
-A*a**10/x + B*b**10*x**10/10 + a**9*(10*A*b + B*a)*log(x) + x**9*(A*b**10 /9 + 10*B*a*b**9/9) + x**8*(5*A*a*b**9/4 + 45*B*a**2*b**8/8) + x**7*(45*A* a**2*b**8/7 + 120*B*a**3*b**7/7) + x**6*(20*A*a**3*b**7 + 35*B*a**4*b**6) + x**5*(42*A*a**4*b**6 + 252*B*a**5*b**5/5) + x**4*(63*A*a**5*b**5 + 105*B *a**6*b**4/2) + x**3*(70*A*a**6*b**4 + 40*B*a**7*b**3) + x**2*(60*A*a**7*b **3 + 45*B*a**8*b**2/2) + x*(45*A*a**8*b**2 + 10*B*a**9*b)
Time = 0.20 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^2} \, dx=\frac {1}{10} \, B b^{10} x^{10} - \frac {A a^{10}}{x} + \frac {1}{9} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{9} + \frac {5}{8} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{8} + \frac {15}{7} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{7} + 5 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{6} + \frac {42}{5} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{5} + \frac {21}{2} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{4} + 10 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{3} + \frac {15}{2} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{2} + 5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x + {\left (B a^{10} + 10 \, A a^{9} b\right )} \log \left (x\right ) \]
1/10*B*b^10*x^10 - A*a^10/x + 1/9*(10*B*a*b^9 + A*b^10)*x^9 + 5/8*(9*B*a^2 *b^8 + 2*A*a*b^9)*x^8 + 15/7*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^7 + 5*(7*B*a^4* b^6 + 4*A*a^3*b^7)*x^6 + 42/5*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^5 + 21/2*(5*B* a^6*b^4 + 6*A*a^5*b^5)*x^4 + 10*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^3 + 15/2*(3* B*a^8*b^2 + 8*A*a^7*b^3)*x^2 + 5*(2*B*a^9*b + 9*A*a^8*b^2)*x + (B*a^10 + 1 0*A*a^9*b)*log(x)
Time = 0.28 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^2} \, dx=\frac {1}{10} \, B b^{10} x^{10} + \frac {10}{9} \, B a b^{9} x^{9} + \frac {1}{9} \, A b^{10} x^{9} + \frac {45}{8} \, B a^{2} b^{8} x^{8} + \frac {5}{4} \, A a b^{9} x^{8} + \frac {120}{7} \, B a^{3} b^{7} x^{7} + \frac {45}{7} \, A a^{2} b^{8} x^{7} + 35 \, B a^{4} b^{6} x^{6} + 20 \, A a^{3} b^{7} x^{6} + \frac {252}{5} \, B a^{5} b^{5} x^{5} + 42 \, A a^{4} b^{6} x^{5} + \frac {105}{2} \, B a^{6} b^{4} x^{4} + 63 \, A a^{5} b^{5} x^{4} + 40 \, B a^{7} b^{3} x^{3} + 70 \, A a^{6} b^{4} x^{3} + \frac {45}{2} \, B a^{8} b^{2} x^{2} + 60 \, A a^{7} b^{3} x^{2} + 10 \, B a^{9} b x + 45 \, A a^{8} b^{2} x - \frac {A a^{10}}{x} + {\left (B a^{10} + 10 \, A a^{9} b\right )} \log \left ({\left | x \right |}\right ) \]
1/10*B*b^10*x^10 + 10/9*B*a*b^9*x^9 + 1/9*A*b^10*x^9 + 45/8*B*a^2*b^8*x^8 + 5/4*A*a*b^9*x^8 + 120/7*B*a^3*b^7*x^7 + 45/7*A*a^2*b^8*x^7 + 35*B*a^4*b^ 6*x^6 + 20*A*a^3*b^7*x^6 + 252/5*B*a^5*b^5*x^5 + 42*A*a^4*b^6*x^5 + 105/2* B*a^6*b^4*x^4 + 63*A*a^5*b^5*x^4 + 40*B*a^7*b^3*x^3 + 70*A*a^6*b^4*x^3 + 4 5/2*B*a^8*b^2*x^2 + 60*A*a^7*b^3*x^2 + 10*B*a^9*b*x + 45*A*a^8*b^2*x - A*a ^10/x + (B*a^10 + 10*A*a^9*b)*log(abs(x))
Time = 0.10 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^2} \, dx=x^9\,\left (\frac {A\,b^{10}}{9}+\frac {10\,B\,a\,b^9}{9}\right )+\ln \left (x\right )\,\left (B\,a^{10}+10\,A\,b\,a^9\right )-\frac {A\,a^{10}}{x}+\frac {B\,b^{10}\,x^{10}}{10}+\frac {15\,a^7\,b^2\,x^2\,\left (8\,A\,b+3\,B\,a\right )}{2}+10\,a^6\,b^3\,x^3\,\left (7\,A\,b+4\,B\,a\right )+\frac {21\,a^5\,b^4\,x^4\,\left (6\,A\,b+5\,B\,a\right )}{2}+\frac {42\,a^4\,b^5\,x^5\,\left (5\,A\,b+6\,B\,a\right )}{5}+5\,a^3\,b^6\,x^6\,\left (4\,A\,b+7\,B\,a\right )+\frac {15\,a^2\,b^7\,x^7\,\left (3\,A\,b+8\,B\,a\right )}{7}+5\,a^8\,b\,x\,\left (9\,A\,b+2\,B\,a\right )+\frac {5\,a\,b^8\,x^8\,\left (2\,A\,b+9\,B\,a\right )}{8} \]
x^9*((A*b^10)/9 + (10*B*a*b^9)/9) + log(x)*(B*a^10 + 10*A*a^9*b) - (A*a^10 )/x + (B*b^10*x^10)/10 + (15*a^7*b^2*x^2*(8*A*b + 3*B*a))/2 + 10*a^6*b^3*x ^3*(7*A*b + 4*B*a) + (21*a^5*b^4*x^4*(6*A*b + 5*B*a))/2 + (42*a^4*b^5*x^5* (5*A*b + 6*B*a))/5 + 5*a^3*b^6*x^6*(4*A*b + 7*B*a) + (15*a^2*b^7*x^7*(3*A* b + 8*B*a))/7 + 5*a^8*b*x*(9*A*b + 2*B*a) + (5*a*b^8*x^8*(2*A*b + 9*B*a))/ 8