Integrand size = 20, antiderivative size = 119 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {a^7 c^6}{12 x^{12}}+\frac {5 a^6 b c^6}{11 x^{11}}-\frac {9 a^5 b^2 c^6}{10 x^{10}}+\frac {5 a^4 b^3 c^6}{9 x^9}+\frac {5 a^3 b^4 c^6}{8 x^8}-\frac {9 a^2 b^5 c^6}{7 x^7}+\frac {5 a b^6 c^6}{6 x^6}-\frac {b^7 c^6}{5 x^5} \] Output:
-1/12*a^7*c^6/x^12+5/11*a^6*b*c^6/x^11-9/10*a^5*b^2*c^6/x^10+5/9*a^4*b^3*c ^6/x^9+5/8*a^3*b^4*c^6/x^8-9/7*a^2*b^5*c^6/x^7+5/6*a*b^6*c^6/x^6-1/5*b^7*c ^6/x^5
Time = 0.01 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {a^7 c^6}{12 x^{12}}+\frac {5 a^6 b c^6}{11 x^{11}}-\frac {9 a^5 b^2 c^6}{10 x^{10}}+\frac {5 a^4 b^3 c^6}{9 x^9}+\frac {5 a^3 b^4 c^6}{8 x^8}-\frac {9 a^2 b^5 c^6}{7 x^7}+\frac {5 a b^6 c^6}{6 x^6}-\frac {b^7 c^6}{5 x^5} \] Input:
Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^13,x]
Output:
-1/12*(a^7*c^6)/x^12 + (5*a^6*b*c^6)/(11*x^11) - (9*a^5*b^2*c^6)/(10*x^10) + (5*a^4*b^3*c^6)/(9*x^9) + (5*a^3*b^4*c^6)/(8*x^8) - (9*a^2*b^5*c^6)/(7* x^7) + (5*a*b^6*c^6)/(6*x^6) - (b^7*c^6)/(5*x^5)
Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {84, 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.
\(\displaystyle \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx\) |
\(\Big \downarrow \) 84 |
\(\displaystyle \int \left (\frac {a^7 c^6}{x^{13}}-\frac {5 a^6 b c^6}{x^{12}}+\frac {9 a^5 b^2 c^6}{x^{11}}-\frac {5 a^4 b^3 c^6}{x^{10}}-\frac {5 a^3 b^4 c^6}{x^9}+\frac {9 a^2 b^5 c^6}{x^8}-\frac {5 a b^6 c^6}{x^7}+\frac {b^7 c^6}{x^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^7 c^6}{12 x^{12}}+\frac {5 a^6 b c^6}{11 x^{11}}-\frac {9 a^5 b^2 c^6}{10 x^{10}}+\frac {5 a^4 b^3 c^6}{9 x^9}+\frac {5 a^3 b^4 c^6}{8 x^8}-\frac {9 a^2 b^5 c^6}{7 x^7}+\frac {5 a b^6 c^6}{6 x^6}-\frac {b^7 c^6}{5 x^5}\) |
Input:
Int[((a + b*x)*(a*c - b*c*x)^6)/x^13,x]
Output:
-1/12*(a^7*c^6)/x^12 + (5*a^6*b*c^6)/(11*x^11) - (9*a^5*b^2*c^6)/(10*x^10) + (5*a^4*b^3*c^6)/(9*x^9) + (5*a^3*b^4*c^6)/(8*x^8) - (9*a^2*b^5*c^6)/(7* x^7) + (5*a*b^6*c^6)/(6*x^6) - (b^7*c^6)/(5*x^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] && EqQ[b*e + a*f, 0] && !(ILtQ[n + p + 2, 0 ] && GtQ[n + 2*p, 0])
Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70
method | result | size |
gosper | \(-\frac {c^{6} \left (5544 b^{7} x^{7}-23100 a \,b^{6} x^{6}+35640 a^{2} b^{5} x^{5}-17325 a^{3} b^{4} x^{4}-15400 a^{4} b^{3} x^{3}+24948 a^{5} b^{2} x^{2}-12600 a^{6} b x +2310 a^{7}\right )}{27720 x^{12}}\) | \(83\) |
default | \(c^{6} \left (-\frac {b^{7}}{5 x^{5}}+\frac {5 a^{6} b}{11 x^{11}}-\frac {9 a^{2} b^{5}}{7 x^{7}}+\frac {5 a^{3} b^{4}}{8 x^{8}}-\frac {9 a^{5} b^{2}}{10 x^{10}}+\frac {5 a \,b^{6}}{6 x^{6}}+\frac {5 a^{4} b^{3}}{9 x^{9}}-\frac {a^{7}}{12 x^{12}}\right )\) | \(84\) |
orering | \(-\frac {\left (5544 b^{7} x^{7}-23100 a \,b^{6} x^{6}+35640 a^{2} b^{5} x^{5}-17325 a^{3} b^{4} x^{4}-15400 a^{4} b^{3} x^{3}+24948 a^{5} b^{2} x^{2}-12600 a^{6} b x +2310 a^{7}\right ) \left (-b c x +a c \right )^{6}}{27720 x^{12} \left (-b x +a \right )^{6}}\) | \(99\) |
norman | \(\frac {-\frac {1}{12} a^{7} c^{6}-\frac {1}{5} b^{7} c^{6} x^{7}+\frac {5}{6} a \,b^{6} c^{6} x^{6}-\frac {9}{7} a^{2} b^{5} c^{6} x^{5}+\frac {5}{8} a^{3} b^{4} c^{6} x^{4}+\frac {5}{9} a^{4} b^{3} c^{6} x^{3}-\frac {9}{10} a^{5} b^{2} c^{6} x^{2}+\frac {5}{11} a^{6} b \,c^{6} x}{x^{12}}\) | \(103\) |
risch | \(\frac {-\frac {1}{12} a^{7} c^{6}-\frac {1}{5} b^{7} c^{6} x^{7}+\frac {5}{6} a \,b^{6} c^{6} x^{6}-\frac {9}{7} a^{2} b^{5} c^{6} x^{5}+\frac {5}{8} a^{3} b^{4} c^{6} x^{4}+\frac {5}{9} a^{4} b^{3} c^{6} x^{3}-\frac {9}{10} a^{5} b^{2} c^{6} x^{2}+\frac {5}{11} a^{6} b \,c^{6} x}{x^{12}}\) | \(103\) |
parallelrisch | \(\frac {-5544 b^{7} c^{6} x^{7}+23100 a \,b^{6} c^{6} x^{6}-35640 a^{2} b^{5} c^{6} x^{5}+17325 a^{3} b^{4} c^{6} x^{4}+15400 a^{4} b^{3} c^{6} x^{3}-24948 a^{5} b^{2} c^{6} x^{2}+12600 a^{6} b \,c^{6} x -2310 a^{7} c^{6}}{27720 x^{12}}\) | \(104\) |
Input:
int((b*x+a)*(-b*c*x+a*c)^6/x^13,x,method=_RETURNVERBOSE)
Output:
-1/27720*c^6*(5544*b^7*x^7-23100*a*b^6*x^6+35640*a^2*b^5*x^5-17325*a^3*b^4 *x^4-15400*a^4*b^3*x^3+24948*a^5*b^2*x^2-12600*a^6*b*x+2310*a^7)/x^12
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {5544 \, b^{7} c^{6} x^{7} - 23100 \, a b^{6} c^{6} x^{6} + 35640 \, a^{2} b^{5} c^{6} x^{5} - 17325 \, a^{3} b^{4} c^{6} x^{4} - 15400 \, a^{4} b^{3} c^{6} x^{3} + 24948 \, a^{5} b^{2} c^{6} x^{2} - 12600 \, a^{6} b c^{6} x + 2310 \, a^{7} c^{6}}{27720 \, x^{12}} \] Input:
integrate((b*x+a)*(-b*c*x+a*c)^6/x^13,x, algorithm="fricas")
Output:
-1/27720*(5544*b^7*c^6*x^7 - 23100*a*b^6*c^6*x^6 + 35640*a^2*b^5*c^6*x^5 - 17325*a^3*b^4*c^6*x^4 - 15400*a^4*b^3*c^6*x^3 + 24948*a^5*b^2*c^6*x^2 - 1 2600*a^6*b*c^6*x + 2310*a^7*c^6)/x^12
Time = 0.38 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=\frac {- 2310 a^{7} c^{6} + 12600 a^{6} b c^{6} x - 24948 a^{5} b^{2} c^{6} x^{2} + 15400 a^{4} b^{3} c^{6} x^{3} + 17325 a^{3} b^{4} c^{6} x^{4} - 35640 a^{2} b^{5} c^{6} x^{5} + 23100 a b^{6} c^{6} x^{6} - 5544 b^{7} c^{6} x^{7}}{27720 x^{12}} \] Input:
integrate((b*x+a)*(-b*c*x+a*c)**6/x**13,x)
Output:
(-2310*a**7*c**6 + 12600*a**6*b*c**6*x - 24948*a**5*b**2*c**6*x**2 + 15400 *a**4*b**3*c**6*x**3 + 17325*a**3*b**4*c**6*x**4 - 35640*a**2*b**5*c**6*x* *5 + 23100*a*b**6*c**6*x**6 - 5544*b**7*c**6*x**7)/(27720*x**12)
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {5544 \, b^{7} c^{6} x^{7} - 23100 \, a b^{6} c^{6} x^{6} + 35640 \, a^{2} b^{5} c^{6} x^{5} - 17325 \, a^{3} b^{4} c^{6} x^{4} - 15400 \, a^{4} b^{3} c^{6} x^{3} + 24948 \, a^{5} b^{2} c^{6} x^{2} - 12600 \, a^{6} b c^{6} x + 2310 \, a^{7} c^{6}}{27720 \, x^{12}} \] Input:
integrate((b*x+a)*(-b*c*x+a*c)^6/x^13,x, algorithm="maxima")
Output:
-1/27720*(5544*b^7*c^6*x^7 - 23100*a*b^6*c^6*x^6 + 35640*a^2*b^5*c^6*x^5 - 17325*a^3*b^4*c^6*x^4 - 15400*a^4*b^3*c^6*x^3 + 24948*a^5*b^2*c^6*x^2 - 1 2600*a^6*b*c^6*x + 2310*a^7*c^6)/x^12
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {5544 \, b^{7} c^{6} x^{7} - 23100 \, a b^{6} c^{6} x^{6} + 35640 \, a^{2} b^{5} c^{6} x^{5} - 17325 \, a^{3} b^{4} c^{6} x^{4} - 15400 \, a^{4} b^{3} c^{6} x^{3} + 24948 \, a^{5} b^{2} c^{6} x^{2} - 12600 \, a^{6} b c^{6} x + 2310 \, a^{7} c^{6}}{27720 \, x^{12}} \] Input:
integrate((b*x+a)*(-b*c*x+a*c)^6/x^13,x, algorithm="giac")
Output:
-1/27720*(5544*b^7*c^6*x^7 - 23100*a*b^6*c^6*x^6 + 35640*a^2*b^5*c^6*x^5 - 17325*a^3*b^4*c^6*x^4 - 15400*a^4*b^3*c^6*x^3 + 24948*a^5*b^2*c^6*x^2 - 1 2600*a^6*b*c^6*x + 2310*a^7*c^6)/x^12
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=-\frac {\frac {a^7\,c^6}{12}-\frac {5\,a^6\,b\,c^6\,x}{11}+\frac {9\,a^5\,b^2\,c^6\,x^2}{10}-\frac {5\,a^4\,b^3\,c^6\,x^3}{9}-\frac {5\,a^3\,b^4\,c^6\,x^4}{8}+\frac {9\,a^2\,b^5\,c^6\,x^5}{7}-\frac {5\,a\,b^6\,c^6\,x^6}{6}+\frac {b^7\,c^6\,x^7}{5}}{x^{12}} \] Input:
int(((a*c - b*c*x)^6*(a + b*x))/x^13,x)
Output:
-((a^7*c^6)/12 + (b^7*c^6*x^7)/5 - (5*a*b^6*c^6*x^6)/6 + (9*a^5*b^2*c^6*x^ 2)/10 - (5*a^4*b^3*c^6*x^3)/9 - (5*a^3*b^4*c^6*x^4)/8 + (9*a^2*b^5*c^6*x^5 )/7 - (5*a^6*b*c^6*x)/11)/x^12
Time = 0.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx=\frac {c^{6} \left (-5544 b^{7} x^{7}+23100 a \,b^{6} x^{6}-35640 a^{2} b^{5} x^{5}+17325 a^{3} b^{4} x^{4}+15400 a^{4} b^{3} x^{3}-24948 a^{5} b^{2} x^{2}+12600 a^{6} b x -2310 a^{7}\right )}{27720 x^{12}} \] Input:
int((b*x+a)*(-b*c*x+a*c)^6/x^13,x)
Output:
(c**6*( - 2310*a**7 + 12600*a**6*b*x - 24948*a**5*b**2*x**2 + 15400*a**4*b **3*x**3 + 17325*a**3*b**4*x**4 - 35640*a**2*b**5*x**5 + 23100*a*b**6*x**6 - 5544*b**7*x**7))/(27720*x**12)