Integrand size = 16, antiderivative size = 87 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {a^2 (A b-a B) (a+b x)^{11}}{11 b^4}-\frac {a (2 A b-3 a B) (a+b x)^{12}}{12 b^4}+\frac {(A b-3 a B) (a+b x)^{13}}{13 b^4}+\frac {B (a+b x)^{14}}{14 b^4} \]
1/11*a^2*(A*b-B*a)*(b*x+a)^11/b^4-1/12*a*(2*A*b-3*B*a)*(b*x+a)^12/b^4+1/13 *(A*b-3*B*a)*(b*x+a)^13/b^4+1/14*B*(b*x+a)^14/b^4
Leaf count is larger than twice the leaf count of optimal. \(226\) vs. \(2(87)=174\).
Time = 0.02 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.60 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {1}{3} a^{10} A x^3+\frac {1}{4} a^9 (10 A b+a B) x^4+a^8 b (9 A b+2 a B) x^5+\frac {5}{2} a^7 b^2 (8 A b+3 a B) x^6+\frac {30}{7} a^6 b^3 (7 A b+4 a B) x^7+\frac {21}{4} a^5 b^4 (6 A b+5 a B) x^8+\frac {14}{3} a^4 b^5 (5 A b+6 a B) x^9+3 a^3 b^6 (4 A b+7 a B) x^{10}+\frac {15}{11} a^2 b^7 (3 A b+8 a B) x^{11}+\frac {5}{12} a b^8 (2 A b+9 a B) x^{12}+\frac {1}{13} b^9 (A b+10 a B) x^{13}+\frac {1}{14} b^{10} B x^{14} \]
(a^10*A*x^3)/3 + (a^9*(10*A*b + a*B)*x^4)/4 + a^8*b*(9*A*b + 2*a*B)*x^5 + (5*a^7*b^2*(8*A*b + 3*a*B)*x^6)/2 + (30*a^6*b^3*(7*A*b + 4*a*B)*x^7)/7 + ( 21*a^5*b^4*(6*A*b + 5*a*B)*x^8)/4 + (14*a^4*b^5*(5*A*b + 6*a*B)*x^9)/3 + 3 *a^3*b^6*(4*A*b + 7*a*B)*x^10 + (15*a^2*b^7*(3*A*b + 8*a*B)*x^11)/11 + (5* a*b^8*(2*A*b + 9*a*B)*x^12)/12 + (b^9*(A*b + 10*a*B)*x^13)/13 + (b^10*B*x^ 14)/14
Time = 0.26 (sec) , antiderivative size = 87, 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.
\(\displaystyle \int x^2 (a+b x)^{10} (A+B x) \, dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (-\frac {a^2 (a+b x)^{10} (a B-A b)}{b^3}+\frac {(a+b x)^{12} (A b-3 a B)}{b^3}+\frac {a (a+b x)^{11} (3 a B-2 A b)}{b^3}+\frac {B (a+b x)^{13}}{b^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 (a+b x)^{11} (A b-a B)}{11 b^4}+\frac {(a+b x)^{13} (A b-3 a B)}{13 b^4}-\frac {a (a+b x)^{12} (2 A b-3 a B)}{12 b^4}+\frac {B (a+b x)^{14}}{14 b^4}\) |
(a^2*(A*b - a*B)*(a + b*x)^11)/(11*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x)^12) /(12*b^4) + ((A*b - 3*a*B)*(a + b*x)^13)/(13*b^4) + (B*(a + b*x)^14)/(14*b ^4)
3.2.45.3.1 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])
Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(79)=158\).
Time = 0.40 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.71
method | result | size |
norman | \(\frac {a^{10} A \,x^{3}}{3}+\left (\frac {5}{2} a^{9} b A +\frac {1}{4} a^{10} B \right ) x^{4}+\left (9 a^{8} b^{2} A +2 a^{9} b B \right ) x^{5}+\left (20 a^{7} b^{3} A +\frac {15}{2} a^{8} b^{2} B \right ) x^{6}+\left (30 a^{6} b^{4} A +\frac {120}{7} a^{7} b^{3} B \right ) x^{7}+\left (\frac {63}{2} a^{5} b^{5} A +\frac {105}{4} a^{6} b^{4} B \right ) x^{8}+\left (\frac {70}{3} a^{4} b^{6} A +28 a^{5} b^{5} B \right ) x^{9}+\left (12 a^{3} b^{7} A +21 a^{4} b^{6} B \right ) x^{10}+\left (\frac {45}{11} a^{2} b^{8} A +\frac {120}{11} a^{3} b^{7} B \right ) x^{11}+\left (\frac {5}{6} a \,b^{9} A +\frac {15}{4} a^{2} b^{8} B \right ) x^{12}+\left (\frac {1}{13} b^{10} A +\frac {10}{13} a \,b^{9} B \right ) x^{13}+\frac {b^{10} B \,x^{14}}{14}\) | \(236\) |
default | \(\frac {b^{10} B \,x^{14}}{14}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{13}}{13}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{12}}{12}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{11}}{11}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{10}}{10}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{9}}{9}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{8}}{8}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{7}}{7}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{6}}{6}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{5}}{5}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{4}}{4}+\frac {a^{10} A \,x^{3}}{3}\) | \(244\) |
gosper | \(\frac {1}{3} a^{10} A \,x^{3}+\frac {5}{2} x^{4} a^{9} b A +\frac {1}{4} x^{4} a^{10} B +9 A \,a^{8} b^{2} x^{5}+2 B \,a^{9} b \,x^{5}+20 x^{6} a^{7} b^{3} A +\frac {15}{2} x^{6} a^{8} b^{2} B +30 x^{7} a^{6} b^{4} A +\frac {120}{7} x^{7} a^{7} b^{3} B +\frac {63}{2} x^{8} a^{5} b^{5} A +\frac {105}{4} x^{8} a^{6} b^{4} B +\frac {70}{3} x^{9} a^{4} b^{6} A +28 x^{9} a^{5} b^{5} B +12 A \,a^{3} b^{7} x^{10}+21 B \,a^{4} b^{6} x^{10}+\frac {45}{11} x^{11} a^{2} b^{8} A +\frac {120}{11} x^{11} a^{3} b^{7} B +\frac {5}{6} x^{12} a \,b^{9} A +\frac {15}{4} x^{12} a^{2} b^{8} B +\frac {1}{13} x^{13} b^{10} A +\frac {10}{13} x^{13} a \,b^{9} B +\frac {1}{14} b^{10} B \,x^{14}\) | \(246\) |
risch | \(\frac {1}{3} a^{10} A \,x^{3}+\frac {5}{2} x^{4} a^{9} b A +\frac {1}{4} x^{4} a^{10} B +9 A \,a^{8} b^{2} x^{5}+2 B \,a^{9} b \,x^{5}+20 x^{6} a^{7} b^{3} A +\frac {15}{2} x^{6} a^{8} b^{2} B +30 x^{7} a^{6} b^{4} A +\frac {120}{7} x^{7} a^{7} b^{3} B +\frac {63}{2} x^{8} a^{5} b^{5} A +\frac {105}{4} x^{8} a^{6} b^{4} B +\frac {70}{3} x^{9} a^{4} b^{6} A +28 x^{9} a^{5} b^{5} B +12 A \,a^{3} b^{7} x^{10}+21 B \,a^{4} b^{6} x^{10}+\frac {45}{11} x^{11} a^{2} b^{8} A +\frac {120}{11} x^{11} a^{3} b^{7} B +\frac {5}{6} x^{12} a \,b^{9} A +\frac {15}{4} x^{12} a^{2} b^{8} B +\frac {1}{13} x^{13} b^{10} A +\frac {10}{13} x^{13} a \,b^{9} B +\frac {1}{14} b^{10} B \,x^{14}\) | \(246\) |
parallelrisch | \(\frac {1}{3} a^{10} A \,x^{3}+\frac {5}{2} x^{4} a^{9} b A +\frac {1}{4} x^{4} a^{10} B +9 A \,a^{8} b^{2} x^{5}+2 B \,a^{9} b \,x^{5}+20 x^{6} a^{7} b^{3} A +\frac {15}{2} x^{6} a^{8} b^{2} B +30 x^{7} a^{6} b^{4} A +\frac {120}{7} x^{7} a^{7} b^{3} B +\frac {63}{2} x^{8} a^{5} b^{5} A +\frac {105}{4} x^{8} a^{6} b^{4} B +\frac {70}{3} x^{9} a^{4} b^{6} A +28 x^{9} a^{5} b^{5} B +12 A \,a^{3} b^{7} x^{10}+21 B \,a^{4} b^{6} x^{10}+\frac {45}{11} x^{11} a^{2} b^{8} A +\frac {120}{11} x^{11} a^{3} b^{7} B +\frac {5}{6} x^{12} a \,b^{9} A +\frac {15}{4} x^{12} a^{2} b^{8} B +\frac {1}{13} x^{13} b^{10} A +\frac {10}{13} x^{13} a \,b^{9} B +\frac {1}{14} b^{10} B \,x^{14}\) | \(246\) |
1/3*a^10*A*x^3+(5/2*a^9*b*A+1/4*a^10*B)*x^4+(9*A*a^8*b^2+2*B*a^9*b)*x^5+(2 0*a^7*b^3*A+15/2*a^8*b^2*B)*x^6+(30*a^6*b^4*A+120/7*a^7*b^3*B)*x^7+(63/2*a ^5*b^5*A+105/4*a^6*b^4*B)*x^8+(70/3*a^4*b^6*A+28*a^5*b^5*B)*x^9+(12*A*a^3* b^7+21*B*a^4*b^6)*x^10+(45/11*a^2*b^8*A+120/11*a^3*b^7*B)*x^11+(5/6*a*b^9* A+15/4*a^2*b^8*B)*x^12+(1/13*b^10*A+10/13*a*b^9*B)*x^13+1/14*b^10*B*x^14
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (80) = 160\).
Time = 0.22 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.78 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {1}{14} \, B b^{10} x^{14} + \frac {1}{3} \, A a^{10} x^{3} + \frac {1}{13} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{13} + \frac {5}{12} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{12} + \frac {15}{11} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{11} + 3 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{10} + \frac {14}{3} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{9} + \frac {21}{4} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{8} + \frac {30}{7} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{7} + \frac {5}{2} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{6} + {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{4} \]
1/14*B*b^10*x^14 + 1/3*A*a^10*x^3 + 1/13*(10*B*a*b^9 + A*b^10)*x^13 + 5/12 *(9*B*a^2*b^8 + 2*A*a*b^9)*x^12 + 15/11*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^11 + 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^10 + 14/3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^9 + 21/4*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^8 + 30/7*(4*B*a^7*b^3 + 7*A*a^6*b^4) *x^7 + 5/2*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^6 + (2*B*a^9*b + 9*A*a^8*b^2)*x^5 + 1/4*(B*a^10 + 10*A*a^9*b)*x^4
Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (80) = 160\).
Time = 0.04 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.01 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {A a^{10} x^{3}}{3} + \frac {B b^{10} x^{14}}{14} + x^{13} \left (\frac {A b^{10}}{13} + \frac {10 B a b^{9}}{13}\right ) + x^{12} \cdot \left (\frac {5 A a b^{9}}{6} + \frac {15 B a^{2} b^{8}}{4}\right ) + x^{11} \cdot \left (\frac {45 A a^{2} b^{8}}{11} + \frac {120 B a^{3} b^{7}}{11}\right ) + x^{10} \cdot \left (12 A a^{3} b^{7} + 21 B a^{4} b^{6}\right ) + x^{9} \cdot \left (\frac {70 A a^{4} b^{6}}{3} + 28 B a^{5} b^{5}\right ) + x^{8} \cdot \left (\frac {63 A a^{5} b^{5}}{2} + \frac {105 B a^{6} b^{4}}{4}\right ) + x^{7} \cdot \left (30 A a^{6} b^{4} + \frac {120 B a^{7} b^{3}}{7}\right ) + x^{6} \cdot \left (20 A a^{7} b^{3} + \frac {15 B a^{8} b^{2}}{2}\right ) + x^{5} \cdot \left (9 A a^{8} b^{2} + 2 B a^{9} b\right ) + x^{4} \cdot \left (\frac {5 A a^{9} b}{2} + \frac {B a^{10}}{4}\right ) \]
A*a**10*x**3/3 + B*b**10*x**14/14 + x**13*(A*b**10/13 + 10*B*a*b**9/13) + x**12*(5*A*a*b**9/6 + 15*B*a**2*b**8/4) + x**11*(45*A*a**2*b**8/11 + 120*B *a**3*b**7/11) + x**10*(12*A*a**3*b**7 + 21*B*a**4*b**6) + x**9*(70*A*a**4 *b**6/3 + 28*B*a**5*b**5) + x**8*(63*A*a**5*b**5/2 + 105*B*a**6*b**4/4) + x**7*(30*A*a**6*b**4 + 120*B*a**7*b**3/7) + x**6*(20*A*a**7*b**3 + 15*B*a* *8*b**2/2) + x**5*(9*A*a**8*b**2 + 2*B*a**9*b) + x**4*(5*A*a**9*b/2 + B*a* *10/4)
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (80) = 160\).
Time = 0.20 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.78 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {1}{14} \, B b^{10} x^{14} + \frac {1}{3} \, A a^{10} x^{3} + \frac {1}{13} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{13} + \frac {5}{12} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{12} + \frac {15}{11} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{11} + 3 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{10} + \frac {14}{3} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{9} + \frac {21}{4} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{8} + \frac {30}{7} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{7} + \frac {5}{2} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{6} + {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{4} \]
1/14*B*b^10*x^14 + 1/3*A*a^10*x^3 + 1/13*(10*B*a*b^9 + A*b^10)*x^13 + 5/12 *(9*B*a^2*b^8 + 2*A*a*b^9)*x^12 + 15/11*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^11 + 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^10 + 14/3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^9 + 21/4*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^8 + 30/7*(4*B*a^7*b^3 + 7*A*a^6*b^4) *x^7 + 5/2*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^6 + (2*B*a^9*b + 9*A*a^8*b^2)*x^5 + 1/4*(B*a^10 + 10*A*a^9*b)*x^4
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (80) = 160\).
Time = 0.29 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.82 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {1}{14} \, B b^{10} x^{14} + \frac {10}{13} \, B a b^{9} x^{13} + \frac {1}{13} \, A b^{10} x^{13} + \frac {15}{4} \, B a^{2} b^{8} x^{12} + \frac {5}{6} \, A a b^{9} x^{12} + \frac {120}{11} \, B a^{3} b^{7} x^{11} + \frac {45}{11} \, A a^{2} b^{8} x^{11} + 21 \, B a^{4} b^{6} x^{10} + 12 \, A a^{3} b^{7} x^{10} + 28 \, B a^{5} b^{5} x^{9} + \frac {70}{3} \, A a^{4} b^{6} x^{9} + \frac {105}{4} \, B a^{6} b^{4} x^{8} + \frac {63}{2} \, A a^{5} b^{5} x^{8} + \frac {120}{7} \, B a^{7} b^{3} x^{7} + 30 \, A a^{6} b^{4} x^{7} + \frac {15}{2} \, B a^{8} b^{2} x^{6} + 20 \, A a^{7} b^{3} x^{6} + 2 \, B a^{9} b x^{5} + 9 \, A a^{8} b^{2} x^{5} + \frac {1}{4} \, B a^{10} x^{4} + \frac {5}{2} \, A a^{9} b x^{4} + \frac {1}{3} \, A a^{10} x^{3} \]
1/14*B*b^10*x^14 + 10/13*B*a*b^9*x^13 + 1/13*A*b^10*x^13 + 15/4*B*a^2*b^8* x^12 + 5/6*A*a*b^9*x^12 + 120/11*B*a^3*b^7*x^11 + 45/11*A*a^2*b^8*x^11 + 2 1*B*a^4*b^6*x^10 + 12*A*a^3*b^7*x^10 + 28*B*a^5*b^5*x^9 + 70/3*A*a^4*b^6*x ^9 + 105/4*B*a^6*b^4*x^8 + 63/2*A*a^5*b^5*x^8 + 120/7*B*a^7*b^3*x^7 + 30*A *a^6*b^4*x^7 + 15/2*B*a^8*b^2*x^6 + 20*A*a^7*b^3*x^6 + 2*B*a^9*b*x^5 + 9*A *a^8*b^2*x^5 + 1/4*B*a^10*x^4 + 5/2*A*a^9*b*x^4 + 1/3*A*a^10*x^3
Time = 0.52 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.41 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=x^4\,\left (\frac {B\,a^{10}}{4}+\frac {5\,A\,b\,a^9}{2}\right )+x^{13}\,\left (\frac {A\,b^{10}}{13}+\frac {10\,B\,a\,b^9}{13}\right )+\frac {A\,a^{10}\,x^3}{3}+\frac {B\,b^{10}\,x^{14}}{14}+\frac {5\,a^7\,b^2\,x^6\,\left (8\,A\,b+3\,B\,a\right )}{2}+\frac {30\,a^6\,b^3\,x^7\,\left (7\,A\,b+4\,B\,a\right )}{7}+\frac {21\,a^5\,b^4\,x^8\,\left (6\,A\,b+5\,B\,a\right )}{4}+\frac {14\,a^4\,b^5\,x^9\,\left (5\,A\,b+6\,B\,a\right )}{3}+3\,a^3\,b^6\,x^{10}\,\left (4\,A\,b+7\,B\,a\right )+\frac {15\,a^2\,b^7\,x^{11}\,\left (3\,A\,b+8\,B\,a\right )}{11}+a^8\,b\,x^5\,\left (9\,A\,b+2\,B\,a\right )+\frac {5\,a\,b^8\,x^{12}\,\left (2\,A\,b+9\,B\,a\right )}{12} \]
x^4*((B*a^10)/4 + (5*A*a^9*b)/2) + x^13*((A*b^10)/13 + (10*B*a*b^9)/13) + (A*a^10*x^3)/3 + (B*b^10*x^14)/14 + (5*a^7*b^2*x^6*(8*A*b + 3*B*a))/2 + (3 0*a^6*b^3*x^7*(7*A*b + 4*B*a))/7 + (21*a^5*b^4*x^8*(6*A*b + 5*B*a))/4 + (1 4*a^4*b^5*x^9*(5*A*b + 6*B*a))/3 + 3*a^3*b^6*x^10*(4*A*b + 7*B*a) + (15*a^ 2*b^7*x^11*(3*A*b + 8*B*a))/11 + a^8*b*x^5*(9*A*b + 2*B*a) + (5*a*b^8*x^12 *(2*A*b + 9*B*a))/12