Integrand size = 16, antiderivative size = 215 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^5} \, dx=-\frac {a^{10} A}{4 x^4}-\frac {a^9 (10 A b+a B)}{3 x^3}-\frac {5 a^8 b (9 A b+2 a B)}{2 x^2}-\frac {15 a^7 b^2 (8 A b+3 a B)}{x}+42 a^5 b^4 (6 A b+5 a B) x+21 a^4 b^5 (5 A b+6 a B) x^2+10 a^3 b^6 (4 A b+7 a B) x^3+\frac {15}{4} a^2 b^7 (3 A b+8 a B) x^4+a b^8 (2 A b+9 a B) x^5+\frac {1}{6} b^9 (A b+10 a B) x^6+\frac {1}{7} b^{10} B x^7+30 a^6 b^3 (7 A b+4 a B) \log (x) \]
-1/4*a^10*A/x^4-1/3*a^9*(10*A*b+B*a)/x^3-5/2*a^8*b*(9*A*b+2*B*a)/x^2-15*a^ 7*b^2*(8*A*b+3*B*a)/x+42*a^5*b^4*(6*A*b+5*B*a)*x+21*a^4*b^5*(5*A*b+6*B*a)* x^2+10*a^3*b^6*(4*A*b+7*B*a)*x^3+15/4*a^2*b^7*(3*A*b+8*B*a)*x^4+a*b^8*(2*A *b+9*B*a)*x^5+1/6*b^9*(A*b+10*B*a)*x^6+1/7*b^10*B*x^7+30*a^6*b^3*(7*A*b+4* B*a)*ln(x)
Time = 0.05 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^5} \, dx=-\frac {120 a^7 A b^3}{x}+210 a^6 b^4 B x+126 a^5 b^5 x (2 A+B x)-\frac {45 a^8 b^2 (A+2 B x)}{2 x^2}+35 a^4 b^6 x^2 (3 A+2 B x)-\frac {5 a^9 b (2 A+3 B x)}{3 x^3}+10 a^3 b^7 x^3 (4 A+3 B x)-\frac {a^{10} (3 A+4 B x)}{12 x^4}+\frac {9}{4} a^2 b^8 x^4 (5 A+4 B x)+\frac {1}{3} a b^9 x^5 (6 A+5 B x)+\frac {1}{42} b^{10} x^6 (7 A+6 B x)+30 a^6 b^3 (7 A b+4 a B) \log (x) \]
(-120*a^7*A*b^3)/x + 210*a^6*b^4*B*x + 126*a^5*b^5*x*(2*A + B*x) - (45*a^8 *b^2*(A + 2*B*x))/(2*x^2) + 35*a^4*b^6*x^2*(3*A + 2*B*x) - (5*a^9*b*(2*A + 3*B*x))/(3*x^3) + 10*a^3*b^7*x^3*(4*A + 3*B*x) - (a^10*(3*A + 4*B*x))/(12 *x^4) + (9*a^2*b^8*x^4*(5*A + 4*B*x))/4 + (a*b^9*x^5*(6*A + 5*B*x))/3 + (b ^10*x^6*(7*A + 6*B*x))/42 + 30*a^6*b^3*(7*A*b + 4*a*B)*Log[x]
Time = 0.37 (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.
| \(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{x^5} \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {a^{10} A}{x^5}+\frac {a^9 (a B+10 A b)}{x^4}+\frac {5 a^8 b (2 a B+9 A b)}{x^3}+\frac {15 a^7 b^2 (3 a B+8 A b)}{x^2}+\frac {30 a^6 b^3 (4 a B+7 A b)}{x}+42 a^5 b^4 (5 a B+6 A b)+42 a^4 b^5 x (6 a B+5 A b)+30 a^3 b^6 x^2 (7 a B+4 A b)+15 a^2 b^7 x^3 (8 a B+3 A b)+b^9 x^5 (10 a B+A b)+5 a b^8 x^4 (9 a B+2 A b)+b^{10} B x^6\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{10} A}{4 x^4}-\frac {a^9 (a B+10 A b)}{3 x^3}-\frac {5 a^8 b (2 a B+9 A b)}{2 x^2}-\frac {15 a^7 b^2 (3 a B+8 A b)}{x}+30 a^6 b^3 \log (x) (4 a B+7 A b)+42 a^5 b^4 x (5 a B+6 A b)+21 a^4 b^5 x^2 (6 a B+5 A b)+10 a^3 b^6 x^3 (7 a B+4 A b)+\frac {15}{4} a^2 b^7 x^4 (8 a B+3 A b)+\frac {1}{6} b^9 x^6 (10 a B+A b)+a b^8 x^5 (9 a B+2 A b)+\frac {1}{7} b^{10} B x^7\) |
-1/4*(a^10*A)/x^4 - (a^9*(10*A*b + a*B))/(3*x^3) - (5*a^8*b*(9*A*b + 2*a*B ))/(2*x^2) - (15*a^7*b^2*(8*A*b + 3*a*B))/x + 42*a^5*b^4*(6*A*b + 5*a*B)*x + 21*a^4*b^5*(5*A*b + 6*a*B)*x^2 + 10*a^3*b^6*(4*A*b + 7*a*B)*x^3 + (15*a ^2*b^7*(3*A*b + 8*a*B)*x^4)/4 + a*b^8*(2*A*b + 9*a*B)*x^5 + (b^9*(A*b + 10 *a*B)*x^6)/6 + (b^10*B*x^7)/7 + 30*a^6*b^3*(7*A*b + 4*a*B)*Log[x]
3.2.52.3.1 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 = 0.40 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {b^{10} B \,x^{7}}{7}+\frac {A \,b^{10} x^{6}}{6}+\frac {5 B a \,b^{9} x^{6}}{3}+2 A a \,b^{9} x^{5}+9 B \,a^{2} b^{8} x^{5}+\frac {45 A \,a^{2} b^{8} x^{4}}{4}+30 B \,a^{3} b^{7} x^{4}+40 A \,a^{3} b^{7} x^{3}+70 B \,a^{4} b^{6} x^{3}+105 A \,a^{4} b^{6} x^{2}+126 B \,a^{5} b^{5} x^{2}+252 A \,a^{5} b^{5} x +210 B \,a^{6} b^{4} x +30 a^{6} b^{3} \left (7 A b +4 B a \right ) \ln \left (x \right )-\frac {a^{9} \left (10 A b +B a \right )}{3 x^{3}}-\frac {15 a^{7} b^{2} \left (8 A b +3 B a \right )}{x}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{2 x^{2}}-\frac {a^{10} A}{4 x^{4}}\) | \(226\) |
| norman | \(\frac {\left (\frac {1}{6} b^{10} A +\frac {5}{3} a \,b^{9} B \right ) x^{10}+\left (\frac {45}{4} a^{2} b^{8} A +30 a^{3} b^{7} B \right ) x^{8}+\left (-\frac {45}{2} a^{8} b^{2} A -5 a^{9} b B \right ) x^{2}+\left (-\frac {10}{3} a^{9} b A -\frac {1}{3} a^{10} B \right ) x +\left (2 a \,b^{9} A +9 a^{2} b^{8} B \right ) x^{9}+\left (40 a^{3} b^{7} A +70 a^{4} b^{6} B \right ) x^{7}+\left (105 a^{4} b^{6} A +126 a^{5} b^{5} B \right ) x^{6}+\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{5}+\left (-120 a^{7} b^{3} A -45 a^{8} b^{2} B \right ) x^{3}-\frac {a^{10} A}{4}+\frac {b^{10} B \,x^{11}}{7}}{x^{4}}+\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) \ln \left (x \right )\) | \(235\) |
| risch | \(\frac {b^{10} B \,x^{7}}{7}+\frac {A \,b^{10} x^{6}}{6}+\frac {5 B a \,b^{9} x^{6}}{3}+2 A a \,b^{9} x^{5}+9 B \,a^{2} b^{8} x^{5}+\frac {45 A \,a^{2} b^{8} x^{4}}{4}+30 B \,a^{3} b^{7} x^{4}+40 A \,a^{3} b^{7} x^{3}+70 B \,a^{4} b^{6} x^{3}+105 A \,a^{4} b^{6} x^{2}+126 B \,a^{5} b^{5} x^{2}+252 A \,a^{5} b^{5} x +210 B \,a^{6} b^{4} x +\frac {\left (-120 a^{7} b^{3} A -45 a^{8} b^{2} B \right ) x^{3}+\left (-\frac {45}{2} a^{8} b^{2} A -5 a^{9} b B \right ) x^{2}+\left (-\frac {10}{3} a^{9} b A -\frac {1}{3} a^{10} B \right ) x -\frac {a^{10} A}{4}}{x^{4}}+210 A \ln \left (x \right ) a^{6} b^{4}+120 B \ln \left (x \right ) a^{7} b^{3}\) | \(237\) |
| parallelrisch | \(\frac {12 b^{10} B \,x^{11}+14 A \,b^{10} x^{10}+140 B a \,b^{9} x^{10}+168 a A \,b^{9} x^{9}+756 B \,a^{2} b^{8} x^{9}+945 a^{2} A \,b^{8} x^{8}+2520 B \,a^{3} b^{7} x^{8}+3360 a^{3} A \,b^{7} x^{7}+5880 B \,a^{4} b^{6} x^{7}+8820 a^{4} A \,b^{6} x^{6}+10584 B \,a^{5} b^{5} x^{6}+17640 A \ln \left (x \right ) x^{4} a^{6} b^{4}+21168 a^{5} A \,b^{5} x^{5}+10080 B \ln \left (x \right ) x^{4} a^{7} b^{3}+17640 B \,a^{6} b^{4} x^{5}-10080 a^{7} A \,b^{3} x^{3}-3780 B \,a^{8} b^{2} x^{3}-1890 a^{8} A \,b^{2} x^{2}-420 B \,a^{9} b \,x^{2}-280 a^{9} A b x -28 a^{10} B x -21 a^{10} A}{84 x^{4}}\) | \(248\) |
1/7*b^10*B*x^7+1/6*A*b^10*x^6+5/3*B*a*b^9*x^6+2*A*a*b^9*x^5+9*B*a^2*b^8*x^ 5+45/4*A*a^2*b^8*x^4+30*B*a^3*b^7*x^4+40*A*a^3*b^7*x^3+70*B*a^4*b^6*x^3+10 5*A*a^4*b^6*x^2+126*B*a^5*b^5*x^2+252*A*a^5*b^5*x+210*B*a^6*b^4*x+30*a^6*b ^3*(7*A*b+4*B*a)*ln(x)-1/3*a^9*(10*A*b+B*a)/x^3-15*a^7*b^2*(8*A*b+3*B*a)/x -5/2*a^8*b*(9*A*b+2*B*a)/x^2-1/4*a^10*A/x^4
Time = 0.21 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^5} \, dx=\frac {12 \, B b^{10} x^{11} - 21 \, A a^{10} + 14 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 84 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 315 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 840 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 1764 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 3528 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 2520 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} \log \left (x\right ) - 1260 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 210 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 28 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{84 \, x^{4}} \]
1/84*(12*B*b^10*x^11 - 21*A*a^10 + 14*(10*B*a*b^9 + A*b^10)*x^10 + 84*(9*B *a^2*b^8 + 2*A*a*b^9)*x^9 + 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 840*(7*B *a^4*b^6 + 4*A*a^3*b^7)*x^7 + 1764*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 3528* (5*B*a^6*b^4 + 6*A*a^5*b^5)*x^5 + 2520*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4*log (x) - 1260*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 - 210*(2*B*a^9*b + 9*A*a^8*b^2) *x^2 - 28*(B*a^10 + 10*A*a^9*b)*x)/x^4
Time = 0.81 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^5} \, dx=\frac {B b^{10} x^{7}}{7} + 30 a^{6} b^{3} \cdot \left (7 A b + 4 B a\right ) \log {\left (x \right )} + x^{6} \left (\frac {A b^{10}}{6} + \frac {5 B a b^{9}}{3}\right ) + x^{5} \cdot \left (2 A a b^{9} + 9 B a^{2} b^{8}\right ) + x^{4} \cdot \left (\frac {45 A a^{2} b^{8}}{4} + 30 B a^{3} b^{7}\right ) + x^{3} \cdot \left (40 A a^{3} b^{7} + 70 B a^{4} b^{6}\right ) + x^{2} \cdot \left (105 A a^{4} b^{6} + 126 B a^{5} b^{5}\right ) + x \left (252 A a^{5} b^{5} + 210 B a^{6} b^{4}\right ) + \frac {- 3 A a^{10} + x^{3} \left (- 1440 A a^{7} b^{3} - 540 B a^{8} b^{2}\right ) + x^{2} \left (- 270 A a^{8} b^{2} - 60 B a^{9} b\right ) + x \left (- 40 A a^{9} b - 4 B a^{10}\right )}{12 x^{4}} \]
B*b**10*x**7/7 + 30*a**6*b**3*(7*A*b + 4*B*a)*log(x) + x**6*(A*b**10/6 + 5 *B*a*b**9/3) + x**5*(2*A*a*b**9 + 9*B*a**2*b**8) + x**4*(45*A*a**2*b**8/4 + 30*B*a**3*b**7) + x**3*(40*A*a**3*b**7 + 70*B*a**4*b**6) + x**2*(105*A*a **4*b**6 + 126*B*a**5*b**5) + x*(252*A*a**5*b**5 + 210*B*a**6*b**4) + (-3* A*a**10 + x**3*(-1440*A*a**7*b**3 - 540*B*a**8*b**2) + x**2*(-270*A*a**8*b **2 - 60*B*a**9*b) + x*(-40*A*a**9*b - 4*B*a**10))/(12*x**4)
Time = 0.19 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^5} \, dx=\frac {1}{7} \, B b^{10} x^{7} + \frac {1}{6} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{6} + {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{5} + \frac {15}{4} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{4} + 10 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{3} + 21 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{2} + 42 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x + 30 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} \log \left (x\right ) - \frac {3 \, A a^{10} + 180 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 30 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12 \, x^{4}} \]
1/7*B*b^10*x^7 + 1/6*(10*B*a*b^9 + A*b^10)*x^6 + (9*B*a^2*b^8 + 2*A*a*b^9) *x^5 + 15/4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^4 + 10*(7*B*a^4*b^6 + 4*A*a^3*b^ 7)*x^3 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^2 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^ 5)*x + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*log(x) - 1/12*(3*A*a^10 + 180*(3*B*a ^8*b^2 + 8*A*a^7*b^3)*x^3 + 30*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 4*(B*a^10 + 10*A*a^9*b)*x)/x^4
Time = 0.30 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^5} \, dx=\frac {1}{7} \, B b^{10} x^{7} + \frac {5}{3} \, B a b^{9} x^{6} + \frac {1}{6} \, A b^{10} x^{6} + 9 \, B a^{2} b^{8} x^{5} + 2 \, A a b^{9} x^{5} + 30 \, B a^{3} b^{7} x^{4} + \frac {45}{4} \, A a^{2} b^{8} x^{4} + 70 \, B a^{4} b^{6} x^{3} + 40 \, A a^{3} b^{7} x^{3} + 126 \, B a^{5} b^{5} x^{2} + 105 \, A a^{4} b^{6} x^{2} + 210 \, B a^{6} b^{4} x + 252 \, A a^{5} b^{5} x + 30 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, A a^{10} + 180 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 30 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12 \, x^{4}} \]
1/7*B*b^10*x^7 + 5/3*B*a*b^9*x^6 + 1/6*A*b^10*x^6 + 9*B*a^2*b^8*x^5 + 2*A* a*b^9*x^5 + 30*B*a^3*b^7*x^4 + 45/4*A*a^2*b^8*x^4 + 70*B*a^4*b^6*x^3 + 40* A*a^3*b^7*x^3 + 126*B*a^5*b^5*x^2 + 105*A*a^4*b^6*x^2 + 210*B*a^6*b^4*x + 252*A*a^5*b^5*x + 30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*log(abs(x)) - 1/12*(3*A*a ^10 + 180*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^3 + 30*(2*B*a^9*b + 9*A*a^8*b^2)*x ^2 + 4*(B*a^10 + 10*A*a^9*b)*x)/x^4
Time = 0.09 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^{10} (A+B x)}{x^5} \, dx=x^6\,\left (\frac {A\,b^{10}}{6}+\frac {5\,B\,a\,b^9}{3}\right )-\frac {x\,\left (\frac {B\,a^{10}}{3}+\frac {10\,A\,b\,a^9}{3}\right )+\frac {A\,a^{10}}{4}+x^2\,\left (5\,B\,a^9\,b+\frac {45\,A\,a^8\,b^2}{2}\right )+x^3\,\left (45\,B\,a^8\,b^2+120\,A\,a^7\,b^3\right )}{x^4}+\ln \left (x\right )\,\left (120\,B\,a^7\,b^3+210\,A\,a^6\,b^4\right )+\frac {B\,b^{10}\,x^7}{7}+21\,a^4\,b^5\,x^2\,\left (5\,A\,b+6\,B\,a\right )+10\,a^3\,b^6\,x^3\,\left (4\,A\,b+7\,B\,a\right )+\frac {15\,a^2\,b^7\,x^4\,\left (3\,A\,b+8\,B\,a\right )}{4}+42\,a^5\,b^4\,x\,\left (6\,A\,b+5\,B\,a\right )+a\,b^8\,x^5\,\left (2\,A\,b+9\,B\,a\right ) \]
x^6*((A*b^10)/6 + (5*B*a*b^9)/3) - (x*((B*a^10)/3 + (10*A*a^9*b)/3) + (A*a ^10)/4 + x^2*((45*A*a^8*b^2)/2 + 5*B*a^9*b) + x^3*(120*A*a^7*b^3 + 45*B*a^ 8*b^2))/x^4 + log(x)*(210*A*a^6*b^4 + 120*B*a^7*b^3) + (B*b^10*x^7)/7 + 21 *a^4*b^5*x^2*(5*A*b + 6*B*a) + 10*a^3*b^6*x^3*(4*A*b + 7*B*a) + (15*a^2*b^ 7*x^4*(3*A*b + 8*B*a))/4 + 42*a^5*b^4*x*(6*A*b + 5*B*a) + a*b^8*x^5*(2*A*b + 9*B*a)