Integrand size = 13, antiderivative size = 81 \[ \int \left (a+\frac {b}{x}\right )^8 x^{12} \, dx=\frac {b^4 (b+a x)^9}{9 a^5}-\frac {2 b^3 (b+a x)^{10}}{5 a^5}+\frac {6 b^2 (b+a x)^{11}}{11 a^5}-\frac {b (b+a x)^{12}}{3 a^5}+\frac {(b+a x)^{13}}{13 a^5} \] Output:
1/9*b^4*(a*x+b)^9/a^5-2/5*b^3*(a*x+b)^10/a^5+6/11*b^2*(a*x+b)^11/a^5-1/3*b *(a*x+b)^12/a^5+1/13*(a*x+b)^13/a^5
Time = 0.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.28 \[ \int \left (a+\frac {b}{x}\right )^8 x^{12} \, dx=\frac {b^8 x^5}{5}+\frac {4}{3} a b^7 x^6+4 a^2 b^6 x^7+7 a^3 b^5 x^8+\frac {70}{9} a^4 b^4 x^9+\frac {28}{5} a^5 b^3 x^{10}+\frac {28}{11} a^6 b^2 x^{11}+\frac {2}{3} a^7 b x^{12}+\frac {a^8 x^{13}}{13} \] Input:
Integrate[(a + b/x)^8*x^12,x]
Output:
(b^8*x^5)/5 + (4*a*b^7*x^6)/3 + 4*a^2*b^6*x^7 + 7*a^3*b^5*x^8 + (70*a^4*b^ 4*x^9)/9 + (28*a^5*b^3*x^10)/5 + (28*a^6*b^2*x^11)/11 + (2*a^7*b*x^12)/3 + (a^8*x^13)/13
Time = 0.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {795, 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.
\(\displaystyle \int x^{12} \left (a+\frac {b}{x}\right )^8 \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int x^4 (a x+b)^8dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {b^4 (a x+b)^8}{a^4}-\frac {4 b^3 (a x+b)^9}{a^4}+\frac {6 b^2 (a x+b)^{10}}{a^4}+\frac {(a x+b)^{12}}{a^4}-\frac {4 b (a x+b)^{11}}{a^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b^4 (a x+b)^9}{9 a^5}-\frac {2 b^3 (a x+b)^{10}}{5 a^5}+\frac {6 b^2 (a x+b)^{11}}{11 a^5}+\frac {(a x+b)^{13}}{13 a^5}-\frac {b (a x+b)^{12}}{3 a^5}\) |
Input:
Int[(a + b/x)^8*x^12,x]
Output:
(b^4*(b + a*x)^9)/(9*a^5) - (2*b^3*(b + a*x)^10)/(5*a^5) + (6*b^2*(b + a*x )^11)/(11*a^5) - (b*(b + a*x)^12)/(3*a^5) + (b + a*x)^13/(13*a^5)
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.12
method | result | size |
gosper | \(\frac {x^{5} \left (495 a^{8} x^{8}+4290 a^{7} b \,x^{7}+16380 a^{6} b^{2} x^{6}+36036 a^{5} b^{3} x^{5}+50050 a^{4} x^{4} b^{4}+45045 a^{3} b^{5} x^{3}+25740 a^{2} b^{6} x^{2}+8580 a \,b^{7} x +1287 b^{8}\right )}{6435}\) | \(91\) |
default | \(\frac {1}{13} a^{8} x^{13}+\frac {2}{3} a^{7} b \,x^{12}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{9} a^{4} b^{4} x^{9}+7 a^{3} b^{5} x^{8}+4 a^{2} b^{6} x^{7}+\frac {4}{3} a \,b^{7} x^{6}+\frac {1}{5} b^{8} x^{5}\) | \(91\) |
risch | \(\frac {1}{13} a^{8} x^{13}+\frac {2}{3} a^{7} b \,x^{12}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{9} a^{4} b^{4} x^{9}+7 a^{3} b^{5} x^{8}+4 a^{2} b^{6} x^{7}+\frac {4}{3} a \,b^{7} x^{6}+\frac {1}{5} b^{8} x^{5}\) | \(91\) |
parallelrisch | \(\frac {1}{13} a^{8} x^{13}+\frac {2}{3} a^{7} b \,x^{12}+\frac {28}{11} a^{6} b^{2} x^{11}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {70}{9} a^{4} b^{4} x^{9}+7 a^{3} b^{5} x^{8}+4 a^{2} b^{6} x^{7}+\frac {4}{3} a \,b^{7} x^{6}+\frac {1}{5} b^{8} x^{5}\) | \(91\) |
norman | \(\frac {\frac {1}{13} a^{8} x^{20}+\frac {1}{5} b^{8} x^{12}+\frac {4}{3} a \,b^{7} x^{13}+4 a^{2} b^{6} x^{14}+7 a^{3} b^{5} x^{15}+\frac {70}{9} a^{4} b^{4} x^{16}+\frac {28}{5} a^{5} b^{3} x^{17}+\frac {28}{11} a^{6} b^{2} x^{18}+\frac {2}{3} a^{7} b \,x^{19}}{x^{7}}\) | \(95\) |
orering | \(\frac {x^{13} \left (495 a^{8} x^{8}+4290 a^{7} b \,x^{7}+16380 a^{6} b^{2} x^{6}+36036 a^{5} b^{3} x^{5}+50050 a^{4} x^{4} b^{4}+45045 a^{3} b^{5} x^{3}+25740 a^{2} b^{6} x^{2}+8580 a \,b^{7} x +1287 b^{8}\right ) \left (a +\frac {b}{x}\right )^{8}}{6435 \left (a x +b \right )^{8}}\) | \(107\) |
Input:
int((a+b/x)^8*x^12,x,method=_RETURNVERBOSE)
Output:
1/6435*x^5*(495*a^8*x^8+4290*a^7*b*x^7+16380*a^6*b^2*x^6+36036*a^5*b^3*x^5 +50050*a^4*b^4*x^4+45045*a^3*b^5*x^3+25740*a^2*b^6*x^2+8580*a*b^7*x+1287*b ^8)
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11 \[ \int \left (a+\frac {b}{x}\right )^8 x^{12} \, dx=\frac {1}{13} \, a^{8} x^{13} + \frac {2}{3} \, a^{7} b x^{12} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {28}{5} \, a^{5} b^{3} x^{10} + \frac {70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac {4}{3} \, a b^{7} x^{6} + \frac {1}{5} \, b^{8} x^{5} \] Input:
integrate((a+b/x)^8*x^12,x, algorithm="fricas")
Output:
1/13*a^8*x^13 + 2/3*a^7*b*x^12 + 28/11*a^6*b^2*x^11 + 28/5*a^5*b^3*x^10 + 70/9*a^4*b^4*x^9 + 7*a^3*b^5*x^8 + 4*a^2*b^6*x^7 + 4/3*a*b^7*x^6 + 1/5*b^8 *x^5
Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.28 \[ \int \left (a+\frac {b}{x}\right )^8 x^{12} \, dx=\frac {a^{8} x^{13}}{13} + \frac {2 a^{7} b x^{12}}{3} + \frac {28 a^{6} b^{2} x^{11}}{11} + \frac {28 a^{5} b^{3} x^{10}}{5} + \frac {70 a^{4} b^{4} x^{9}}{9} + 7 a^{3} b^{5} x^{8} + 4 a^{2} b^{6} x^{7} + \frac {4 a b^{7} x^{6}}{3} + \frac {b^{8} x^{5}}{5} \] Input:
integrate((a+b/x)**8*x**12,x)
Output:
a**8*x**13/13 + 2*a**7*b*x**12/3 + 28*a**6*b**2*x**11/11 + 28*a**5*b**3*x* *10/5 + 70*a**4*b**4*x**9/9 + 7*a**3*b**5*x**8 + 4*a**2*b**6*x**7 + 4*a*b* *7*x**6/3 + b**8*x**5/5
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11 \[ \int \left (a+\frac {b}{x}\right )^8 x^{12} \, dx=\frac {1}{13} \, a^{8} x^{13} + \frac {2}{3} \, a^{7} b x^{12} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {28}{5} \, a^{5} b^{3} x^{10} + \frac {70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac {4}{3} \, a b^{7} x^{6} + \frac {1}{5} \, b^{8} x^{5} \] Input:
integrate((a+b/x)^8*x^12,x, algorithm="maxima")
Output:
1/13*a^8*x^13 + 2/3*a^7*b*x^12 + 28/11*a^6*b^2*x^11 + 28/5*a^5*b^3*x^10 + 70/9*a^4*b^4*x^9 + 7*a^3*b^5*x^8 + 4*a^2*b^6*x^7 + 4/3*a*b^7*x^6 + 1/5*b^8 *x^5
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11 \[ \int \left (a+\frac {b}{x}\right )^8 x^{12} \, dx=\frac {1}{13} \, a^{8} x^{13} + \frac {2}{3} \, a^{7} b x^{12} + \frac {28}{11} \, a^{6} b^{2} x^{11} + \frac {28}{5} \, a^{5} b^{3} x^{10} + \frac {70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac {4}{3} \, a b^{7} x^{6} + \frac {1}{5} \, b^{8} x^{5} \] Input:
integrate((a+b/x)^8*x^12,x, algorithm="giac")
Output:
1/13*a^8*x^13 + 2/3*a^7*b*x^12 + 28/11*a^6*b^2*x^11 + 28/5*a^5*b^3*x^10 + 70/9*a^4*b^4*x^9 + 7*a^3*b^5*x^8 + 4*a^2*b^6*x^7 + 4/3*a*b^7*x^6 + 1/5*b^8 *x^5
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11 \[ \int \left (a+\frac {b}{x}\right )^8 x^{12} \, dx=\frac {a^8\,x^{13}}{13}+\frac {2\,a^7\,b\,x^{12}}{3}+\frac {28\,a^6\,b^2\,x^{11}}{11}+\frac {28\,a^5\,b^3\,x^{10}}{5}+\frac {70\,a^4\,b^4\,x^9}{9}+7\,a^3\,b^5\,x^8+4\,a^2\,b^6\,x^7+\frac {4\,a\,b^7\,x^6}{3}+\frac {b^8\,x^5}{5} \] Input:
int(x^12*(a + b/x)^8,x)
Output:
(a^8*x^13)/13 + (b^8*x^5)/5 + (4*a*b^7*x^6)/3 + (2*a^7*b*x^12)/3 + 4*a^2*b ^6*x^7 + 7*a^3*b^5*x^8 + (70*a^4*b^4*x^9)/9 + (28*a^5*b^3*x^10)/5 + (28*a^ 6*b^2*x^11)/11
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11 \[ \int \left (a+\frac {b}{x}\right )^8 x^{12} \, dx=\frac {x^{5} \left (495 a^{8} x^{8}+4290 a^{7} b \,x^{7}+16380 a^{6} b^{2} x^{6}+36036 a^{5} b^{3} x^{5}+50050 a^{4} b^{4} x^{4}+45045 a^{3} b^{5} x^{3}+25740 a^{2} b^{6} x^{2}+8580 a \,b^{7} x +1287 b^{8}\right )}{6435} \] Input:
int((a+b/x)^8*x^12,x)
Output:
(x**5*(495*a**8*x**8 + 4290*a**7*b*x**7 + 16380*a**6*b**2*x**6 + 36036*a** 5*b**3*x**5 + 50050*a**4*b**4*x**4 + 45045*a**3*b**5*x**3 + 25740*a**2*b** 6*x**2 + 8580*a*b**7*x + 1287*b**8))/6435