Integrand size = 20, antiderivative size = 89 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{11}} \, dx=-\frac {c^6 (a-b x)^7}{10 x^{10}}-\frac {13 b c^6 (a-b x)^7}{90 a x^9}-\frac {13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}-\frac {13 b^3 c^6 (a-b x)^7}{2520 a^3 x^7} \] Output:
-1/10*c^6*(-b*x+a)^7/x^10-13/90*b*c^6*(-b*x+a)^7/a/x^9-13/360*b^2*c^6*(-b* x+a)^7/a^2/x^8-13/2520*b^3*c^6*(-b*x+a)^7/a^3/x^7
Time = 0.01 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{11}} \, dx=-\frac {a^7 c^6}{10 x^{10}}+\frac {5 a^6 b c^6}{9 x^9}-\frac {9 a^5 b^2 c^6}{8 x^8}+\frac {5 a^4 b^3 c^6}{7 x^7}+\frac {5 a^3 b^4 c^6}{6 x^6}-\frac {9 a^2 b^5 c^6}{5 x^5}+\frac {5 a b^6 c^6}{4 x^4}-\frac {b^7 c^6}{3 x^3} \] Input:
Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^11,x]
Output:
-1/10*(a^7*c^6)/x^10 + (5*a^6*b*c^6)/(9*x^9) - (9*a^5*b^2*c^6)/(8*x^8) + ( 5*a^4*b^3*c^6)/(7*x^7) + (5*a^3*b^4*c^6)/(6*x^6) - (9*a^2*b^5*c^6)/(5*x^5) + (5*a*b^6*c^6)/(4*x^4) - (b^7*c^6)/(3*x^3)
Time = 0.17 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {87, 27, 55, 55, 48}
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.
\(\displaystyle \int \frac {(a+b x) (a c-b c x)^6}{x^{11}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {13}{10} b \int \frac {c^6 (a-b x)^6}{x^{10}}dx-\frac {c^6 (a-b x)^7}{10 x^{10}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {13}{10} b c^6 \int \frac {(a-b x)^6}{x^{10}}dx-\frac {c^6 (a-b x)^7}{10 x^{10}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {13}{10} b c^6 \left (\frac {2 b \int \frac {(a-b x)^6}{x^9}dx}{9 a}-\frac {(a-b x)^7}{9 a x^9}\right )-\frac {c^6 (a-b x)^7}{10 x^{10}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {13}{10} b c^6 \left (\frac {2 b \left (\frac {b \int \frac {(a-b x)^6}{x^8}dx}{8 a}-\frac {(a-b x)^7}{8 a x^8}\right )}{9 a}-\frac {(a-b x)^7}{9 a x^9}\right )-\frac {c^6 (a-b x)^7}{10 x^{10}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {13}{10} b c^6 \left (\frac {2 b \left (-\frac {b (a-b x)^7}{56 a^2 x^7}-\frac {(a-b x)^7}{8 a x^8}\right )}{9 a}-\frac {(a-b x)^7}{9 a x^9}\right )-\frac {c^6 (a-b x)^7}{10 x^{10}}\) |
Input:
Int[((a + b*x)*(a*c - b*c*x)^6)/x^11,x]
Output:
-1/10*(c^6*(a - b*x)^7)/x^10 + (13*b*c^6*(-1/9*(a - b*x)^7/(a*x^9) + (2*b* (-1/8*(a - b*x)^7/(a*x^8) - (b*(a - b*x)^7)/(56*a^2*x^7)))/(9*a)))/10
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(-\frac {c^{6} \left (840 b^{7} x^{7}-3150 a \,b^{6} x^{6}+4536 a^{2} b^{5} x^{5}-2100 a^{3} b^{4} x^{4}-1800 a^{4} b^{3} x^{3}+2835 a^{5} b^{2} x^{2}-1400 a^{6} b x +252 a^{7}\right )}{2520 x^{10}}\) | \(83\) |
default | \(c^{6} \left (-\frac {b^{7}}{3 x^{3}}-\frac {9 a^{2} b^{5}}{5 x^{5}}+\frac {5 a^{4} b^{3}}{7 x^{7}}+\frac {5 a \,b^{6}}{4 x^{4}}-\frac {9 a^{5} b^{2}}{8 x^{8}}-\frac {a^{7}}{10 x^{10}}+\frac {5 a^{3} b^{4}}{6 x^{6}}+\frac {5 a^{6} b}{9 x^{9}}\right )\) | \(84\) |
orering | \(-\frac {\left (840 b^{7} x^{7}-3150 a \,b^{6} x^{6}+4536 a^{2} b^{5} x^{5}-2100 a^{3} b^{4} x^{4}-1800 a^{4} b^{3} x^{3}+2835 a^{5} b^{2} x^{2}-1400 a^{6} b x +252 a^{7}\right ) \left (-b c x +a c \right )^{6}}{2520 x^{10} \left (-b x +a \right )^{6}}\) | \(99\) |
norman | \(\frac {-\frac {1}{10} a^{7} c^{6}-\frac {1}{3} b^{7} c^{6} x^{7}+\frac {5}{4} a \,b^{6} c^{6} x^{6}-\frac {9}{5} a^{2} b^{5} c^{6} x^{5}+\frac {5}{6} a^{3} b^{4} c^{6} x^{4}+\frac {5}{7} a^{4} b^{3} c^{6} x^{3}-\frac {9}{8} a^{5} b^{2} c^{6} x^{2}+\frac {5}{9} a^{6} b \,c^{6} x}{x^{10}}\) | \(103\) |
risch | \(\frac {-\frac {1}{10} a^{7} c^{6}-\frac {1}{3} b^{7} c^{6} x^{7}+\frac {5}{4} a \,b^{6} c^{6} x^{6}-\frac {9}{5} a^{2} b^{5} c^{6} x^{5}+\frac {5}{6} a^{3} b^{4} c^{6} x^{4}+\frac {5}{7} a^{4} b^{3} c^{6} x^{3}-\frac {9}{8} a^{5} b^{2} c^{6} x^{2}+\frac {5}{9} a^{6} b \,c^{6} x}{x^{10}}\) | \(103\) |
parallelrisch | \(\frac {-840 b^{7} c^{6} x^{7}+3150 a \,b^{6} c^{6} x^{6}-4536 a^{2} b^{5} c^{6} x^{5}+2100 a^{3} b^{4} c^{6} x^{4}+1800 a^{4} b^{3} c^{6} x^{3}-2835 a^{5} b^{2} c^{6} x^{2}+1400 a^{6} b \,c^{6} x -252 a^{7} c^{6}}{2520 x^{10}}\) | \(104\) |
Input:
int((b*x+a)*(-b*c*x+a*c)^6/x^11,x,method=_RETURNVERBOSE)
Output:
-1/2520*c^6*(840*b^7*x^7-3150*a*b^6*x^6+4536*a^2*b^5*x^5-2100*a^3*b^4*x^4- 1800*a^4*b^3*x^3+2835*a^5*b^2*x^2-1400*a^6*b*x+252*a^7)/x^10
Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{11}} \, dx=-\frac {840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \] Input:
integrate((b*x+a)*(-b*c*x+a*c)^6/x^11,x, algorithm="fricas")
Output:
-1/2520*(840*b^7*c^6*x^7 - 3150*a*b^6*c^6*x^6 + 4536*a^2*b^5*c^6*x^5 - 210 0*a^3*b^4*c^6*x^4 - 1800*a^4*b^3*c^6*x^3 + 2835*a^5*b^2*c^6*x^2 - 1400*a^6 *b*c^6*x + 252*a^7*c^6)/x^10
Time = 0.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{11}} \, dx=\frac {- 252 a^{7} c^{6} + 1400 a^{6} b c^{6} x - 2835 a^{5} b^{2} c^{6} x^{2} + 1800 a^{4} b^{3} c^{6} x^{3} + 2100 a^{3} b^{4} c^{6} x^{4} - 4536 a^{2} b^{5} c^{6} x^{5} + 3150 a b^{6} c^{6} x^{6} - 840 b^{7} c^{6} x^{7}}{2520 x^{10}} \] Input:
integrate((b*x+a)*(-b*c*x+a*c)**6/x**11,x)
Output:
(-252*a**7*c**6 + 1400*a**6*b*c**6*x - 2835*a**5*b**2*c**6*x**2 + 1800*a** 4*b**3*c**6*x**3 + 2100*a**3*b**4*c**6*x**4 - 4536*a**2*b**5*c**6*x**5 + 3 150*a*b**6*c**6*x**6 - 840*b**7*c**6*x**7)/(2520*x**10)
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{11}} \, dx=-\frac {840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \] Input:
integrate((b*x+a)*(-b*c*x+a*c)^6/x^11,x, algorithm="maxima")
Output:
-1/2520*(840*b^7*c^6*x^7 - 3150*a*b^6*c^6*x^6 + 4536*a^2*b^5*c^6*x^5 - 210 0*a^3*b^4*c^6*x^4 - 1800*a^4*b^3*c^6*x^3 + 2835*a^5*b^2*c^6*x^2 - 1400*a^6 *b*c^6*x + 252*a^7*c^6)/x^10
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{11}} \, dx=-\frac {840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \] Input:
integrate((b*x+a)*(-b*c*x+a*c)^6/x^11,x, algorithm="giac")
Output:
-1/2520*(840*b^7*c^6*x^7 - 3150*a*b^6*c^6*x^6 + 4536*a^2*b^5*c^6*x^5 - 210 0*a^3*b^4*c^6*x^4 - 1800*a^4*b^3*c^6*x^3 + 2835*a^5*b^2*c^6*x^2 - 1400*a^6 *b*c^6*x + 252*a^7*c^6)/x^10
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{11}} \, dx=-\frac {\frac {a^7\,c^6}{10}-\frac {5\,a^6\,b\,c^6\,x}{9}+\frac {9\,a^5\,b^2\,c^6\,x^2}{8}-\frac {5\,a^4\,b^3\,c^6\,x^3}{7}-\frac {5\,a^3\,b^4\,c^6\,x^4}{6}+\frac {9\,a^2\,b^5\,c^6\,x^5}{5}-\frac {5\,a\,b^6\,c^6\,x^6}{4}+\frac {b^7\,c^6\,x^7}{3}}{x^{10}} \] Input:
int(((a*c - b*c*x)^6*(a + b*x))/x^11,x)
Output:
-((a^7*c^6)/10 + (b^7*c^6*x^7)/3 - (5*a*b^6*c^6*x^6)/4 + (9*a^5*b^2*c^6*x^ 2)/8 - (5*a^4*b^3*c^6*x^3)/7 - (5*a^3*b^4*c^6*x^4)/6 + (9*a^2*b^5*c^6*x^5) /5 - (5*a^6*b*c^6*x)/9)/x^10
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x) (a c-b c x)^6}{x^{11}} \, dx=\frac {c^{6} \left (-840 b^{7} x^{7}+3150 a \,b^{6} x^{6}-4536 a^{2} b^{5} x^{5}+2100 a^{3} b^{4} x^{4}+1800 a^{4} b^{3} x^{3}-2835 a^{5} b^{2} x^{2}+1400 a^{6} b x -252 a^{7}\right )}{2520 x^{10}} \] Input:
int((b*x+a)*(-b*c*x+a*c)^6/x^11,x)
Output:
(c**6*( - 252*a**7 + 1400*a**6*b*x - 2835*a**5*b**2*x**2 + 1800*a**4*b**3* x**3 + 2100*a**3*b**4*x**4 - 4536*a**2*b**5*x**5 + 3150*a*b**6*x**6 - 840* b**7*x**7))/(2520*x**10)