Integrand size = 11, antiderivative size = 117 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=-\frac {a^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5}+252 a^5 b^5 \log (x) \] Output:
-1/5*a^10/x^5-5/2*a^9*b/x^4-15*a^8*b^2/x^3-60*a^7*b^3/x^2-210*a^6*b^4/x+21 0*a^4*b^6*x+60*a^3*b^7*x^2+15*a^2*b^8*x^3+5/2*a*b^9*x^4+1/5*b^10*x^5+252*a ^5*b^5*ln(x)
Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=-\frac {a^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5}+252 a^5 b^5 \log (x) \] Input:
Integrate[(a + b*x)^10/x^6,x]
Output:
-1/5*a^10/x^5 - (5*a^9*b)/(2*x^4) - (15*a^8*b^2)/x^3 - (60*a^7*b^3)/x^2 - (210*a^6*b^4)/x + 210*a^4*b^6*x + 60*a^3*b^7*x^2 + 15*a^2*b^8*x^3 + (5*a*b ^9*x^4)/2 + (b^10*x^5)/5 + 252*a^5*b^5*Log[x]
Time = 0.21 (sec) , antiderivative size = 117, 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.
\(\displaystyle \int \frac {(a+b x)^{10}}{x^6} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {a^{10}}{x^6}+\frac {10 a^9 b}{x^5}+\frac {45 a^8 b^2}{x^4}+\frac {120 a^7 b^3}{x^3}+\frac {210 a^6 b^4}{x^2}+\frac {252 a^5 b^5}{x}+210 a^4 b^6+120 a^3 b^7 x+45 a^2 b^8 x^2+10 a b^9 x^3+b^{10} x^4\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+252 a^5 b^5 \log (x)+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5}\) |
Input:
Int[(a + b*x)^10/x^6,x]
Output:
-1/5*a^10/x^5 - (5*a^9*b)/(2*x^4) - (15*a^8*b^2)/x^3 - (60*a^7*b^3)/x^2 - (210*a^6*b^4)/x + 210*a^4*b^6*x + 60*a^3*b^7*x^2 + 15*a^2*b^8*x^3 + (5*a*b ^9*x^4)/2 + (b^10*x^5)/5 + 252*a^5*b^5*Log[x]
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 = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {a^{10}}{5 x^{5}}-\frac {5 a^{9} b}{2 x^{4}}-\frac {15 a^{8} b^{2}}{x^{3}}-\frac {60 a^{7} b^{3}}{x^{2}}-\frac {210 a^{6} b^{4}}{x}+210 a^{4} b^{6} x +60 a^{3} b^{7} x^{2}+15 a^{2} b^{8} x^{3}+\frac {5 a \,b^{9} x^{4}}{2}+\frac {b^{10} x^{5}}{5}+252 a^{5} b^{5} \ln \left (x \right )\) | \(110\) |
risch | \(\frac {b^{10} x^{5}}{5}+\frac {5 a \,b^{9} x^{4}}{2}+15 a^{2} b^{8} x^{3}+60 a^{3} b^{7} x^{2}+210 a^{4} b^{6} x +\frac {-210 a^{6} b^{4} x^{4}-60 a^{7} b^{3} x^{3}-15 a^{8} b^{2} x^{2}-\frac {5}{2} a^{9} b x -\frac {1}{5} a^{10}}{x^{5}}+252 a^{5} b^{5} \ln \left (x \right )\) | \(110\) |
norman | \(\frac {-\frac {1}{5} a^{10}+\frac {1}{5} b^{10} x^{10}+\frac {5}{2} a \,b^{9} x^{9}+15 a^{2} b^{8} x^{8}+60 a^{3} b^{7} x^{7}+210 a^{4} b^{6} x^{6}-210 a^{6} b^{4} x^{4}-60 a^{7} b^{3} x^{3}-15 a^{8} b^{2} x^{2}-\frac {5}{2} a^{9} b x}{x^{5}}+252 a^{5} b^{5} \ln \left (x \right )\) | \(112\) |
parallelrisch | \(\frac {2 b^{10} x^{10}+25 a \,b^{9} x^{9}+150 a^{2} b^{8} x^{8}+600 a^{3} b^{7} x^{7}+2520 a^{5} b^{5} \ln \left (x \right ) x^{5}+2100 a^{4} b^{6} x^{6}-2100 a^{6} b^{4} x^{4}-600 a^{7} b^{3} x^{3}-150 a^{8} b^{2} x^{2}-25 a^{9} b x -2 a^{10}}{10 x^{5}}\) | \(115\) |
Input:
int((b*x+a)^10/x^6,x,method=_RETURNVERBOSE)
Output:
-1/5*a^10/x^5-5/2*a^9*b/x^4-15*a^8*b^2/x^3-60*a^7*b^3/x^2-210*a^6*b^4/x+21 0*a^4*b^6*x+60*a^3*b^7*x^2+15*a^2*b^8*x^3+5/2*a*b^9*x^4+1/5*b^10*x^5+252*a ^5*b^5*ln(x)
Time = 0.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=\frac {2 \, b^{10} x^{10} + 25 \, a b^{9} x^{9} + 150 \, a^{2} b^{8} x^{8} + 600 \, a^{3} b^{7} x^{7} + 2100 \, a^{4} b^{6} x^{6} + 2520 \, a^{5} b^{5} x^{5} \log \left (x\right ) - 2100 \, a^{6} b^{4} x^{4} - 600 \, a^{7} b^{3} x^{3} - 150 \, a^{8} b^{2} x^{2} - 25 \, a^{9} b x - 2 \, a^{10}}{10 \, x^{5}} \] Input:
integrate((b*x+a)^10/x^6,x, algorithm="fricas")
Output:
1/10*(2*b^10*x^10 + 25*a*b^9*x^9 + 150*a^2*b^8*x^8 + 600*a^3*b^7*x^7 + 210 0*a^4*b^6*x^6 + 2520*a^5*b^5*x^5*log(x) - 2100*a^6*b^4*x^4 - 600*a^7*b^3*x ^3 - 150*a^8*b^2*x^2 - 25*a^9*b*x - 2*a^10)/x^5
Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=252 a^{5} b^{5} \log {\left (x \right )} + 210 a^{4} b^{6} x + 60 a^{3} b^{7} x^{2} + 15 a^{2} b^{8} x^{3} + \frac {5 a b^{9} x^{4}}{2} + \frac {b^{10} x^{5}}{5} + \frac {- 2 a^{10} - 25 a^{9} b x - 150 a^{8} b^{2} x^{2} - 600 a^{7} b^{3} x^{3} - 2100 a^{6} b^{4} x^{4}}{10 x^{5}} \] Input:
integrate((b*x+a)**10/x**6,x)
Output:
252*a**5*b**5*log(x) + 210*a**4*b**6*x + 60*a**3*b**7*x**2 + 15*a**2*b**8* x**3 + 5*a*b**9*x**4/2 + b**10*x**5/5 + (-2*a**10 - 25*a**9*b*x - 150*a**8 *b**2*x**2 - 600*a**7*b**3*x**3 - 2100*a**6*b**4*x**4)/(10*x**5)
Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=\frac {1}{5} \, b^{10} x^{5} + \frac {5}{2} \, a b^{9} x^{4} + 15 \, a^{2} b^{8} x^{3} + 60 \, a^{3} b^{7} x^{2} + 210 \, a^{4} b^{6} x + 252 \, a^{5} b^{5} \log \left (x\right ) - \frac {2100 \, a^{6} b^{4} x^{4} + 600 \, a^{7} b^{3} x^{3} + 150 \, a^{8} b^{2} x^{2} + 25 \, a^{9} b x + 2 \, a^{10}}{10 \, x^{5}} \] Input:
integrate((b*x+a)^10/x^6,x, algorithm="maxima")
Output:
1/5*b^10*x^5 + 5/2*a*b^9*x^4 + 15*a^2*b^8*x^3 + 60*a^3*b^7*x^2 + 210*a^4*b ^6*x + 252*a^5*b^5*log(x) - 1/10*(2100*a^6*b^4*x^4 + 600*a^7*b^3*x^3 + 150 *a^8*b^2*x^2 + 25*a^9*b*x + 2*a^10)/x^5
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=\frac {1}{5} \, b^{10} x^{5} + \frac {5}{2} \, a b^{9} x^{4} + 15 \, a^{2} b^{8} x^{3} + 60 \, a^{3} b^{7} x^{2} + 210 \, a^{4} b^{6} x + 252 \, a^{5} b^{5} \log \left ({\left | x \right |}\right ) - \frac {2100 \, a^{6} b^{4} x^{4} + 600 \, a^{7} b^{3} x^{3} + 150 \, a^{8} b^{2} x^{2} + 25 \, a^{9} b x + 2 \, a^{10}}{10 \, x^{5}} \] Input:
integrate((b*x+a)^10/x^6,x, algorithm="giac")
Output:
1/5*b^10*x^5 + 5/2*a*b^9*x^4 + 15*a^2*b^8*x^3 + 60*a^3*b^7*x^2 + 210*a^4*b ^6*x + 252*a^5*b^5*log(abs(x)) - 1/10*(2100*a^6*b^4*x^4 + 600*a^7*b^3*x^3 + 150*a^8*b^2*x^2 + 25*a^9*b*x + 2*a^10)/x^5
Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=\frac {b^{10}\,x^5}{5}-\frac {\frac {a^{10}}{5}+\frac {5\,a^9\,b\,x}{2}+15\,a^8\,b^2\,x^2+60\,a^7\,b^3\,x^3+210\,a^6\,b^4\,x^4}{x^5}+210\,a^4\,b^6\,x+\frac {5\,a\,b^9\,x^4}{2}+60\,a^3\,b^7\,x^2+15\,a^2\,b^8\,x^3+252\,a^5\,b^5\,\ln \left (x\right ) \] Input:
int((a + b*x)^10/x^6,x)
Output:
(b^10*x^5)/5 - (a^10/5 + 15*a^8*b^2*x^2 + 60*a^7*b^3*x^3 + 210*a^6*b^4*x^4 + (5*a^9*b*x)/2)/x^5 + 210*a^4*b^6*x + (5*a*b^9*x^4)/2 + 60*a^3*b^7*x^2 + 15*a^2*b^8*x^3 + 252*a^5*b^5*log(x)
Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^6} \, dx=\frac {2520 \,\mathrm {log}\left (x \right ) a^{5} b^{5} x^{5}-2 a^{10}-25 a^{9} b x -150 a^{8} b^{2} x^{2}-600 a^{7} b^{3} x^{3}-2100 a^{6} b^{4} x^{4}+2100 a^{4} b^{6} x^{6}+600 a^{3} b^{7} x^{7}+150 a^{2} b^{8} x^{8}+25 a \,b^{9} x^{9}+2 b^{10} x^{10}}{10 x^{5}} \] Input:
int((b*x+a)^10/x^6,x)
Output:
(2520*log(x)*a**5*b**5*x**5 - 2*a**10 - 25*a**9*b*x - 150*a**8*b**2*x**2 - 600*a**7*b**3*x**3 - 2100*a**6*b**4*x**4 + 2100*a**4*b**6*x**6 + 600*a**3 *b**7*x**7 + 150*a**2*b**8*x**8 + 25*a*b**9*x**9 + 2*b**10*x**10)/(10*x**5 )