Integrand size = 16, antiderivative size = 215 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{10}} \, dx=-\frac {a^{10} A}{9 x^9}-\frac {a^9 (10 A b+a B)}{8 x^8}-\frac {5 a^8 b (9 A b+2 a B)}{7 x^7}-\frac {5 a^7 b^2 (8 A b+3 a B)}{2 x^6}-\frac {6 a^6 b^3 (7 A b+4 a B)}{x^5}-\frac {21 a^5 b^4 (6 A b+5 a B)}{2 x^4}-\frac {14 a^4 b^5 (5 A b+6 a B)}{x^3}-\frac {15 a^3 b^6 (4 A b+7 a B)}{x^2}-\frac {15 a^2 b^7 (3 A b+8 a B)}{x}+b^9 (A b+10 a B) x+\frac {1}{2} b^{10} B x^2+5 a b^8 (2 A b+9 a B) \log (x) \]
-1/9*a^10*A/x^9-1/8*a^9*(10*A*b+B*a)/x^8-5/7*a^8*b*(9*A*b+2*B*a)/x^7-5/2*a ^7*b^2*(8*A*b+3*B*a)/x^6-6*a^6*b^3*(7*A*b+4*B*a)/x^5-21/2*a^5*b^4*(6*A*b+5 *B*a)/x^4-14*a^4*b^5*(5*A*b+6*B*a)/x^3-15*a^3*b^6*(4*A*b+7*B*a)/x^2-15*a^2 *b^7*(3*A*b+8*B*a)/x+b^9*(A*b+10*B*a)*x+1/2*b^10*B*x^2+5*a*b^8*(2*A*b+9*B* a)*ln(x)
Time = 0.06 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{10}} \, dx=-\frac {45 a^2 A b^8}{x}+10 a b^9 B x+\frac {1}{2} b^{10} x (2 A+B x)-\frac {60 a^3 b^7 (A+2 B x)}{x^2}-\frac {35 a^4 b^6 (2 A+3 B x)}{x^3}-\frac {21 a^5 b^5 (3 A+4 B x)}{x^4}-\frac {21 a^6 b^4 (4 A+5 B x)}{2 x^5}-\frac {4 a^7 b^3 (5 A+6 B x)}{x^6}-\frac {15 a^8 b^2 (6 A+7 B x)}{14 x^7}-\frac {5 a^9 b (7 A+8 B x)}{28 x^8}-\frac {a^{10} (8 A+9 B x)}{72 x^9}+5 a b^8 (2 A b+9 a B) \log (x) \]
(-45*a^2*A*b^8)/x + 10*a*b^9*B*x + (b^10*x*(2*A + B*x))/2 - (60*a^3*b^7*(A + 2*B*x))/x^2 - (35*a^4*b^6*(2*A + 3*B*x))/x^3 - (21*a^5*b^5*(3*A + 4*B*x ))/x^4 - (21*a^6*b^4*(4*A + 5*B*x))/(2*x^5) - (4*a^7*b^3*(5*A + 6*B*x))/x^ 6 - (15*a^8*b^2*(6*A + 7*B*x))/(14*x^7) - (5*a^9*b*(7*A + 8*B*x))/(28*x^8) - (a^10*(8*A + 9*B*x))/(72*x^9) + 5*a*b^8*(2*A*b + 9*a*B)*Log[x]
Time = 0.40 (sec) , antiderivative size = 215, 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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {a^{10} A}{x^{10}}+\frac {a^9 (a B+10 A b)}{x^9}+\frac {5 a^8 b (2 a B+9 A b)}{x^8}+\frac {15 a^7 b^2 (3 a B+8 A b)}{x^7}+\frac {30 a^6 b^3 (4 a B+7 A b)}{x^6}+\frac {42 a^5 b^4 (5 a B+6 A b)}{x^5}+\frac {42 a^4 b^5 (6 a B+5 A b)}{x^4}+\frac {30 a^3 b^6 (7 a B+4 A b)}{x^3}+\frac {15 a^2 b^7 (8 a B+3 A b)}{x^2}+b^9 (10 a B+A b)+\frac {5 a b^8 (9 a B+2 A b)}{x}+b^{10} B x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{10} A}{9 x^9}-\frac {a^9 (a B+10 A b)}{8 x^8}-\frac {5 a^8 b (2 a B+9 A b)}{7 x^7}-\frac {5 a^7 b^2 (3 a B+8 A b)}{2 x^6}-\frac {6 a^6 b^3 (4 a B+7 A b)}{x^5}-\frac {21 a^5 b^4 (5 a B+6 A b)}{2 x^4}-\frac {14 a^4 b^5 (6 a B+5 A b)}{x^3}-\frac {15 a^3 b^6 (7 a B+4 A b)}{x^2}-\frac {15 a^2 b^7 (8 a B+3 A b)}{x}+b^9 x (10 a B+A b)+5 a b^8 \log (x) (9 a B+2 A b)+\frac {1}{2} b^{10} B x^2\) |
-1/9*(a^10*A)/x^9 - (a^9*(10*A*b + a*B))/(8*x^8) - (5*a^8*b*(9*A*b + 2*a*B ))/(7*x^7) - (5*a^7*b^2*(8*A*b + 3*a*B))/(2*x^6) - (6*a^6*b^3*(7*A*b + 4*a *B))/x^5 - (21*a^5*b^4*(6*A*b + 5*a*B))/(2*x^4) - (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^2 - (15*a^2*b^7*(3*A*b + 8*a*B)) /x + b^9*(A*b + 10*a*B)*x + (b^10*B*x^2)/2 + 5*a*b^8*(2*A*b + 9*a*B)*Log[x ]
3.2.57.3.1 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 = 205, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(\frac {b^{10} B \,x^{2}}{2}+A \,b^{10} x +10 B a \,b^{9} x +5 a \,b^{8} \left (2 A b +9 B a \right ) \ln \left (x \right )-\frac {5 a^{7} b^{2} \left (8 A b +3 B a \right )}{2 x^{6}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{7 x^{7}}-\frac {a^{9} \left (10 A b +B a \right )}{8 x^{8}}-\frac {14 a^{4} b^{5} \left (5 A b +6 B a \right )}{x^{3}}-\frac {15 a^{2} b^{7} \left (3 A b +8 B a \right )}{x}-\frac {15 a^{3} b^{6} \left (4 A b +7 B a \right )}{x^{2}}-\frac {21 a^{5} b^{4} \left (6 A b +5 B a \right )}{2 x^{4}}-\frac {6 a^{6} b^{3} \left (7 A b +4 B a \right )}{x^{5}}-\frac {a^{10} A}{9 x^{9}}\) | \(205\) |
| risch | \(\frac {b^{10} B \,x^{2}}{2}+A \,b^{10} x +10 B a \,b^{9} x +\frac {\left (-45 a^{2} b^{8} A -120 a^{3} b^{7} B \right ) x^{8}+\left (-60 a^{3} b^{7} A -105 a^{4} b^{6} B \right ) x^{7}+\left (-70 a^{4} b^{6} A -84 a^{5} b^{5} B \right ) x^{6}+\left (-63 a^{5} b^{5} A -\frac {105}{2} a^{6} b^{4} B \right ) x^{5}+\left (-42 a^{6} b^{4} A -24 a^{7} b^{3} B \right ) x^{4}+\left (-20 a^{7} b^{3} A -\frac {15}{2} a^{8} b^{2} B \right ) x^{3}+\left (-\frac {45}{7} a^{8} b^{2} A -\frac {10}{7} a^{9} b B \right ) x^{2}+\left (-\frac {5}{4} a^{9} b A -\frac {1}{8} a^{10} B \right ) x -\frac {a^{10} A}{9}}{x^{9}}+10 A \ln \left (x \right ) a \,b^{9}+45 B \ln \left (x \right ) a^{2} b^{8}\) | \(231\) |
| norman | \(\frac {\left (-63 a^{5} b^{5} A -\frac {105}{2} a^{6} b^{4} B \right ) x^{5}+\left (-20 a^{7} b^{3} A -\frac {15}{2} a^{8} b^{2} B \right ) x^{3}+\left (-\frac {45}{7} a^{8} b^{2} A -\frac {10}{7} a^{9} b B \right ) x^{2}+\left (-\frac {5}{4} a^{9} b A -\frac {1}{8} a^{10} B \right ) x +\left (b^{10} A +10 a \,b^{9} B \right ) x^{10}+\left (-45 a^{2} b^{8} A -120 a^{3} b^{7} B \right ) x^{8}+\left (-60 a^{3} b^{7} A -105 a^{4} b^{6} B \right ) x^{7}+\left (-70 a^{4} b^{6} A -84 a^{5} b^{5} B \right ) x^{6}+\left (-42 a^{6} b^{4} A -24 a^{7} b^{3} B \right ) x^{4}-\frac {a^{10} A}{9}+\frac {b^{10} B \,x^{11}}{2}}{x^{9}}+\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) \ln \left (x \right )\) | \(234\) |
| parallelrisch | \(\frac {252 b^{10} B \,x^{11}+5040 A \ln \left (x \right ) x^{9} a \,b^{9}+504 A \,b^{10} x^{10}+22680 B \ln \left (x \right ) x^{9} a^{2} b^{8}+5040 B a \,b^{9} x^{10}-22680 a^{2} A \,b^{8} x^{8}-60480 B \,a^{3} b^{7} x^{8}-30240 a^{3} A \,b^{7} x^{7}-52920 B \,a^{4} b^{6} x^{7}-35280 a^{4} A \,b^{6} x^{6}-42336 B \,a^{5} b^{5} x^{6}-31752 a^{5} A \,b^{5} x^{5}-26460 B \,a^{6} b^{4} x^{5}-21168 a^{6} A \,b^{4} x^{4}-12096 B \,a^{7} b^{3} x^{4}-10080 a^{7} A \,b^{3} x^{3}-3780 B \,a^{8} b^{2} x^{3}-3240 a^{8} A \,b^{2} x^{2}-720 B \,a^{9} b \,x^{2}-630 a^{9} A b x -63 a^{10} B x -56 a^{10} A}{504 x^{9}}\) | \(248\) |
1/2*b^10*B*x^2+A*b^10*x+10*B*a*b^9*x+5*a*b^8*(2*A*b+9*B*a)*ln(x)-5/2*a^7*b ^2*(8*A*b+3*B*a)/x^6-5/7*a^8*b*(9*A*b+2*B*a)/x^7-1/8*a^9*(10*A*b+B*a)/x^8- 14*a^4*b^5*(5*A*b+6*B*a)/x^3-15*a^2*b^7*(3*A*b+8*B*a)/x-15*a^3*b^6*(4*A*b+ 7*B*a)/x^2-21/2*a^5*b^4*(6*A*b+5*B*a)/x^4-6*a^6*b^3*(7*A*b+4*B*a)/x^5-1/9* a^10*A/x^9
Time = 0.22 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{10}} \, dx=\frac {252 \, B b^{10} x^{11} - 56 \, A a^{10} + 504 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 2520 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} \log \left (x\right ) - 7560 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 7560 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 7056 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 5292 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 3024 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 1260 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 360 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 63 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{9}} \]
1/504*(252*B*b^10*x^11 - 56*A*a^10 + 504*(10*B*a*b^9 + A*b^10)*x^10 + 2520 *(9*B*a^2*b^8 + 2*A*a*b^9)*x^9*log(x) - 7560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x ^8 - 7560*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 - 7056*(6*B*a^5*b^5 + 5*A*a^4*b^ 6)*x^6 - 5292*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 - 3024*(4*B*a^7*b^3 + 7*A*a^ 6*b^4)*x^4 - 1260*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 - 360*(2*B*a^9*b + 9*A*a ^8*b^2)*x^2 - 63*(B*a^10 + 10*A*a^9*b)*x)/x^9
Time = 6.40 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{10}} \, dx=\frac {B b^{10} x^{2}}{2} + 5 a b^{8} \cdot \left (2 A b + 9 B a\right ) \log {\left (x \right )} + x \left (A b^{10} + 10 B a b^{9}\right ) + \frac {- 56 A a^{10} + x^{8} \left (- 22680 A a^{2} b^{8} - 60480 B a^{3} b^{7}\right ) + x^{7} \left (- 30240 A a^{3} b^{7} - 52920 B a^{4} b^{6}\right ) + x^{6} \left (- 35280 A a^{4} b^{6} - 42336 B a^{5} b^{5}\right ) + x^{5} \left (- 31752 A a^{5} b^{5} - 26460 B a^{6} b^{4}\right ) + x^{4} \left (- 21168 A a^{6} b^{4} - 12096 B a^{7} b^{3}\right ) + x^{3} \left (- 10080 A a^{7} b^{3} - 3780 B a^{8} b^{2}\right ) + x^{2} \left (- 3240 A a^{8} b^{2} - 720 B a^{9} b\right ) + x \left (- 630 A a^{9} b - 63 B a^{10}\right )}{504 x^{9}} \]
B*b**10*x**2/2 + 5*a*b**8*(2*A*b + 9*B*a)*log(x) + x*(A*b**10 + 10*B*a*b** 9) + (-56*A*a**10 + x**8*(-22680*A*a**2*b**8 - 60480*B*a**3*b**7) + x**7*( -30240*A*a**3*b**7 - 52920*B*a**4*b**6) + x**6*(-35280*A*a**4*b**6 - 42336 *B*a**5*b**5) + x**5*(-31752*A*a**5*b**5 - 26460*B*a**6*b**4) + x**4*(-211 68*A*a**6*b**4 - 12096*B*a**7*b**3) + x**3*(-10080*A*a**7*b**3 - 3780*B*a* *8*b**2) + x**2*(-3240*A*a**8*b**2 - 720*B*a**9*b) + x*(-630*A*a**9*b - 63 *B*a**10))/(504*x**9)
Time = 0.19 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{10}} \, dx=\frac {1}{2} \, B b^{10} x^{2} + {\left (10 \, B a b^{9} + A b^{10}\right )} x + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} \log \left (x\right ) - \frac {56 \, A a^{10} + 7560 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 7560 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7056 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 5292 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 3024 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 1260 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 360 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 63 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{9}} \]
1/2*B*b^10*x^2 + (10*B*a*b^9 + A*b^10)*x + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*log (x) - 1/504*(56*A*a^10 + 7560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 7560*(7*B* a^4*b^6 + 4*A*a^3*b^7)*x^7 + 7056*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 5292*( 5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 3024*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 12 60*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 360*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 6 3*(B*a^10 + 10*A*a^9*b)*x)/x^9
Time = 0.33 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{10}} \, dx=\frac {1}{2} \, B b^{10} x^{2} + 10 \, B a b^{9} x + A b^{10} x + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} \log \left ({\left | x \right |}\right ) - \frac {56 \, A a^{10} + 7560 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 7560 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7056 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 5292 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 3024 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 1260 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 360 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 63 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{9}} \]
1/2*B*b^10*x^2 + 10*B*a*b^9*x + A*b^10*x + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*log (abs(x)) - 1/504*(56*A*a^10 + 7560*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 7560* (7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 7056*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 5 292*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 3024*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4 + 1260*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 360*(2*B*a^9*b + 9*A*a^8*b^2)*x^ 2 + 63*(B*a^10 + 10*A*a^9*b)*x)/x^9
Time = 0.37 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^{10}} \, dx=x\,\left (A\,b^{10}+10\,B\,a\,b^9\right )+\ln \left (x\right )\,\left (45\,B\,a^2\,b^8+10\,A\,a\,b^9\right )-\frac {x\,\left (\frac {B\,a^{10}}{8}+\frac {5\,A\,b\,a^9}{4}\right )+\frac {A\,a^{10}}{9}+x^2\,\left (\frac {10\,B\,a^9\,b}{7}+\frac {45\,A\,a^8\,b^2}{7}\right )+x^3\,\left (\frac {15\,B\,a^8\,b^2}{2}+20\,A\,a^7\,b^3\right )+x^4\,\left (24\,B\,a^7\,b^3+42\,A\,a^6\,b^4\right )+x^6\,\left (84\,B\,a^5\,b^5+70\,A\,a^4\,b^6\right )+x^7\,\left (105\,B\,a^4\,b^6+60\,A\,a^3\,b^7\right )+x^8\,\left (120\,B\,a^3\,b^7+45\,A\,a^2\,b^8\right )+x^5\,\left (\frac {105\,B\,a^6\,b^4}{2}+63\,A\,a^5\,b^5\right )}{x^9}+\frac {B\,b^{10}\,x^2}{2} \]
x*(A*b^10 + 10*B*a*b^9) + log(x)*(45*B*a^2*b^8 + 10*A*a*b^9) - (x*((B*a^10 )/8 + (5*A*a^9*b)/4) + (A*a^10)/9 + x^2*((45*A*a^8*b^2)/7 + (10*B*a^9*b)/7 ) + x^3*(20*A*a^7*b^3 + (15*B*a^8*b^2)/2) + x^4*(42*A*a^6*b^4 + 24*B*a^7*b ^3) + x^6*(70*A*a^4*b^6 + 84*B*a^5*b^5) + x^7*(60*A*a^3*b^7 + 105*B*a^4*b^ 6) + x^8*(45*A*a^2*b^8 + 120*B*a^3*b^7) + x^5*(63*A*a^5*b^5 + (105*B*a^6*b ^4)/2))/x^9 + (B*b^10*x^2)/2