Integrand size = 16, antiderivative size = 216 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^3} \, dx=-\frac {a^{10} A}{2 x^2}-\frac {a^9 (10 A b+a B)}{x}+15 a^7 b^2 (8 A b+3 a B) x+15 a^6 b^3 (7 A b+4 a B) x^2+14 a^5 b^4 (6 A b+5 a B) x^3+\frac {21}{2} a^4 b^5 (5 A b+6 a B) x^4+6 a^3 b^6 (4 A b+7 a B) x^5+\frac {5}{2} a^2 b^7 (3 A b+8 a B) x^6+\frac {5}{7} a b^8 (2 A b+9 a B) x^7+\frac {1}{8} b^9 (A b+10 a B) x^8+\frac {1}{9} b^{10} B x^9+5 a^8 b (9 A b+2 a B) \log (x) \]
-1/2*a^10*A/x^2-a^9*(10*A*b+B*a)/x+15*a^7*b^2*(8*A*b+3*B*a)*x+15*a^6*b^3*( 7*A*b+4*B*a)*x^2+14*a^5*b^4*(6*A*b+5*B*a)*x^3+21/2*a^4*b^5*(5*A*b+6*B*a)*x ^4+6*a^3*b^6*(4*A*b+7*B*a)*x^5+5/2*a^2*b^7*(3*A*b+8*B*a)*x^6+5/7*a*b^8*(2* A*b+9*B*a)*x^7+1/8*b^9*(A*b+10*B*a)*x^8+1/9*b^10*B*x^9+5*a^8*b*(9*A*b+2*B* a)*ln(x)
Time = 0.06 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^3} \, dx=-\frac {10 a^9 A b}{x}+45 a^8 b^2 B x+60 a^7 b^3 x (2 A+B x)-\frac {a^{10} (A+2 B x)}{2 x^2}+35 a^6 b^4 x^2 (3 A+2 B x)+21 a^5 b^5 x^3 (4 A+3 B x)+\frac {21}{2} a^4 b^6 x^4 (5 A+4 B x)+4 a^3 b^7 x^5 (6 A+5 B x)+\frac {15}{14} a^2 b^8 x^6 (7 A+6 B x)+\frac {5}{28} a b^9 x^7 (8 A+7 B x)+\frac {1}{72} b^{10} x^8 (9 A+8 B x)+5 a^8 b (9 A b+2 a B) \log (x) \]
(-10*a^9*A*b)/x + 45*a^8*b^2*B*x + 60*a^7*b^3*x*(2*A + B*x) - (a^10*(A + 2 *B*x))/(2*x^2) + 35*a^6*b^4*x^2*(3*A + 2*B*x) + 21*a^5*b^5*x^3*(4*A + 3*B* x) + (21*a^4*b^6*x^4*(5*A + 4*B*x))/2 + 4*a^3*b^7*x^5*(6*A + 5*B*x) + (15* a^2*b^8*x^6*(7*A + 6*B*x))/14 + (5*a*b^9*x^7*(8*A + 7*B*x))/28 + (b^10*x^8 *(9*A + 8*B*x))/72 + 5*a^8*b*(9*A*b + 2*a*B)*Log[x]
Time = 0.37 (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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{x^3} \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {a^{10} A}{x^3}+\frac {a^9 (a B+10 A b)}{x^2}+\frac {5 a^8 b (2 a B+9 A b)}{x}+15 a^7 b^2 (3 a B+8 A b)+30 a^6 b^3 x (4 a B+7 A b)+42 a^5 b^4 x^2 (5 a B+6 A b)+42 a^4 b^5 x^3 (6 a B+5 A b)+30 a^3 b^6 x^4 (7 a B+4 A b)+15 a^2 b^7 x^5 (8 a B+3 A b)+b^9 x^7 (10 a B+A b)+5 a b^8 x^6 (9 a B+2 A b)+b^{10} B x^8\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{10} A}{2 x^2}-\frac {a^9 (a B+10 A b)}{x}+5 a^8 b \log (x) (2 a B+9 A b)+15 a^7 b^2 x (3 a B+8 A b)+15 a^6 b^3 x^2 (4 a B+7 A b)+14 a^5 b^4 x^3 (5 a B+6 A b)+\frac {21}{2} a^4 b^5 x^4 (6 a B+5 A b)+6 a^3 b^6 x^5 (7 a B+4 A b)+\frac {5}{2} a^2 b^7 x^6 (8 a B+3 A b)+\frac {1}{8} b^9 x^8 (10 a B+A b)+\frac {5}{7} a b^8 x^7 (9 a B+2 A b)+\frac {1}{9} b^{10} B x^9\) |
-1/2*(a^10*A)/x^2 - (a^9*(10*A*b + a*B))/x + 15*a^7*b^2*(8*A*b + 3*a*B)*x + 15*a^6*b^3*(7*A*b + 4*a*B)*x^2 + 14*a^5*b^4*(6*A*b + 5*a*B)*x^3 + (21*a^ 4*b^5*(5*A*b + 6*a*B)*x^4)/2 + 6*a^3*b^6*(4*A*b + 7*a*B)*x^5 + (5*a^2*b^7* (3*A*b + 8*a*B)*x^6)/2 + (5*a*b^8*(2*A*b + 9*a*B)*x^7)/7 + (b^9*(A*b + 10* a*B)*x^8)/8 + (b^10*B*x^9)/9 + 5*a^8*b*(9*A*b + 2*a*B)*Log[x]
3.2.50.3.1 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 = 234, normalized size of antiderivative = 1.08
| method | result | size |
| default | \(\frac {b^{10} B \,x^{9}}{9}+\frac {A \,b^{10} x^{8}}{8}+\frac {5 B a \,b^{9} x^{8}}{4}+\frac {10 A a \,b^{9} x^{7}}{7}+\frac {45 B \,a^{2} b^{8} x^{7}}{7}+\frac {15 A \,a^{2} b^{8} x^{6}}{2}+20 B \,a^{3} b^{7} x^{6}+24 A \,a^{3} b^{7} x^{5}+42 B \,a^{4} b^{6} x^{5}+\frac {105 A \,a^{4} b^{6} x^{4}}{2}+63 B \,a^{5} b^{5} x^{4}+84 A \,a^{5} b^{5} x^{3}+70 B \,a^{6} b^{4} x^{3}+105 A \,a^{6} b^{4} x^{2}+60 B \,a^{7} b^{3} x^{2}+120 a^{7} b^{3} A x +45 a^{8} b^{2} B x +5 a^{8} b \left (9 A b +2 B a \right ) \ln \left (x \right )-\frac {a^{9} \left (10 A b +B a \right )}{x}-\frac {a^{10} A}{2 x^{2}}\) | \(234\) |
| norman | \(\frac {\left (\frac {1}{8} b^{10} A +\frac {5}{4} a \,b^{9} B \right ) x^{10}+\left (\frac {10}{7} a \,b^{9} A +\frac {45}{7} a^{2} b^{8} B \right ) x^{9}+\left (\frac {15}{2} a^{2} b^{8} A +20 a^{3} b^{7} B \right ) x^{8}+\left (\frac {105}{2} a^{4} b^{6} A +63 a^{5} b^{5} B \right ) x^{6}+\left (24 a^{3} b^{7} A +42 a^{4} b^{6} B \right ) x^{7}+\left (84 a^{5} b^{5} A +70 a^{6} b^{4} B \right ) x^{5}+\left (105 a^{6} b^{4} A +60 a^{7} b^{3} B \right ) x^{4}+\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{3}+\left (-10 a^{9} b A -a^{10} B \right ) x -\frac {a^{10} A}{2}+\frac {b^{10} B \,x^{11}}{9}}{x^{2}}+\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) \ln \left (x \right )\) | \(235\) |
| risch | \(\frac {b^{10} B \,x^{9}}{9}+\frac {A \,b^{10} x^{8}}{8}+\frac {5 B a \,b^{9} x^{8}}{4}+\frac {10 A a \,b^{9} x^{7}}{7}+\frac {45 B \,a^{2} b^{8} x^{7}}{7}+\frac {15 A \,a^{2} b^{8} x^{6}}{2}+20 B \,a^{3} b^{7} x^{6}+24 A \,a^{3} b^{7} x^{5}+42 B \,a^{4} b^{6} x^{5}+\frac {105 A \,a^{4} b^{6} x^{4}}{2}+63 B \,a^{5} b^{5} x^{4}+84 A \,a^{5} b^{5} x^{3}+70 B \,a^{6} b^{4} x^{3}+105 A \,a^{6} b^{4} x^{2}+60 B \,a^{7} b^{3} x^{2}+120 a^{7} b^{3} A x +45 a^{8} b^{2} B x +\frac {\left (-10 a^{9} b A -a^{10} B \right ) x -\frac {a^{10} A}{2}}{x^{2}}+45 A \ln \left (x \right ) a^{8} b^{2}+10 B \ln \left (x \right ) a^{9} b\) | \(239\) |
| parallelrisch | \(\frac {56 b^{10} B \,x^{11}+63 A \,b^{10} x^{10}+630 B a \,b^{9} x^{10}+720 a A \,b^{9} x^{9}+3240 B \,a^{2} b^{8} x^{9}+3780 a^{2} A \,b^{8} x^{8}+10080 B \,a^{3} b^{7} x^{8}+12096 a^{3} A \,b^{7} x^{7}+21168 B \,a^{4} b^{6} x^{7}+26460 a^{4} A \,b^{6} x^{6}+31752 B \,a^{5} b^{5} x^{6}+42336 a^{5} A \,b^{5} x^{5}+35280 B \,a^{6} b^{4} x^{5}+52920 a^{6} A \,b^{4} x^{4}+30240 B \,a^{7} b^{3} x^{4}+22680 A \ln \left (x \right ) x^{2} a^{8} b^{2}+60480 a^{7} A \,b^{3} x^{3}+5040 B \ln \left (x \right ) x^{2} a^{9} b +22680 B \,a^{8} b^{2} x^{3}-5040 a^{9} A b x -504 a^{10} B x -252 a^{10} A}{504 x^{2}}\) | \(248\) |
1/9*b^10*B*x^9+1/8*A*b^10*x^8+5/4*B*a*b^9*x^8+10/7*A*a*b^9*x^7+45/7*B*a^2* b^8*x^7+15/2*A*a^2*b^8*x^6+20*B*a^3*b^7*x^6+24*A*a^3*b^7*x^5+42*B*a^4*b^6* x^5+105/2*A*a^4*b^6*x^4+63*B*a^5*b^5*x^4+84*A*a^5*b^5*x^3+70*B*a^6*b^4*x^3 +105*A*a^6*b^4*x^2+60*B*a^7*b^3*x^2+120*a^7*b^3*A*x+45*a^8*b^2*B*x+5*a^8*b *(9*A*b+2*B*a)*ln(x)-a^9*(10*A*b+B*a)/x-1/2*a^10*A/x^2
Time = 0.22 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^3} \, dx=\frac {56 \, B b^{10} x^{11} - 252 \, A a^{10} + 63 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 360 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 1260 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 3024 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 5292 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7056 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 7560 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 7560 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 2520 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} \log \left (x\right ) - 504 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{2}} \]
1/504*(56*B*b^10*x^11 - 252*A*a^10 + 63*(10*B*a*b^9 + A*b^10)*x^10 + 360*( 9*B*a^2*b^8 + 2*A*a*b^9)*x^9 + 1260*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 3024 *(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^7 + 5292*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 7056*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 7560*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^ 4 + 7560*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 2520*(2*B*a^9*b + 9*A*a^8*b^2)* x^2*log(x) - 504*(B*a^10 + 10*A*a^9*b)*x)/x^2
Time = 0.33 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^3} \, dx=\frac {B b^{10} x^{9}}{9} + 5 a^{8} b \left (9 A b + 2 B a\right ) \log {\left (x \right )} + x^{8} \left (\frac {A b^{10}}{8} + \frac {5 B a b^{9}}{4}\right ) + x^{7} \cdot \left (\frac {10 A a b^{9}}{7} + \frac {45 B a^{2} b^{8}}{7}\right ) + x^{6} \cdot \left (\frac {15 A a^{2} b^{8}}{2} + 20 B a^{3} b^{7}\right ) + x^{5} \cdot \left (24 A a^{3} b^{7} + 42 B a^{4} b^{6}\right ) + x^{4} \cdot \left (\frac {105 A a^{4} b^{6}}{2} + 63 B a^{5} b^{5}\right ) + x^{3} \cdot \left (84 A a^{5} b^{5} + 70 B a^{6} b^{4}\right ) + x^{2} \cdot \left (105 A a^{6} b^{4} + 60 B a^{7} b^{3}\right ) + x \left (120 A a^{7} b^{3} + 45 B a^{8} b^{2}\right ) + \frac {- A a^{10} + x \left (- 20 A a^{9} b - 2 B a^{10}\right )}{2 x^{2}} \]
B*b**10*x**9/9 + 5*a**8*b*(9*A*b + 2*B*a)*log(x) + x**8*(A*b**10/8 + 5*B*a *b**9/4) + x**7*(10*A*a*b**9/7 + 45*B*a**2*b**8/7) + x**6*(15*A*a**2*b**8/ 2 + 20*B*a**3*b**7) + x**5*(24*A*a**3*b**7 + 42*B*a**4*b**6) + x**4*(105*A *a**4*b**6/2 + 63*B*a**5*b**5) + x**3*(84*A*a**5*b**5 + 70*B*a**6*b**4) + x**2*(105*A*a**6*b**4 + 60*B*a**7*b**3) + x*(120*A*a**7*b**3 + 45*B*a**8*b **2) + (-A*a**10 + x*(-20*A*a**9*b - 2*B*a**10))/(2*x**2)
Time = 0.19 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^3} \, dx=\frac {1}{9} \, B b^{10} x^{9} + \frac {1}{8} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{8} + \frac {5}{7} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{7} + \frac {5}{2} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{6} + 6 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{5} + \frac {21}{2} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{4} + 14 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{3} + 15 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{2} + 15 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x + 5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} \log \left (x\right ) - \frac {A a^{10} + 2 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2 \, x^{2}} \]
1/9*B*b^10*x^9 + 1/8*(10*B*a*b^9 + A*b^10)*x^8 + 5/7*(9*B*a^2*b^8 + 2*A*a* b^9)*x^7 + 5/2*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^6 + 6*(7*B*a^4*b^6 + 4*A*a^3* b^7)*x^5 + 21/2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^4 + 14*(5*B*a^6*b^4 + 6*A*a^ 5*b^5)*x^3 + 15*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^2 + 15*(3*B*a^8*b^2 + 8*A*a^ 7*b^3)*x + 5*(2*B*a^9*b + 9*A*a^8*b^2)*log(x) - 1/2*(A*a^10 + 2*(B*a^10 + 10*A*a^9*b)*x)/x^2
Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^3} \, dx=\frac {1}{9} \, B b^{10} x^{9} + \frac {5}{4} \, B a b^{9} x^{8} + \frac {1}{8} \, A b^{10} x^{8} + \frac {45}{7} \, B a^{2} b^{8} x^{7} + \frac {10}{7} \, A a b^{9} x^{7} + 20 \, B a^{3} b^{7} x^{6} + \frac {15}{2} \, A a^{2} b^{8} x^{6} + 42 \, B a^{4} b^{6} x^{5} + 24 \, A a^{3} b^{7} x^{5} + 63 \, B a^{5} b^{5} x^{4} + \frac {105}{2} \, A a^{4} b^{6} x^{4} + 70 \, B a^{6} b^{4} x^{3} + 84 \, A a^{5} b^{5} x^{3} + 60 \, B a^{7} b^{3} x^{2} + 105 \, A a^{6} b^{4} x^{2} + 45 \, B a^{8} b^{2} x + 120 \, A a^{7} b^{3} x + 5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{10} + 2 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2 \, x^{2}} \]
1/9*B*b^10*x^9 + 5/4*B*a*b^9*x^8 + 1/8*A*b^10*x^8 + 45/7*B*a^2*b^8*x^7 + 1 0/7*A*a*b^9*x^7 + 20*B*a^3*b^7*x^6 + 15/2*A*a^2*b^8*x^6 + 42*B*a^4*b^6*x^5 + 24*A*a^3*b^7*x^5 + 63*B*a^5*b^5*x^4 + 105/2*A*a^4*b^6*x^4 + 70*B*a^6*b^ 4*x^3 + 84*A*a^5*b^5*x^3 + 60*B*a^7*b^3*x^2 + 105*A*a^6*b^4*x^2 + 45*B*a^8 *b^2*x + 120*A*a^7*b^3*x + 5*(2*B*a^9*b + 9*A*a^8*b^2)*log(abs(x)) - 1/2*( A*a^10 + 2*(B*a^10 + 10*A*a^9*b)*x)/x^2
Time = 0.36 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^3} \, dx=\ln \left (x\right )\,\left (10\,B\,a^9\,b+45\,A\,a^8\,b^2\right )-\frac {x\,\left (B\,a^{10}+10\,A\,b\,a^9\right )+\frac {A\,a^{10}}{2}}{x^2}+x^8\,\left (\frac {A\,b^{10}}{8}+\frac {5\,B\,a\,b^9}{4}\right )+\frac {B\,b^{10}\,x^9}{9}+15\,a^6\,b^3\,x^2\,\left (7\,A\,b+4\,B\,a\right )+14\,a^5\,b^4\,x^3\,\left (6\,A\,b+5\,B\,a\right )+\frac {21\,a^4\,b^5\,x^4\,\left (5\,A\,b+6\,B\,a\right )}{2}+6\,a^3\,b^6\,x^5\,\left (4\,A\,b+7\,B\,a\right )+\frac {5\,a^2\,b^7\,x^6\,\left (3\,A\,b+8\,B\,a\right )}{2}+15\,a^7\,b^2\,x\,\left (8\,A\,b+3\,B\,a\right )+\frac {5\,a\,b^8\,x^7\,\left (2\,A\,b+9\,B\,a\right )}{7} \]
log(x)*(45*A*a^8*b^2 + 10*B*a^9*b) - (x*(B*a^10 + 10*A*a^9*b) + (A*a^10)/2 )/x^2 + x^8*((A*b^10)/8 + (5*B*a*b^9)/4) + (B*b^10*x^9)/9 + 15*a^6*b^3*x^2 *(7*A*b + 4*B*a) + 14*a^5*b^4*x^3*(6*A*b + 5*B*a) + (21*a^4*b^5*x^4*(5*A*b + 6*B*a))/2 + 6*a^3*b^6*x^5*(4*A*b + 7*B*a) + (5*a^2*b^7*x^6*(3*A*b + 8*B *a))/2 + 15*a^7*b^2*x*(8*A*b + 3*B*a) + (5*a*b^8*x^7*(2*A*b + 9*B*a))/7