Integrand size = 16, antiderivative size = 139 \[ \int x^4 (a+b x)^{10} (A+B x) \, dx=\frac {a^4 (A b-a B) (a+b x)^{11}}{11 b^6}-\frac {a^3 (4 A b-5 a B) (a+b x)^{12}}{12 b^6}+\frac {2 a^2 (3 A b-5 a B) (a+b x)^{13}}{13 b^6}-\frac {a (2 A b-5 a B) (a+b x)^{14}}{7 b^6}+\frac {(A b-5 a B) (a+b x)^{15}}{15 b^6}+\frac {B (a+b x)^{16}}{16 b^6} \] Output:
1/11*a^4*(A*b-B*a)*(b*x+a)^11/b^6-1/12*a^3*(4*A*b-5*B*a)*(b*x+a)^12/b^6+2/ 13*a^2*(3*A*b-5*B*a)*(b*x+a)^13/b^6-1/7*a*(2*A*b-5*B*a)*(b*x+a)^14/b^6+1/1 5*(A*b-5*B*a)*(b*x+a)^15/b^6+1/16*B*(b*x+a)^16/b^6
Time = 0.02 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.66 \[ \int x^4 (a+b x)^{10} (A+B x) \, dx=\frac {1}{5} a^{10} A x^5+\frac {1}{6} a^9 (10 A b+a B) x^6+\frac {5}{7} a^8 b (9 A b+2 a B) x^7+\frac {15}{8} a^7 b^2 (8 A b+3 a B) x^8+\frac {10}{3} a^6 b^3 (7 A b+4 a B) x^9+\frac {21}{5} a^5 b^4 (6 A b+5 a B) x^{10}+\frac {42}{11} a^4 b^5 (5 A b+6 a B) x^{11}+\frac {5}{2} a^3 b^6 (4 A b+7 a B) x^{12}+\frac {15}{13} a^2 b^7 (3 A b+8 a B) x^{13}+\frac {5}{14} a b^8 (2 A b+9 a B) x^{14}+\frac {1}{15} b^9 (A b+10 a B) x^{15}+\frac {1}{16} b^{10} B x^{16} \] Input:
Integrate[x^4*(a + b*x)^10*(A + B*x),x]
Output:
(a^10*A*x^5)/5 + (a^9*(10*A*b + a*B)*x^6)/6 + (5*a^8*b*(9*A*b + 2*a*B)*x^7 )/7 + (15*a^7*b^2*(8*A*b + 3*a*B)*x^8)/8 + (10*a^6*b^3*(7*A*b + 4*a*B)*x^9 )/3 + (21*a^5*b^4*(6*A*b + 5*a*B)*x^10)/5 + (42*a^4*b^5*(5*A*b + 6*a*B)*x^ 11)/11 + (5*a^3*b^6*(4*A*b + 7*a*B)*x^12)/2 + (15*a^2*b^7*(3*A*b + 8*a*B)* x^13)/13 + (5*a*b^8*(2*A*b + 9*a*B)*x^14)/14 + (b^9*(A*b + 10*a*B)*x^15)/1 5 + (b^10*B*x^16)/16
Time = 0.33 (sec) , antiderivative size = 139, 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.
| \(\displaystyle \int x^4 (a+b x)^{10} (A+B x) \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (-\frac {a^4 (a+b x)^{10} (a B-A b)}{b^5}+\frac {a^3 (a+b x)^{11} (5 a B-4 A b)}{b^5}-\frac {2 a^2 (a+b x)^{12} (5 a B-3 A b)}{b^5}+\frac {(a+b x)^{14} (A b-5 a B)}{b^5}+\frac {2 a (a+b x)^{13} (5 a B-2 A b)}{b^5}+\frac {B (a+b x)^{15}}{b^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^4 (a+b x)^{11} (A b-a B)}{11 b^6}-\frac {a^3 (a+b x)^{12} (4 A b-5 a B)}{12 b^6}+\frac {2 a^2 (a+b x)^{13} (3 A b-5 a B)}{13 b^6}+\frac {(a+b x)^{15} (A b-5 a B)}{15 b^6}-\frac {a (a+b x)^{14} (2 A b-5 a B)}{7 b^6}+\frac {B (a+b x)^{16}}{16 b^6}\) |
Input:
Int[x^4*(a + b*x)^10*(A + B*x),x]
Output:
(a^4*(A*b - a*B)*(a + b*x)^11)/(11*b^6) - (a^3*(4*A*b - 5*a*B)*(a + b*x)^1 2)/(12*b^6) + (2*a^2*(3*A*b - 5*a*B)*(a + b*x)^13)/(13*b^6) - (a*(2*A*b - 5*a*B)*(a + b*x)^14)/(7*b^6) + ((A*b - 5*a*B)*(a + b*x)^15)/(15*b^6) + (B* (a + b*x)^16)/(16*b^6)
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 = 0.11 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.70
| method | result | size |
| norman | \(\frac {a^{10} A \,x^{5}}{5}+\left (\frac {5}{3} a^{9} b A +\frac {1}{6} a^{10} B \right ) x^{6}+\left (\frac {45}{7} a^{8} b^{2} A +\frac {10}{7} a^{9} b B \right ) x^{7}+\left (15 a^{7} b^{3} A +\frac {45}{8} a^{8} b^{2} B \right ) x^{8}+\left (\frac {70}{3} a^{6} b^{4} A +\frac {40}{3} a^{7} b^{3} B \right ) x^{9}+\left (\frac {126}{5} a^{5} b^{5} A +21 a^{6} b^{4} B \right ) x^{10}+\left (\frac {210}{11} a^{4} b^{6} A +\frac {252}{11} a^{5} b^{5} B \right ) x^{11}+\left (10 a^{3} b^{7} A +\frac {35}{2} a^{4} b^{6} B \right ) x^{12}+\left (\frac {45}{13} a^{2} b^{8} A +\frac {120}{13} a^{3} b^{7} B \right ) x^{13}+\left (\frac {5}{7} a \,b^{9} A +\frac {45}{14} a^{2} b^{8} B \right ) x^{14}+\left (\frac {1}{15} b^{10} A +\frac {2}{3} a \,b^{9} B \right ) x^{15}+\frac {b^{10} B \,x^{16}}{16}\) | \(236\) |
| default | \(\frac {b^{10} B \,x^{16}}{16}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{15}}{15}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{14}}{14}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{13}}{13}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{12}}{12}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{11}}{11}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{10}}{10}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{9}}{9}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{8}}{8}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{7}}{7}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{6}}{6}+\frac {a^{10} A \,x^{5}}{5}\) | \(244\) |
| orering | \(\frac {x^{5} \left (15015 B \,b^{10} x^{11}+16016 A \,b^{10} x^{10}+160160 B a \,b^{9} x^{10}+171600 a A \,b^{9} x^{9}+772200 B \,a^{2} b^{8} x^{9}+831600 a^{2} A \,b^{8} x^{8}+2217600 B \,a^{3} b^{7} x^{8}+2402400 a^{3} A \,b^{7} x^{7}+4204200 B \,a^{4} b^{6} x^{7}+4586400 a^{4} A \,b^{6} x^{6}+5503680 B \,a^{5} b^{5} x^{6}+6054048 a^{5} A \,b^{5} x^{5}+5045040 B \,a^{6} b^{4} x^{5}+5605600 a^{6} A \,b^{4} x^{4}+3203200 B \,a^{7} b^{3} x^{4}+3603600 a^{7} A \,b^{3} x^{3}+1351350 B \,a^{8} b^{2} x^{3}+1544400 a^{8} A \,b^{2} x^{2}+343200 B \,a^{9} b \,x^{2}+400400 a^{9} A b x +40040 B \,a^{10} x +48048 a^{10} A \right )}{240240}\) | \(244\) |
| gosper | \(\frac {1}{5} a^{10} A \,x^{5}+\frac {5}{3} x^{6} a^{9} b A +\frac {1}{6} x^{6} a^{10} B +\frac {45}{7} x^{7} a^{8} b^{2} A +\frac {10}{7} x^{7} a^{9} b B +15 x^{8} a^{7} b^{3} A +\frac {45}{8} x^{8} a^{8} b^{2} B +\frac {70}{3} x^{9} a^{6} b^{4} A +\frac {40}{3} x^{9} a^{7} b^{3} B +\frac {126}{5} x^{10} a^{5} b^{5} A +21 x^{10} a^{6} b^{4} B +\frac {210}{11} x^{11} a^{4} b^{6} A +\frac {252}{11} x^{11} a^{5} b^{5} B +10 x^{12} a^{3} b^{7} A +\frac {35}{2} x^{12} a^{4} b^{6} B +\frac {45}{13} x^{13} a^{2} b^{8} A +\frac {120}{13} x^{13} a^{3} b^{7} B +\frac {5}{7} x^{14} a \,b^{9} A +\frac {45}{14} x^{14} a^{2} b^{8} B +\frac {1}{15} x^{15} b^{10} A +\frac {2}{3} x^{15} a \,b^{9} B +\frac {1}{16} b^{10} B \,x^{16}\) | \(246\) |
| risch | \(\frac {1}{5} a^{10} A \,x^{5}+\frac {5}{3} x^{6} a^{9} b A +\frac {1}{6} x^{6} a^{10} B +\frac {45}{7} x^{7} a^{8} b^{2} A +\frac {10}{7} x^{7} a^{9} b B +15 x^{8} a^{7} b^{3} A +\frac {45}{8} x^{8} a^{8} b^{2} B +\frac {70}{3} x^{9} a^{6} b^{4} A +\frac {40}{3} x^{9} a^{7} b^{3} B +\frac {126}{5} x^{10} a^{5} b^{5} A +21 x^{10} a^{6} b^{4} B +\frac {210}{11} x^{11} a^{4} b^{6} A +\frac {252}{11} x^{11} a^{5} b^{5} B +10 x^{12} a^{3} b^{7} A +\frac {35}{2} x^{12} a^{4} b^{6} B +\frac {45}{13} x^{13} a^{2} b^{8} A +\frac {120}{13} x^{13} a^{3} b^{7} B +\frac {5}{7} x^{14} a \,b^{9} A +\frac {45}{14} x^{14} a^{2} b^{8} B +\frac {1}{15} x^{15} b^{10} A +\frac {2}{3} x^{15} a \,b^{9} B +\frac {1}{16} b^{10} B \,x^{16}\) | \(246\) |
| parallelrisch | \(\frac {1}{5} a^{10} A \,x^{5}+\frac {5}{3} x^{6} a^{9} b A +\frac {1}{6} x^{6} a^{10} B +\frac {45}{7} x^{7} a^{8} b^{2} A +\frac {10}{7} x^{7} a^{9} b B +15 x^{8} a^{7} b^{3} A +\frac {45}{8} x^{8} a^{8} b^{2} B +\frac {70}{3} x^{9} a^{6} b^{4} A +\frac {40}{3} x^{9} a^{7} b^{3} B +\frac {126}{5} x^{10} a^{5} b^{5} A +21 x^{10} a^{6} b^{4} B +\frac {210}{11} x^{11} a^{4} b^{6} A +\frac {252}{11} x^{11} a^{5} b^{5} B +10 x^{12} a^{3} b^{7} A +\frac {35}{2} x^{12} a^{4} b^{6} B +\frac {45}{13} x^{13} a^{2} b^{8} A +\frac {120}{13} x^{13} a^{3} b^{7} B +\frac {5}{7} x^{14} a \,b^{9} A +\frac {45}{14} x^{14} a^{2} b^{8} B +\frac {1}{15} x^{15} b^{10} A +\frac {2}{3} x^{15} a \,b^{9} B +\frac {1}{16} b^{10} B \,x^{16}\) | \(246\) |
Input:
int(x^4*(b*x+a)^10*(B*x+A),x,method=_RETURNVERBOSE)
Output:
1/5*a^10*A*x^5+(5/3*a^9*b*A+1/6*a^10*B)*x^6+(45/7*a^8*b^2*A+10/7*a^9*b*B)* x^7+(15*a^7*b^3*A+45/8*a^8*b^2*B)*x^8+(70/3*a^6*b^4*A+40/3*a^7*b^3*B)*x^9+ (126/5*a^5*b^5*A+21*a^6*b^4*B)*x^10+(210/11*a^4*b^6*A+252/11*a^5*b^5*B)*x^ 11+(10*a^3*b^7*A+35/2*a^4*b^6*B)*x^12+(45/13*a^2*b^8*A+120/13*a^3*b^7*B)*x ^13+(5/7*a*b^9*A+45/14*a^2*b^8*B)*x^14+(1/15*b^10*A+2/3*a*b^9*B)*x^15+1/16 *b^10*B*x^16
Time = 0.07 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.75 \[ \int x^4 (a+b x)^{10} (A+B x) \, dx=\frac {1}{16} \, B b^{10} x^{16} + \frac {1}{5} \, A a^{10} x^{5} + \frac {1}{15} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{15} + \frac {5}{14} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{14} + \frac {15}{13} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{13} + \frac {5}{2} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{12} + \frac {42}{11} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{11} + \frac {21}{5} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{10} + \frac {10}{3} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{9} + \frac {15}{8} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{8} + \frac {5}{7} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{6} \] Input:
integrate(x^4*(b*x+a)^10*(B*x+A),x, algorithm="fricas")
Output:
1/16*B*b^10*x^16 + 1/5*A*a^10*x^5 + 1/15*(10*B*a*b^9 + A*b^10)*x^15 + 5/14 *(9*B*a^2*b^8 + 2*A*a*b^9)*x^14 + 15/13*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^13 + 5/2*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^12 + 42/11*(6*B*a^5*b^5 + 5*A*a^4*b^6)* x^11 + 21/5*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^10 + 10/3*(4*B*a^7*b^3 + 7*A*a^6 *b^4)*x^9 + 15/8*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^8 + 5/7*(2*B*a^9*b + 9*A*a^ 8*b^2)*x^7 + 1/6*(B*a^10 + 10*A*a^9*b)*x^6
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (131) = 262\).
Time = 0.05 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.94 \[ \int x^4 (a+b x)^{10} (A+B x) \, dx=\frac {A a^{10} x^{5}}{5} + \frac {B b^{10} x^{16}}{16} + x^{15} \left (\frac {A b^{10}}{15} + \frac {2 B a b^{9}}{3}\right ) + x^{14} \cdot \left (\frac {5 A a b^{9}}{7} + \frac {45 B a^{2} b^{8}}{14}\right ) + x^{13} \cdot \left (\frac {45 A a^{2} b^{8}}{13} + \frac {120 B a^{3} b^{7}}{13}\right ) + x^{12} \cdot \left (10 A a^{3} b^{7} + \frac {35 B a^{4} b^{6}}{2}\right ) + x^{11} \cdot \left (\frac {210 A a^{4} b^{6}}{11} + \frac {252 B a^{5} b^{5}}{11}\right ) + x^{10} \cdot \left (\frac {126 A a^{5} b^{5}}{5} + 21 B a^{6} b^{4}\right ) + x^{9} \cdot \left (\frac {70 A a^{6} b^{4}}{3} + \frac {40 B a^{7} b^{3}}{3}\right ) + x^{8} \cdot \left (15 A a^{7} b^{3} + \frac {45 B a^{8} b^{2}}{8}\right ) + x^{7} \cdot \left (\frac {45 A a^{8} b^{2}}{7} + \frac {10 B a^{9} b}{7}\right ) + x^{6} \cdot \left (\frac {5 A a^{9} b}{3} + \frac {B a^{10}}{6}\right ) \] Input:
integrate(x**4*(b*x+a)**10*(B*x+A),x)
Output:
A*a**10*x**5/5 + B*b**10*x**16/16 + x**15*(A*b**10/15 + 2*B*a*b**9/3) + x* *14*(5*A*a*b**9/7 + 45*B*a**2*b**8/14) + x**13*(45*A*a**2*b**8/13 + 120*B* a**3*b**7/13) + x**12*(10*A*a**3*b**7 + 35*B*a**4*b**6/2) + x**11*(210*A*a **4*b**6/11 + 252*B*a**5*b**5/11) + x**10*(126*A*a**5*b**5/5 + 21*B*a**6*b **4) + x**9*(70*A*a**6*b**4/3 + 40*B*a**7*b**3/3) + x**8*(15*A*a**7*b**3 + 45*B*a**8*b**2/8) + x**7*(45*A*a**8*b**2/7 + 10*B*a**9*b/7) + x**6*(5*A*a **9*b/3 + B*a**10/6)
Time = 0.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.75 \[ \int x^4 (a+b x)^{10} (A+B x) \, dx=\frac {1}{16} \, B b^{10} x^{16} + \frac {1}{5} \, A a^{10} x^{5} + \frac {1}{15} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{15} + \frac {5}{14} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{14} + \frac {15}{13} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{13} + \frac {5}{2} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{12} + \frac {42}{11} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{11} + \frac {21}{5} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{10} + \frac {10}{3} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{9} + \frac {15}{8} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{8} + \frac {5}{7} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{6} \] Input:
integrate(x^4*(b*x+a)^10*(B*x+A),x, algorithm="maxima")
Output:
1/16*B*b^10*x^16 + 1/5*A*a^10*x^5 + 1/15*(10*B*a*b^9 + A*b^10)*x^15 + 5/14 *(9*B*a^2*b^8 + 2*A*a*b^9)*x^14 + 15/13*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^13 + 5/2*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^12 + 42/11*(6*B*a^5*b^5 + 5*A*a^4*b^6)* x^11 + 21/5*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^10 + 10/3*(4*B*a^7*b^3 + 7*A*a^6 *b^4)*x^9 + 15/8*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^8 + 5/7*(2*B*a^9*b + 9*A*a^ 8*b^2)*x^7 + 1/6*(B*a^10 + 10*A*a^9*b)*x^6
Time = 0.12 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.76 \[ \int x^4 (a+b x)^{10} (A+B x) \, dx=\frac {1}{16} \, B b^{10} x^{16} + \frac {2}{3} \, B a b^{9} x^{15} + \frac {1}{15} \, A b^{10} x^{15} + \frac {45}{14} \, B a^{2} b^{8} x^{14} + \frac {5}{7} \, A a b^{9} x^{14} + \frac {120}{13} \, B a^{3} b^{7} x^{13} + \frac {45}{13} \, A a^{2} b^{8} x^{13} + \frac {35}{2} \, B a^{4} b^{6} x^{12} + 10 \, A a^{3} b^{7} x^{12} + \frac {252}{11} \, B a^{5} b^{5} x^{11} + \frac {210}{11} \, A a^{4} b^{6} x^{11} + 21 \, B a^{6} b^{4} x^{10} + \frac {126}{5} \, A a^{5} b^{5} x^{10} + \frac {40}{3} \, B a^{7} b^{3} x^{9} + \frac {70}{3} \, A a^{6} b^{4} x^{9} + \frac {45}{8} \, B a^{8} b^{2} x^{8} + 15 \, A a^{7} b^{3} x^{8} + \frac {10}{7} \, B a^{9} b x^{7} + \frac {45}{7} \, A a^{8} b^{2} x^{7} + \frac {1}{6} \, B a^{10} x^{6} + \frac {5}{3} \, A a^{9} b x^{6} + \frac {1}{5} \, A a^{10} x^{5} \] Input:
integrate(x^4*(b*x+a)^10*(B*x+A),x, algorithm="giac")
Output:
1/16*B*b^10*x^16 + 2/3*B*a*b^9*x^15 + 1/15*A*b^10*x^15 + 45/14*B*a^2*b^8*x ^14 + 5/7*A*a*b^9*x^14 + 120/13*B*a^3*b^7*x^13 + 45/13*A*a^2*b^8*x^13 + 35 /2*B*a^4*b^6*x^12 + 10*A*a^3*b^7*x^12 + 252/11*B*a^5*b^5*x^11 + 210/11*A*a ^4*b^6*x^11 + 21*B*a^6*b^4*x^10 + 126/5*A*a^5*b^5*x^10 + 40/3*B*a^7*b^3*x^ 9 + 70/3*A*a^6*b^4*x^9 + 45/8*B*a^8*b^2*x^8 + 15*A*a^7*b^3*x^8 + 10/7*B*a^ 9*b*x^7 + 45/7*A*a^8*b^2*x^7 + 1/6*B*a^10*x^6 + 5/3*A*a^9*b*x^6 + 1/5*A*a^ 10*x^5
Time = 0.06 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.52 \[ \int x^4 (a+b x)^{10} (A+B x) \, dx=x^6\,\left (\frac {B\,a^{10}}{6}+\frac {5\,A\,b\,a^9}{3}\right )+x^{15}\,\left (\frac {A\,b^{10}}{15}+\frac {2\,B\,a\,b^9}{3}\right )+\frac {A\,a^{10}\,x^5}{5}+\frac {B\,b^{10}\,x^{16}}{16}+\frac {15\,a^7\,b^2\,x^8\,\left (8\,A\,b+3\,B\,a\right )}{8}+\frac {10\,a^6\,b^3\,x^9\,\left (7\,A\,b+4\,B\,a\right )}{3}+\frac {21\,a^5\,b^4\,x^{10}\,\left (6\,A\,b+5\,B\,a\right )}{5}+\frac {42\,a^4\,b^5\,x^{11}\,\left (5\,A\,b+6\,B\,a\right )}{11}+\frac {5\,a^3\,b^6\,x^{12}\,\left (4\,A\,b+7\,B\,a\right )}{2}+\frac {15\,a^2\,b^7\,x^{13}\,\left (3\,A\,b+8\,B\,a\right )}{13}+\frac {5\,a^8\,b\,x^7\,\left (9\,A\,b+2\,B\,a\right )}{7}+\frac {5\,a\,b^8\,x^{14}\,\left (2\,A\,b+9\,B\,a\right )}{14} \] Input:
int(x^4*(A + B*x)*(a + b*x)^10,x)
Output:
x^6*((B*a^10)/6 + (5*A*a^9*b)/3) + x^15*((A*b^10)/15 + (2*B*a*b^9)/3) + (A *a^10*x^5)/5 + (B*b^10*x^16)/16 + (15*a^7*b^2*x^8*(8*A*b + 3*B*a))/8 + (10 *a^6*b^3*x^9*(7*A*b + 4*B*a))/3 + (21*a^5*b^4*x^10*(6*A*b + 5*B*a))/5 + (4 2*a^4*b^5*x^11*(5*A*b + 6*B*a))/11 + (5*a^3*b^6*x^12*(4*A*b + 7*B*a))/2 + (15*a^2*b^7*x^13*(3*A*b + 8*B*a))/13 + (5*a^8*b*x^7*(9*A*b + 2*B*a))/7 + ( 5*a*b^8*x^14*(2*A*b + 9*B*a))/14
Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.88 \[ \int x^4 (a+b x)^{10} (A+B x) \, dx=\frac {x^{5} \left (1365 b^{11} x^{11}+16016 a \,b^{10} x^{10}+85800 a^{2} b^{9} x^{9}+277200 a^{3} b^{8} x^{8}+600600 a^{4} b^{7} x^{7}+917280 a^{5} b^{6} x^{6}+1009008 a^{6} b^{5} x^{5}+800800 a^{7} b^{4} x^{4}+450450 a^{8} b^{3} x^{3}+171600 a^{9} b^{2} x^{2}+40040 a^{10} b x +4368 a^{11}\right )}{21840} \] Input:
int(x^4*(b*x+a)^10*(B*x+A),x)
Output:
(x**5*(4368*a**11 + 40040*a**10*b*x + 171600*a**9*b**2*x**2 + 450450*a**8* b**3*x**3 + 800800*a**7*b**4*x**4 + 1009008*a**6*b**5*x**5 + 917280*a**5*b **6*x**6 + 600600*a**4*b**7*x**7 + 277200*a**3*b**8*x**8 + 85800*a**2*b**9 *x**9 + 16016*a*b**10*x**10 + 1365*b**11*x**11))/21840