Integrand size = 16, antiderivative size = 112 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=-\frac {a^3 (A b-a B) (a+b x)^{11}}{11 b^5}+\frac {a^2 (3 A b-4 a B) (a+b x)^{12}}{12 b^5}-\frac {3 a (A b-2 a B) (a+b x)^{13}}{13 b^5}+\frac {(A b-4 a B) (a+b x)^{14}}{14 b^5}+\frac {B (a+b x)^{15}}{15 b^5} \] Output:
-1/11*a^3*(A*b-B*a)*(b*x+a)^11/b^5+1/12*a^2*(3*A*b-4*B*a)*(b*x+a)^12/b^5-3 /13*a*(A*b-2*B*a)*(b*x+a)^13/b^5+1/14*(A*b-4*B*a)*(b*x+a)^14/b^5+1/15*B*(b *x+a)^15/b^5
Leaf count is larger than twice the leaf count of optimal. \(231\) vs. \(2(112)=224\).
Time = 0.02 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.06 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {1}{4} a^{10} A x^4+\frac {1}{5} a^9 (10 A b+a B) x^5+\frac {5}{6} a^8 b (9 A b+2 a B) x^6+\frac {15}{7} a^7 b^2 (8 A b+3 a B) x^7+\frac {15}{4} a^6 b^3 (7 A b+4 a B) x^8+\frac {14}{3} a^5 b^4 (6 A b+5 a B) x^9+\frac {21}{5} a^4 b^5 (5 A b+6 a B) x^{10}+\frac {30}{11} a^3 b^6 (4 A b+7 a B) x^{11}+\frac {5}{4} a^2 b^7 (3 A b+8 a B) x^{12}+\frac {5}{13} a b^8 (2 A b+9 a B) x^{13}+\frac {1}{14} b^9 (A b+10 a B) x^{14}+\frac {1}{15} b^{10} B x^{15} \] Input:
Integrate[x^3*(a + b*x)^10*(A + B*x),x]
Output:
(a^10*A*x^4)/4 + (a^9*(10*A*b + a*B)*x^5)/5 + (5*a^8*b*(9*A*b + 2*a*B)*x^6 )/6 + (15*a^7*b^2*(8*A*b + 3*a*B)*x^7)/7 + (15*a^6*b^3*(7*A*b + 4*a*B)*x^8 )/4 + (14*a^5*b^4*(6*A*b + 5*a*B)*x^9)/3 + (21*a^4*b^5*(5*A*b + 6*a*B)*x^1 0)/5 + (30*a^3*b^6*(4*A*b + 7*a*B)*x^11)/11 + (5*a^2*b^7*(3*A*b + 8*a*B)*x ^12)/4 + (5*a*b^8*(2*A*b + 9*a*B)*x^13)/13 + (b^9*(A*b + 10*a*B)*x^14)/14 + (b^10*B*x^15)/15
Time = 0.30 (sec) , antiderivative size = 112, 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.
| \(\displaystyle \int x^3 (a+b x)^{10} (A+B x) \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {a^3 (a+b x)^{10} (a B-A b)}{b^4}-\frac {a^2 (a+b x)^{11} (4 a B-3 A b)}{b^4}+\frac {(a+b x)^{13} (A b-4 a B)}{b^4}+\frac {3 a (a+b x)^{12} (2 a B-A b)}{b^4}+\frac {B (a+b x)^{14}}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 (a+b x)^{11} (A b-a B)}{11 b^5}+\frac {a^2 (a+b x)^{12} (3 A b-4 a B)}{12 b^5}+\frac {(a+b x)^{14} (A b-4 a B)}{14 b^5}-\frac {3 a (a+b x)^{13} (A b-2 a B)}{13 b^5}+\frac {B (a+b x)^{15}}{15 b^5}\) |
Input:
Int[x^3*(a + b*x)^10*(A + B*x),x]
Output:
-1/11*(a^3*(A*b - a*B)*(a + b*x)^11)/b^5 + (a^2*(3*A*b - 4*a*B)*(a + b*x)^ 12)/(12*b^5) - (3*a*(A*b - 2*a*B)*(a + b*x)^13)/(13*b^5) + ((A*b - 4*a*B)* (a + b*x)^14)/(14*b^5) + (B*(a + b*x)^15)/(15*b^5)
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])
Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(102)=204\).
Time = 0.10 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.11
| method | result | size |
| norman | \(\frac {a^{10} A \,x^{4}}{4}+\left (2 a^{9} b A +\frac {1}{5} a^{10} B \right ) x^{5}+\left (\frac {15}{2} a^{8} b^{2} A +\frac {5}{3} a^{9} b B \right ) x^{6}+\left (\frac {120}{7} a^{7} b^{3} A +\frac {45}{7} a^{8} b^{2} B \right ) x^{7}+\left (\frac {105}{4} a^{6} b^{4} A +15 a^{7} b^{3} B \right ) x^{8}+\left (28 a^{5} b^{5} A +\frac {70}{3} a^{6} b^{4} B \right ) x^{9}+\left (21 a^{4} b^{6} A +\frac {126}{5} a^{5} b^{5} B \right ) x^{10}+\left (\frac {120}{11} a^{3} b^{7} A +\frac {210}{11} a^{4} b^{6} B \right ) x^{11}+\left (\frac {15}{4} a^{2} b^{8} A +10 a^{3} b^{7} B \right ) x^{12}+\left (\frac {10}{13} a \,b^{9} A +\frac {45}{13} a^{2} b^{8} B \right ) x^{13}+\left (\frac {1}{14} b^{10} A +\frac {5}{7} a \,b^{9} B \right ) x^{14}+\frac {b^{10} B \,x^{15}}{15}\) | \(236\) |
| default | \(\frac {b^{10} B \,x^{15}}{15}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{14}}{14}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{13}}{13}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{12}}{12}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{11}}{11}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{10}}{10}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{9}}{9}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{8}}{8}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{7}}{7}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{6}}{6}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{5}}{5}+\frac {a^{10} A \,x^{4}}{4}\) | \(244\) |
| orering | \(\frac {x^{4} \left (4004 B \,b^{10} x^{11}+4290 A \,b^{10} x^{10}+42900 B a \,b^{9} x^{10}+46200 a A \,b^{9} x^{9}+207900 B \,a^{2} b^{8} x^{9}+225225 a^{2} A \,b^{8} x^{8}+600600 B \,a^{3} b^{7} x^{8}+655200 a^{3} A \,b^{7} x^{7}+1146600 B \,a^{4} b^{6} x^{7}+1261260 a^{4} A \,b^{6} x^{6}+1513512 B \,a^{5} b^{5} x^{6}+1681680 a^{5} A \,b^{5} x^{5}+1401400 B \,a^{6} b^{4} x^{5}+1576575 a^{6} A \,b^{4} x^{4}+900900 B \,a^{7} b^{3} x^{4}+1029600 a^{7} A \,b^{3} x^{3}+386100 B \,a^{8} b^{2} x^{3}+450450 a^{8} A \,b^{2} x^{2}+100100 B \,a^{9} b \,x^{2}+120120 a^{9} A b x +12012 B \,a^{10} x +15015 a^{10} A \right )}{60060}\) | \(244\) |
| gosper | \(\frac {1}{4} a^{10} A \,x^{4}+2 x^{5} a^{9} b A +\frac {1}{5} x^{5} a^{10} B +\frac {15}{2} x^{6} a^{8} b^{2} A +\frac {5}{3} x^{6} a^{9} b B +\frac {120}{7} x^{7} a^{7} b^{3} A +\frac {45}{7} x^{7} a^{8} b^{2} B +\frac {105}{4} x^{8} a^{6} b^{4} A +15 x^{8} a^{7} b^{3} B +28 x^{9} a^{5} b^{5} A +\frac {70}{3} x^{9} a^{6} b^{4} B +21 x^{10} a^{4} b^{6} A +\frac {126}{5} x^{10} a^{5} b^{5} B +\frac {120}{11} x^{11} a^{3} b^{7} A +\frac {210}{11} x^{11} a^{4} b^{6} B +\frac {15}{4} x^{12} a^{2} b^{8} A +10 x^{12} a^{3} b^{7} B +\frac {10}{13} x^{13} a \,b^{9} A +\frac {45}{13} x^{13} a^{2} b^{8} B +\frac {1}{14} x^{14} b^{10} A +\frac {5}{7} x^{14} a \,b^{9} B +\frac {1}{15} b^{10} B \,x^{15}\) | \(246\) |
| risch | \(\frac {1}{4} a^{10} A \,x^{4}+2 x^{5} a^{9} b A +\frac {1}{5} x^{5} a^{10} B +\frac {15}{2} x^{6} a^{8} b^{2} A +\frac {5}{3} x^{6} a^{9} b B +\frac {120}{7} x^{7} a^{7} b^{3} A +\frac {45}{7} x^{7} a^{8} b^{2} B +\frac {105}{4} x^{8} a^{6} b^{4} A +15 x^{8} a^{7} b^{3} B +28 x^{9} a^{5} b^{5} A +\frac {70}{3} x^{9} a^{6} b^{4} B +21 x^{10} a^{4} b^{6} A +\frac {126}{5} x^{10} a^{5} b^{5} B +\frac {120}{11} x^{11} a^{3} b^{7} A +\frac {210}{11} x^{11} a^{4} b^{6} B +\frac {15}{4} x^{12} a^{2} b^{8} A +10 x^{12} a^{3} b^{7} B +\frac {10}{13} x^{13} a \,b^{9} A +\frac {45}{13} x^{13} a^{2} b^{8} B +\frac {1}{14} x^{14} b^{10} A +\frac {5}{7} x^{14} a \,b^{9} B +\frac {1}{15} b^{10} B \,x^{15}\) | \(246\) |
| parallelrisch | \(\frac {1}{4} a^{10} A \,x^{4}+2 x^{5} a^{9} b A +\frac {1}{5} x^{5} a^{10} B +\frac {15}{2} x^{6} a^{8} b^{2} A +\frac {5}{3} x^{6} a^{9} b B +\frac {120}{7} x^{7} a^{7} b^{3} A +\frac {45}{7} x^{7} a^{8} b^{2} B +\frac {105}{4} x^{8} a^{6} b^{4} A +15 x^{8} a^{7} b^{3} B +28 x^{9} a^{5} b^{5} A +\frac {70}{3} x^{9} a^{6} b^{4} B +21 x^{10} a^{4} b^{6} A +\frac {126}{5} x^{10} a^{5} b^{5} B +\frac {120}{11} x^{11} a^{3} b^{7} A +\frac {210}{11} x^{11} a^{4} b^{6} B +\frac {15}{4} x^{12} a^{2} b^{8} A +10 x^{12} a^{3} b^{7} B +\frac {10}{13} x^{13} a \,b^{9} A +\frac {45}{13} x^{13} a^{2} b^{8} B +\frac {1}{14} x^{14} b^{10} A +\frac {5}{7} x^{14} a \,b^{9} B +\frac {1}{15} b^{10} B \,x^{15}\) | \(246\) |
Input:
int(x^3*(b*x+a)^10*(B*x+A),x,method=_RETURNVERBOSE)
Output:
1/4*a^10*A*x^4+(2*a^9*b*A+1/5*a^10*B)*x^5+(15/2*a^8*b^2*A+5/3*a^9*b*B)*x^6 +(120/7*a^7*b^3*A+45/7*a^8*b^2*B)*x^7+(105/4*a^6*b^4*A+15*a^7*b^3*B)*x^8+( 28*a^5*b^5*A+70/3*a^6*b^4*B)*x^9+(21*a^4*b^6*A+126/5*a^5*b^5*B)*x^10+(120/ 11*a^3*b^7*A+210/11*a^4*b^6*B)*x^11+(15/4*a^2*b^8*A+10*a^3*b^7*B)*x^12+(10 /13*a*b^9*A+45/13*a^2*b^8*B)*x^13+(1/14*b^10*A+5/7*a*b^9*B)*x^14+1/15*b^10 *B*x^15
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (104) = 208\).
Time = 0.07 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.17 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {1}{15} \, B b^{10} x^{15} + \frac {1}{4} \, A a^{10} x^{4} + \frac {1}{14} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{14} + \frac {5}{13} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{13} + \frac {5}{4} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{12} + \frac {30}{11} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{11} + \frac {21}{5} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{10} + \frac {14}{3} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{9} + \frac {15}{4} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{8} + \frac {15}{7} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{7} + \frac {5}{6} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{5} \] Input:
integrate(x^3*(b*x+a)^10*(B*x+A),x, algorithm="fricas")
Output:
1/15*B*b^10*x^15 + 1/4*A*a^10*x^4 + 1/14*(10*B*a*b^9 + A*b^10)*x^14 + 5/13 *(9*B*a^2*b^8 + 2*A*a*b^9)*x^13 + 5/4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^12 + 3 0/11*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^11 + 21/5*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x ^10 + 14/3*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^9 + 15/4*(4*B*a^7*b^3 + 7*A*a^6*b ^4)*x^8 + 15/7*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^7 + 5/6*(2*B*a^9*b + 9*A*a^8* b^2)*x^6 + 1/5*(B*a^10 + 10*A*a^9*b)*x^5
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (105) = 210\).
Time = 0.04 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.37 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {A a^{10} x^{4}}{4} + \frac {B b^{10} x^{15}}{15} + x^{14} \left (\frac {A b^{10}}{14} + \frac {5 B a b^{9}}{7}\right ) + x^{13} \cdot \left (\frac {10 A a b^{9}}{13} + \frac {45 B a^{2} b^{8}}{13}\right ) + x^{12} \cdot \left (\frac {15 A a^{2} b^{8}}{4} + 10 B a^{3} b^{7}\right ) + x^{11} \cdot \left (\frac {120 A a^{3} b^{7}}{11} + \frac {210 B a^{4} b^{6}}{11}\right ) + x^{10} \cdot \left (21 A a^{4} b^{6} + \frac {126 B a^{5} b^{5}}{5}\right ) + x^{9} \cdot \left (28 A a^{5} b^{5} + \frac {70 B a^{6} b^{4}}{3}\right ) + x^{8} \cdot \left (\frac {105 A a^{6} b^{4}}{4} + 15 B a^{7} b^{3}\right ) + x^{7} \cdot \left (\frac {120 A a^{7} b^{3}}{7} + \frac {45 B a^{8} b^{2}}{7}\right ) + x^{6} \cdot \left (\frac {15 A a^{8} b^{2}}{2} + \frac {5 B a^{9} b}{3}\right ) + x^{5} \cdot \left (2 A a^{9} b + \frac {B a^{10}}{5}\right ) \] Input:
integrate(x**3*(b*x+a)**10*(B*x+A),x)
Output:
A*a**10*x**4/4 + B*b**10*x**15/15 + x**14*(A*b**10/14 + 5*B*a*b**9/7) + x* *13*(10*A*a*b**9/13 + 45*B*a**2*b**8/13) + x**12*(15*A*a**2*b**8/4 + 10*B* a**3*b**7) + x**11*(120*A*a**3*b**7/11 + 210*B*a**4*b**6/11) + x**10*(21*A *a**4*b**6 + 126*B*a**5*b**5/5) + x**9*(28*A*a**5*b**5 + 70*B*a**6*b**4/3) + x**8*(105*A*a**6*b**4/4 + 15*B*a**7*b**3) + x**7*(120*A*a**7*b**3/7 + 4 5*B*a**8*b**2/7) + x**6*(15*A*a**8*b**2/2 + 5*B*a**9*b/3) + x**5*(2*A*a**9 *b + B*a**10/5)
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (104) = 208\).
Time = 0.05 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.17 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {1}{15} \, B b^{10} x^{15} + \frac {1}{4} \, A a^{10} x^{4} + \frac {1}{14} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{14} + \frac {5}{13} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{13} + \frac {5}{4} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{12} + \frac {30}{11} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{11} + \frac {21}{5} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{10} + \frac {14}{3} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{9} + \frac {15}{4} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{8} + \frac {15}{7} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{7} + \frac {5}{6} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{5} \] Input:
integrate(x^3*(b*x+a)^10*(B*x+A),x, algorithm="maxima")
Output:
1/15*B*b^10*x^15 + 1/4*A*a^10*x^4 + 1/14*(10*B*a*b^9 + A*b^10)*x^14 + 5/13 *(9*B*a^2*b^8 + 2*A*a*b^9)*x^13 + 5/4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^12 + 3 0/11*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^11 + 21/5*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x ^10 + 14/3*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^9 + 15/4*(4*B*a^7*b^3 + 7*A*a^6*b ^4)*x^8 + 15/7*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^7 + 5/6*(2*B*a^9*b + 9*A*a^8* b^2)*x^6 + 1/5*(B*a^10 + 10*A*a^9*b)*x^5
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (104) = 208\).
Time = 0.12 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.19 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {1}{15} \, B b^{10} x^{15} + \frac {5}{7} \, B a b^{9} x^{14} + \frac {1}{14} \, A b^{10} x^{14} + \frac {45}{13} \, B a^{2} b^{8} x^{13} + \frac {10}{13} \, A a b^{9} x^{13} + 10 \, B a^{3} b^{7} x^{12} + \frac {15}{4} \, A a^{2} b^{8} x^{12} + \frac {210}{11} \, B a^{4} b^{6} x^{11} + \frac {120}{11} \, A a^{3} b^{7} x^{11} + \frac {126}{5} \, B a^{5} b^{5} x^{10} + 21 \, A a^{4} b^{6} x^{10} + \frac {70}{3} \, B a^{6} b^{4} x^{9} + 28 \, A a^{5} b^{5} x^{9} + 15 \, B a^{7} b^{3} x^{8} + \frac {105}{4} \, A a^{6} b^{4} x^{8} + \frac {45}{7} \, B a^{8} b^{2} x^{7} + \frac {120}{7} \, A a^{7} b^{3} x^{7} + \frac {5}{3} \, B a^{9} b x^{6} + \frac {15}{2} \, A a^{8} b^{2} x^{6} + \frac {1}{5} \, B a^{10} x^{5} + 2 \, A a^{9} b x^{5} + \frac {1}{4} \, A a^{10} x^{4} \] Input:
integrate(x^3*(b*x+a)^10*(B*x+A),x, algorithm="giac")
Output:
1/15*B*b^10*x^15 + 5/7*B*a*b^9*x^14 + 1/14*A*b^10*x^14 + 45/13*B*a^2*b^8*x ^13 + 10/13*A*a*b^9*x^13 + 10*B*a^3*b^7*x^12 + 15/4*A*a^2*b^8*x^12 + 210/1 1*B*a^4*b^6*x^11 + 120/11*A*a^3*b^7*x^11 + 126/5*B*a^5*b^5*x^10 + 21*A*a^4 *b^6*x^10 + 70/3*B*a^6*b^4*x^9 + 28*A*a^5*b^5*x^9 + 15*B*a^7*b^3*x^8 + 105 /4*A*a^6*b^4*x^8 + 45/7*B*a^8*b^2*x^7 + 120/7*A*a^7*b^3*x^7 + 5/3*B*a^9*b* x^6 + 15/2*A*a^8*b^2*x^6 + 1/5*B*a^10*x^5 + 2*A*a^9*b*x^5 + 1/4*A*a^10*x^4
Time = 0.06 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.88 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=x^5\,\left (\frac {B\,a^{10}}{5}+2\,A\,b\,a^9\right )+x^{14}\,\left (\frac {A\,b^{10}}{14}+\frac {5\,B\,a\,b^9}{7}\right )+\frac {A\,a^{10}\,x^4}{4}+\frac {B\,b^{10}\,x^{15}}{15}+\frac {15\,a^7\,b^2\,x^7\,\left (8\,A\,b+3\,B\,a\right )}{7}+\frac {15\,a^6\,b^3\,x^8\,\left (7\,A\,b+4\,B\,a\right )}{4}+\frac {14\,a^5\,b^4\,x^9\,\left (6\,A\,b+5\,B\,a\right )}{3}+\frac {21\,a^4\,b^5\,x^{10}\,\left (5\,A\,b+6\,B\,a\right )}{5}+\frac {30\,a^3\,b^6\,x^{11}\,\left (4\,A\,b+7\,B\,a\right )}{11}+\frac {5\,a^2\,b^7\,x^{12}\,\left (3\,A\,b+8\,B\,a\right )}{4}+\frac {5\,a^8\,b\,x^6\,\left (9\,A\,b+2\,B\,a\right )}{6}+\frac {5\,a\,b^8\,x^{13}\,\left (2\,A\,b+9\,B\,a\right )}{13} \] Input:
int(x^3*(A + B*x)*(a + b*x)^10,x)
Output:
x^5*((B*a^10)/5 + 2*A*a^9*b) + x^14*((A*b^10)/14 + (5*B*a*b^9)/7) + (A*a^1 0*x^4)/4 + (B*b^10*x^15)/15 + (15*a^7*b^2*x^7*(8*A*b + 3*B*a))/7 + (15*a^6 *b^3*x^8*(7*A*b + 4*B*a))/4 + (14*a^5*b^4*x^9*(6*A*b + 5*B*a))/3 + (21*a^4 *b^5*x^10*(5*A*b + 6*B*a))/5 + (30*a^3*b^6*x^11*(4*A*b + 7*B*a))/11 + (5*a ^2*b^7*x^12*(3*A*b + 8*B*a))/4 + (5*a^8*b*x^6*(9*A*b + 2*B*a))/6 + (5*a*b^ 8*x^13*(2*A*b + 9*B*a))/13
Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.10 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {x^{4} \left (364 b^{11} x^{11}+4290 a \,b^{10} x^{10}+23100 a^{2} b^{9} x^{9}+75075 a^{3} b^{8} x^{8}+163800 a^{4} b^{7} x^{7}+252252 a^{5} b^{6} x^{6}+280280 a^{6} b^{5} x^{5}+225225 a^{7} b^{4} x^{4}+128700 a^{8} b^{3} x^{3}+50050 a^{9} b^{2} x^{2}+12012 a^{10} b x +1365 a^{11}\right )}{5460} \] Input:
int(x^3*(b*x+a)^10*(B*x+A),x)
Output:
(x**4*(1365*a**11 + 12012*a**10*b*x + 50050*a**9*b**2*x**2 + 128700*a**8*b **3*x**3 + 225225*a**7*b**4*x**4 + 280280*a**6*b**5*x**5 + 252252*a**5*b** 6*x**6 + 163800*a**4*b**7*x**7 + 75075*a**3*b**8*x**8 + 23100*a**2*b**9*x* *9 + 4290*a*b**10*x**10 + 364*b**11*x**11))/5460