Integrand size = 11, antiderivative size = 98 \[ \int x^5 (a+b x)^{10} \, dx=-\frac {a^5 (a+b x)^{11}}{11 b^6}+\frac {5 a^4 (a+b x)^{12}}{12 b^6}-\frac {10 a^3 (a+b x)^{13}}{13 b^6}+\frac {5 a^2 (a+b x)^{14}}{7 b^6}-\frac {a (a+b x)^{15}}{3 b^6}+\frac {(a+b x)^{16}}{16 b^6} \] Output:
-1/11*a^5*(b*x+a)^11/b^6+5/12*a^4*(b*x+a)^12/b^6-10/13*a^3*(b*x+a)^13/b^6+ 5/7*a^2*(b*x+a)^14/b^6-1/3*a*(b*x+a)^15/b^6+1/16*(b*x+a)^16/b^6
Time = 0.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.35 \[ \int x^5 (a+b x)^{10} \, dx=\frac {a^{10} x^6}{6}+\frac {10}{7} a^9 b x^7+\frac {45}{8} a^8 b^2 x^8+\frac {40}{3} a^7 b^3 x^9+21 a^6 b^4 x^{10}+\frac {252}{11} a^5 b^5 x^{11}+\frac {35}{2} a^4 b^6 x^{12}+\frac {120}{13} a^3 b^7 x^{13}+\frac {45}{14} a^2 b^8 x^{14}+\frac {2}{3} a b^9 x^{15}+\frac {b^{10} x^{16}}{16} \] Input:
Integrate[x^5*(a + b*x)^10,x]
Output:
(a^10*x^6)/6 + (10*a^9*b*x^7)/7 + (45*a^8*b^2*x^8)/8 + (40*a^7*b^3*x^9)/3 + 21*a^6*b^4*x^10 + (252*a^5*b^5*x^11)/11 + (35*a^4*b^6*x^12)/2 + (120*a^3 *b^7*x^13)/13 + (45*a^2*b^8*x^14)/14 + (2*a*b^9*x^15)/3 + (b^10*x^16)/16
Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {49, 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.
\(\displaystyle \int x^5 (a+b x)^{10} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (-\frac {a^5 (a+b x)^{10}}{b^5}+\frac {5 a^4 (a+b x)^{11}}{b^5}-\frac {10 a^3 (a+b x)^{12}}{b^5}+\frac {10 a^2 (a+b x)^{13}}{b^5}+\frac {(a+b x)^{15}}{b^5}-\frac {5 a (a+b x)^{14}}{b^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^5 (a+b x)^{11}}{11 b^6}+\frac {5 a^4 (a+b x)^{12}}{12 b^6}-\frac {10 a^3 (a+b x)^{13}}{13 b^6}+\frac {5 a^2 (a+b x)^{14}}{7 b^6}+\frac {(a+b x)^{16}}{16 b^6}-\frac {a (a+b x)^{15}}{3 b^6}\) |
Input:
Int[x^5*(a + b*x)^10,x]
Output:
-1/11*(a^5*(a + b*x)^11)/b^6 + (5*a^4*(a + b*x)^12)/(12*b^6) - (10*a^3*(a + b*x)^13)/(13*b^6) + (5*a^2*(a + b*x)^14)/(7*b^6) - (a*(a + b*x)^15)/(3*b ^6) + (a + b*x)^16/(16*b^6)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.15
method | result | size |
gosper | \(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\) | \(113\) |
default | \(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\) | \(113\) |
norman | \(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\) | \(113\) |
risch | \(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\) | \(113\) |
parallelrisch | \(\frac {1}{6} a^{10} x^{6}+\frac {10}{7} a^{9} b \,x^{7}+\frac {45}{8} a^{8} b^{2} x^{8}+\frac {40}{3} a^{7} b^{3} x^{9}+21 a^{6} b^{4} x^{10}+\frac {252}{11} a^{5} b^{5} x^{11}+\frac {35}{2} a^{4} b^{6} x^{12}+\frac {120}{13} a^{3} b^{7} x^{13}+\frac {45}{14} a^{2} b^{8} x^{14}+\frac {2}{3} a \,b^{9} x^{15}+\frac {1}{16} b^{10} x^{16}\) | \(113\) |
orering | \(\frac {x^{6} \left (3003 b^{10} x^{10}+32032 a \,b^{9} x^{9}+154440 a^{2} b^{8} x^{8}+443520 a^{3} b^{7} x^{7}+840840 a^{4} b^{6} x^{6}+1100736 a^{5} b^{5} x^{5}+1009008 a^{6} b^{4} x^{4}+640640 a^{7} b^{3} x^{3}+270270 a^{8} b^{2} x^{2}+68640 a^{9} b x +8008 a^{10}\right )}{48048}\) | \(113\) |
Input:
int(x^5*(b*x+a)^10,x,method=_RETURNVERBOSE)
Output:
1/6*a^10*x^6+10/7*a^9*b*x^7+45/8*a^8*b^2*x^8+40/3*a^7*b^3*x^9+21*a^6*b^4*x ^10+252/11*a^5*b^5*x^11+35/2*a^4*b^6*x^12+120/13*a^3*b^7*x^13+45/14*a^2*b^ 8*x^14+2/3*a*b^9*x^15+1/16*b^10*x^16
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^{10} \, dx=\frac {1}{16} \, b^{10} x^{16} + \frac {2}{3} \, a b^{9} x^{15} + \frac {45}{14} \, a^{2} b^{8} x^{14} + \frac {120}{13} \, a^{3} b^{7} x^{13} + \frac {35}{2} \, a^{4} b^{6} x^{12} + \frac {252}{11} \, a^{5} b^{5} x^{11} + 21 \, a^{6} b^{4} x^{10} + \frac {40}{3} \, a^{7} b^{3} x^{9} + \frac {45}{8} \, a^{8} b^{2} x^{8} + \frac {10}{7} \, a^{9} b x^{7} + \frac {1}{6} \, a^{10} x^{6} \] Input:
integrate(x^5*(b*x+a)^10,x, algorithm="fricas")
Output:
1/16*b^10*x^16 + 2/3*a*b^9*x^15 + 45/14*a^2*b^8*x^14 + 120/13*a^3*b^7*x^13 + 35/2*a^4*b^6*x^12 + 252/11*a^5*b^5*x^11 + 21*a^6*b^4*x^10 + 40/3*a^7*b^ 3*x^9 + 45/8*a^8*b^2*x^8 + 10/7*a^9*b*x^7 + 1/6*a^10*x^6
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.36 \[ \int x^5 (a+b x)^{10} \, dx=\frac {a^{10} x^{6}}{6} + \frac {10 a^{9} b x^{7}}{7} + \frac {45 a^{8} b^{2} x^{8}}{8} + \frac {40 a^{7} b^{3} x^{9}}{3} + 21 a^{6} b^{4} x^{10} + \frac {252 a^{5} b^{5} x^{11}}{11} + \frac {35 a^{4} b^{6} x^{12}}{2} + \frac {120 a^{3} b^{7} x^{13}}{13} + \frac {45 a^{2} b^{8} x^{14}}{14} + \frac {2 a b^{9} x^{15}}{3} + \frac {b^{10} x^{16}}{16} \] Input:
integrate(x**5*(b*x+a)**10,x)
Output:
a**10*x**6/6 + 10*a**9*b*x**7/7 + 45*a**8*b**2*x**8/8 + 40*a**7*b**3*x**9/ 3 + 21*a**6*b**4*x**10 + 252*a**5*b**5*x**11/11 + 35*a**4*b**6*x**12/2 + 1 20*a**3*b**7*x**13/13 + 45*a**2*b**8*x**14/14 + 2*a*b**9*x**15/3 + b**10*x **16/16
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^{10} \, dx=\frac {1}{16} \, b^{10} x^{16} + \frac {2}{3} \, a b^{9} x^{15} + \frac {45}{14} \, a^{2} b^{8} x^{14} + \frac {120}{13} \, a^{3} b^{7} x^{13} + \frac {35}{2} \, a^{4} b^{6} x^{12} + \frac {252}{11} \, a^{5} b^{5} x^{11} + 21 \, a^{6} b^{4} x^{10} + \frac {40}{3} \, a^{7} b^{3} x^{9} + \frac {45}{8} \, a^{8} b^{2} x^{8} + \frac {10}{7} \, a^{9} b x^{7} + \frac {1}{6} \, a^{10} x^{6} \] Input:
integrate(x^5*(b*x+a)^10,x, algorithm="maxima")
Output:
1/16*b^10*x^16 + 2/3*a*b^9*x^15 + 45/14*a^2*b^8*x^14 + 120/13*a^3*b^7*x^13 + 35/2*a^4*b^6*x^12 + 252/11*a^5*b^5*x^11 + 21*a^6*b^4*x^10 + 40/3*a^7*b^ 3*x^9 + 45/8*a^8*b^2*x^8 + 10/7*a^9*b*x^7 + 1/6*a^10*x^6
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^{10} \, dx=\frac {1}{16} \, b^{10} x^{16} + \frac {2}{3} \, a b^{9} x^{15} + \frac {45}{14} \, a^{2} b^{8} x^{14} + \frac {120}{13} \, a^{3} b^{7} x^{13} + \frac {35}{2} \, a^{4} b^{6} x^{12} + \frac {252}{11} \, a^{5} b^{5} x^{11} + 21 \, a^{6} b^{4} x^{10} + \frac {40}{3} \, a^{7} b^{3} x^{9} + \frac {45}{8} \, a^{8} b^{2} x^{8} + \frac {10}{7} \, a^{9} b x^{7} + \frac {1}{6} \, a^{10} x^{6} \] Input:
integrate(x^5*(b*x+a)^10,x, algorithm="giac")
Output:
1/16*b^10*x^16 + 2/3*a*b^9*x^15 + 45/14*a^2*b^8*x^14 + 120/13*a^3*b^7*x^13 + 35/2*a^4*b^6*x^12 + 252/11*a^5*b^5*x^11 + 21*a^6*b^4*x^10 + 40/3*a^7*b^ 3*x^9 + 45/8*a^8*b^2*x^8 + 10/7*a^9*b*x^7 + 1/6*a^10*x^6
Time = 0.05 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^{10} \, dx=\frac {a^{10}\,x^6}{6}+\frac {10\,a^9\,b\,x^7}{7}+\frac {45\,a^8\,b^2\,x^8}{8}+\frac {40\,a^7\,b^3\,x^9}{3}+21\,a^6\,b^4\,x^{10}+\frac {252\,a^5\,b^5\,x^{11}}{11}+\frac {35\,a^4\,b^6\,x^{12}}{2}+\frac {120\,a^3\,b^7\,x^{13}}{13}+\frac {45\,a^2\,b^8\,x^{14}}{14}+\frac {2\,a\,b^9\,x^{15}}{3}+\frac {b^{10}\,x^{16}}{16} \] Input:
int(x^5*(a + b*x)^10,x)
Output:
(a^10*x^6)/6 + (b^10*x^16)/16 + (10*a^9*b*x^7)/7 + (2*a*b^9*x^15)/3 + (45* a^8*b^2*x^8)/8 + (40*a^7*b^3*x^9)/3 + 21*a^6*b^4*x^10 + (252*a^5*b^5*x^11) /11 + (35*a^4*b^6*x^12)/2 + (120*a^3*b^7*x^13)/13 + (45*a^2*b^8*x^14)/14
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^{10} \, dx=\frac {x^{6} \left (3003 b^{10} x^{10}+32032 a \,b^{9} x^{9}+154440 a^{2} b^{8} x^{8}+443520 a^{3} b^{7} x^{7}+840840 a^{4} b^{6} x^{6}+1100736 a^{5} b^{5} x^{5}+1009008 a^{6} b^{4} x^{4}+640640 a^{7} b^{3} x^{3}+270270 a^{8} b^{2} x^{2}+68640 a^{9} b x +8008 a^{10}\right )}{48048} \] Input:
int(x^5*(b*x+a)^10,x)
Output:
(x**6*(8008*a**10 + 68640*a**9*b*x + 270270*a**8*b**2*x**2 + 640640*a**7*b **3*x**3 + 1009008*a**6*b**4*x**4 + 1100736*a**5*b**5*x**5 + 840840*a**4*b **6*x**6 + 443520*a**3*b**7*x**7 + 154440*a**2*b**8*x**8 + 32032*a*b**9*x* *9 + 3003*b**10*x**10))/48048