Integrand size = 15, antiderivative size = 186 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {d^7 x}{b^7}-\frac {(b c-a d)^7}{6 b^8 (a+b x)^6}-\frac {7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac {21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}-\frac {35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac {35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac {21 d^5 (b c-a d)^2}{b^8 (a+b x)}+\frac {7 d^6 (b c-a d) \log (a+b x)}{b^8} \] Output:
d^7*x/b^7-1/6*(-a*d+b*c)^7/b^8/(b*x+a)^6-7/5*d*(-a*d+b*c)^6/b^8/(b*x+a)^5- 21/4*d^2*(-a*d+b*c)^5/b^8/(b*x+a)^4-35/3*d^3*(-a*d+b*c)^4/b^8/(b*x+a)^3-35 /2*d^4*(-a*d+b*c)^3/b^8/(b*x+a)^2-21*d^5*(-a*d+b*c)^2/b^8/(b*x+a)+7*d^6*(- a*d+b*c)*ln(b*x+a)/b^8
Leaf count is larger than twice the leaf count of optimal. \(390\) vs. \(2(186)=372\).
Time = 0.12 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.10 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=-\frac {669 a^7 d^7+3 a^6 b d^6 (-343 c+1198 d x)+3 a^5 b^2 d^5 \left (70 c^2-1918 c d x+2575 d^2 x^2\right )+5 a^4 b^3 d^4 \left (14 c^3+252 c^2 d x-2625 c d^2 x^2+1640 d^3 x^3\right )+5 a^3 b^4 d^3 \left (7 c^4+84 c^3 d x+630 c^2 d^2 x^2-3080 c d^3 x^3+810 d^4 x^4\right )+3 a^2 b^5 d^2 \left (7 c^5+70 c^4 d x+350 c^3 d^2 x^2+1400 c^2 d^3 x^3-3150 c d^4 x^4+120 d^5 x^5\right )+a b^6 d \left (14 c^6+126 c^5 d x+525 c^4 d^2 x^2+1400 c^3 d^3 x^3+3150 c^2 d^4 x^4-2520 c d^5 x^5-360 d^6 x^6\right )+b^7 \left (10 c^7+84 c^6 d x+315 c^5 d^2 x^2+700 c^4 d^3 x^3+1050 c^3 d^4 x^4+1260 c^2 d^5 x^5-60 d^7 x^7\right )+420 d^6 (-b c+a d) (a+b x)^6 \log (a+b x)}{60 b^8 (a+b x)^6} \] Input:
Integrate[(c + d*x)^7/(a + b*x)^7,x]
Output:
-1/60*(669*a^7*d^7 + 3*a^6*b*d^6*(-343*c + 1198*d*x) + 3*a^5*b^2*d^5*(70*c ^2 - 1918*c*d*x + 2575*d^2*x^2) + 5*a^4*b^3*d^4*(14*c^3 + 252*c^2*d*x - 26 25*c*d^2*x^2 + 1640*d^3*x^3) + 5*a^3*b^4*d^3*(7*c^4 + 84*c^3*d*x + 630*c^2 *d^2*x^2 - 3080*c*d^3*x^3 + 810*d^4*x^4) + 3*a^2*b^5*d^2*(7*c^5 + 70*c^4*d *x + 350*c^3*d^2*x^2 + 1400*c^2*d^3*x^3 - 3150*c*d^4*x^4 + 120*d^5*x^5) + a*b^6*d*(14*c^6 + 126*c^5*d*x + 525*c^4*d^2*x^2 + 1400*c^3*d^3*x^3 + 3150* c^2*d^4*x^4 - 2520*c*d^5*x^5 - 360*d^6*x^6) + b^7*(10*c^7 + 84*c^6*d*x + 3 15*c^5*d^2*x^2 + 700*c^4*d^3*x^3 + 1050*c^3*d^4*x^4 + 1260*c^2*d^5*x^5 - 6 0*d^7*x^7) + 420*d^6*(-(b*c) + a*d)*(a + b*x)^6*Log[a + b*x])/(b^8*(a + b* x)^6)
Time = 0.37 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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.
| \(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^7} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {7 d^6 (b c-a d)}{b^7 (a+b x)}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^2}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^3}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^4}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^5}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^6}+\frac {(b c-a d)^7}{b^7 (a+b x)^7}+\frac {d^7}{b^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7 d^6 (b c-a d) \log (a+b x)}{b^8}-\frac {21 d^5 (b c-a d)^2}{b^8 (a+b x)}-\frac {35 d^4 (b c-a d)^3}{2 b^8 (a+b x)^2}-\frac {35 d^3 (b c-a d)^4}{3 b^8 (a+b x)^3}-\frac {21 d^2 (b c-a d)^5}{4 b^8 (a+b x)^4}-\frac {7 d (b c-a d)^6}{5 b^8 (a+b x)^5}-\frac {(b c-a d)^7}{6 b^8 (a+b x)^6}+\frac {d^7 x}{b^7}\) |
Input:
Int[(c + d*x)^7/(a + b*x)^7,x]
Output:
(d^7*x)/b^7 - (b*c - a*d)^7/(6*b^8*(a + b*x)^6) - (7*d*(b*c - a*d)^6)/(5*b ^8*(a + b*x)^5) - (21*d^2*(b*c - a*d)^5)/(4*b^8*(a + b*x)^4) - (35*d^3*(b* c - a*d)^4)/(3*b^8*(a + b*x)^3) - (35*d^4*(b*c - a*d)^3)/(2*b^8*(a + b*x)^ 2) - (21*d^5*(b*c - a*d)^2)/(b^8*(a + b*x)) + (7*d^6*(b*c - a*d)*Log[a + b *x])/b^8
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]
Leaf count of result is larger than twice the leaf count of optimal. \(451\) vs. \(2(176)=352\).
Time = 0.13 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.43
| method | result | size |
| risch | \(\frac {d^{7} x}{b^{7}}+\frac {\left (-21 a^{2} b^{4} d^{7}+42 a \,b^{5} c \,d^{6}-21 b^{6} c^{2} d^{5}\right ) x^{5}-\frac {35 b^{3} d^{4} \left (5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{4}}{2}-\frac {35 b^{2} d^{3} \left (13 a^{4} d^{4}-22 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) x^{3}}{3}-\frac {7 b \,d^{2} \left (77 a^{5} d^{5}-125 a^{4} b c \,d^{4}+30 a^{3} b^{2} c^{2} d^{3}+10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +3 c^{5} b^{5}\right ) x^{2}}{4}-\frac {7 d \left (87 a^{6} d^{6}-137 a^{5} b c \,d^{5}+30 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+5 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +2 c^{6} b^{6}\right ) x}{10}-\frac {669 a^{7} d^{7}-1029 a^{6} b c \,d^{6}+210 a^{5} b^{2} c^{2} d^{5}+70 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+14 a \,b^{6} c^{6} d +10 b^{7} c^{7}}{60 b}}{b^{7} \left (b x +a \right )^{6}}-\frac {7 d^{7} \ln \left (b x +a \right ) a}{b^{8}}+\frac {7 d^{6} \ln \left (b x +a \right ) c}{b^{7}}\) | \(452\) |
| default | \(\frac {d^{7} x}{b^{7}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{5 b^{8} \left (b x +a \right )^{5}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{4 b^{8} \left (b x +a \right )^{4}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b^{8} \left (b x +a \right )^{2}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{8} \left (b x +a \right )}-\frac {7 d^{6} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{8}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{3 b^{8} \left (b x +a \right )^{3}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{6 b^{8} \left (b x +a \right )^{6}}\) | \(456\) |
| norman | \(\frac {\frac {d^{7} x^{7}}{b}-\frac {1029 a^{7} d^{7}-1029 a^{6} b c \,d^{6}+210 a^{5} b^{2} c^{2} d^{5}+70 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+14 a \,b^{6} c^{6} d +10 b^{7} c^{7}}{60 b^{8}}-\frac {3 \left (14 a^{2} d^{7}-14 a b c \,d^{6}+7 b^{2} c^{2} d^{5}\right ) x^{5}}{b^{3}}-\frac {5 \left (63 a^{3} d^{7}-63 a^{2} b c \,d^{6}+21 a \,b^{2} c^{2} d^{5}+7 b^{3} c^{3} d^{4}\right ) x^{4}}{2 b^{4}}-\frac {5 \left (154 a^{4} d^{7}-154 a^{3} b c \,d^{6}+42 a^{2} b^{2} c^{2} d^{5}+14 a \,b^{3} c^{3} d^{4}+7 b^{4} c^{4} d^{3}\right ) x^{3}}{3 b^{5}}-\frac {\left (875 a^{5} d^{7}-875 a^{4} b c \,d^{6}+210 a^{3} b^{2} c^{2} d^{5}+70 a^{2} b^{3} c^{3} d^{4}+35 a \,b^{4} c^{4} d^{3}+21 b^{5} c^{5} d^{2}\right ) x^{2}}{4 b^{6}}-\frac {\left (959 a^{6} d^{7}-959 a^{5} b c \,d^{6}+210 a^{4} b^{2} c^{2} d^{5}+70 a^{3} b^{3} c^{3} d^{4}+35 a^{2} b^{4} c^{4} d^{3}+21 a \,b^{5} c^{5} d^{2}+14 b^{6} c^{6} d \right ) x}{10 b^{7}}}{\left (b x +a \right )^{6}}-\frac {7 d^{6} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{8}}\) | \(457\) |
| parallelrisch | \(-\frac {-1029 a^{6} b c \,d^{6}+210 a^{5} b^{2} c^{2} d^{5}+70 a^{4} b^{3} c^{3} d^{4}-9450 x^{4} a^{2} b^{5} c \,d^{6}+3150 x^{4} a \,b^{6} c^{2} d^{5}-2520 x^{5} a \,b^{6} c \,d^{6}-420 \ln \left (b x +a \right ) a^{6} b c \,d^{6}-60 x^{7} d^{7} b^{7}+5754 x \,a^{6} b \,d^{7}+84 x \,b^{7} c^{6} d +13125 x^{2} a^{5} b^{2} d^{7}+315 x^{2} b^{7} c^{5} d^{2}+15400 x^{3} a^{4} b^{3} d^{7}+700 x^{3} b^{7} c^{4} d^{3}+9450 x^{4} a^{3} b^{4} d^{7}+1050 x^{4} b^{7} c^{3} d^{4}+2520 x^{5} a^{2} b^{5} d^{7}+1260 x^{5} b^{7} c^{2} d^{5}+10 b^{7} c^{7}+1029 a^{7} d^{7}+8400 \ln \left (b x +a \right ) x^{3} a^{4} b^{3} d^{7}+6300 \ln \left (b x +a \right ) x^{2} a^{5} b^{2} d^{7}+2520 \ln \left (b x +a \right ) x^{5} a^{2} b^{5} d^{7}+35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+14 a \,b^{6} c^{6} d -2520 \ln \left (b x +a \right ) x^{5} a \,b^{6} c \,d^{6}-5754 x \,a^{5} b^{2} c \,d^{6}+1260 x \,a^{4} b^{3} c^{2} d^{5}+420 x \,a^{3} b^{4} c^{3} d^{4}+210 x \,a^{2} b^{5} c^{4} d^{3}+126 x a \,b^{6} c^{5} d^{2}-13125 x^{2} a^{4} b^{3} c \,d^{6}+3150 x^{2} a^{3} b^{4} c^{2} d^{5}+1050 x^{2} a^{2} b^{5} c^{3} d^{4}+525 x^{2} a \,b^{6} c^{4} d^{3}-15400 x^{3} a^{3} b^{4} c \,d^{6}+4200 x^{3} a^{2} b^{5} c^{2} d^{5}+1400 x^{3} a \,b^{6} c^{3} d^{4}-6300 \ln \left (b x +a \right ) x^{2} a^{4} b^{3} c \,d^{6}+2520 \ln \left (b x +a \right ) x \,a^{6} b \,d^{7}+420 \ln \left (b x +a \right ) x^{6} a \,b^{6} d^{7}-420 \ln \left (b x +a \right ) x^{6} b^{7} c \,d^{6}-2520 \ln \left (b x +a \right ) x \,a^{5} b^{2} c \,d^{6}+420 \ln \left (b x +a \right ) a^{7} d^{7}+6300 \ln \left (b x +a \right ) x^{4} a^{3} b^{4} d^{7}-6300 \ln \left (b x +a \right ) x^{4} a^{2} b^{5} c \,d^{6}-8400 \ln \left (b x +a \right ) x^{3} a^{3} b^{4} c \,d^{6}}{60 b^{8} \left (b x +a \right )^{6}}\) | \(737\) |
Input:
int((d*x+c)^7/(b*x+a)^7,x,method=_RETURNVERBOSE)
Output:
d^7*x/b^7+((-21*a^2*b^4*d^7+42*a*b^5*c*d^6-21*b^6*c^2*d^5)*x^5-35/2*b^3*d^ 4*(5*a^3*d^3-9*a^2*b*c*d^2+3*a*b^2*c^2*d+b^3*c^3)*x^4-35/3*b^2*d^3*(13*a^4 *d^4-22*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2+2*a*b^3*c^3*d+b^4*c^4)*x^3-7/4*b*d^2 *(77*a^5*d^5-125*a^4*b*c*d^4+30*a^3*b^2*c^2*d^3+10*a^2*b^3*c^3*d^2+5*a*b^4 *c^4*d+3*b^5*c^5)*x^2-7/10*d*(87*a^6*d^6-137*a^5*b*c*d^5+30*a^4*b^2*c^2*d^ 4+10*a^3*b^3*c^3*d^3+5*a^2*b^4*c^4*d^2+3*a*b^5*c^5*d+2*b^6*c^6)*x-1/60/b*( 669*a^7*d^7-1029*a^6*b*c*d^6+210*a^5*b^2*c^2*d^5+70*a^4*b^3*c^3*d^4+35*a^3 *b^4*c^4*d^3+21*a^2*b^5*c^5*d^2+14*a*b^6*c^6*d+10*b^7*c^7))/b^7/(b*x+a)^6- 7/b^8*d^7*ln(b*x+a)*a+7/b^7*d^6*ln(b*x+a)*c
Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (176) = 352\).
Time = 0.07 (sec) , antiderivative size = 692, normalized size of antiderivative = 3.72 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {60 \, b^{7} d^{7} x^{7} + 360 \, a b^{6} d^{7} x^{6} - 10 \, b^{7} c^{7} - 14 \, a b^{6} c^{6} d - 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} - 70 \, a^{4} b^{3} c^{3} d^{4} - 210 \, a^{5} b^{2} c^{2} d^{5} + 1029 \, a^{6} b c d^{6} - 669 \, a^{7} d^{7} - 180 \, {\left (7 \, b^{7} c^{2} d^{5} - 14 \, a b^{6} c d^{6} + 2 \, a^{2} b^{5} d^{7}\right )} x^{5} - 150 \, {\left (7 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} - 63 \, a^{2} b^{5} c d^{6} + 27 \, a^{3} b^{4} d^{7}\right )} x^{4} - 100 \, {\left (7 \, b^{7} c^{4} d^{3} + 14 \, a b^{6} c^{3} d^{4} + 42 \, a^{2} b^{5} c^{2} d^{5} - 154 \, a^{3} b^{4} c d^{6} + 82 \, a^{4} b^{3} d^{7}\right )} x^{3} - 15 \, {\left (21 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 70 \, a^{2} b^{5} c^{3} d^{4} + 210 \, a^{3} b^{4} c^{2} d^{5} - 875 \, a^{4} b^{3} c d^{6} + 515 \, a^{5} b^{2} d^{7}\right )} x^{2} - 6 \, {\left (14 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 70 \, a^{3} b^{4} c^{3} d^{4} + 210 \, a^{4} b^{3} c^{2} d^{5} - 959 \, a^{5} b^{2} c d^{6} + 599 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{6} b c d^{6} - a^{7} d^{7} + {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 6 \, {\left (a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 15 \, {\left (a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 20 \, {\left (a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{3} + 15 \, {\left (a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 6 \, {\left (a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} \] Input:
integrate((d*x+c)^7/(b*x+a)^7,x, algorithm="fricas")
Output:
1/60*(60*b^7*d^7*x^7 + 360*a*b^6*d^7*x^6 - 10*b^7*c^7 - 14*a*b^6*c^6*d - 2 1*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 - 70*a^4*b^3*c^3*d^4 - 210*a^5*b^2* c^2*d^5 + 1029*a^6*b*c*d^6 - 669*a^7*d^7 - 180*(7*b^7*c^2*d^5 - 14*a*b^6*c *d^6 + 2*a^2*b^5*d^7)*x^5 - 150*(7*b^7*c^3*d^4 + 21*a*b^6*c^2*d^5 - 63*a^2 *b^5*c*d^6 + 27*a^3*b^4*d^7)*x^4 - 100*(7*b^7*c^4*d^3 + 14*a*b^6*c^3*d^4 + 42*a^2*b^5*c^2*d^5 - 154*a^3*b^4*c*d^6 + 82*a^4*b^3*d^7)*x^3 - 15*(21*b^7 *c^5*d^2 + 35*a*b^6*c^4*d^3 + 70*a^2*b^5*c^3*d^4 + 210*a^3*b^4*c^2*d^5 - 8 75*a^4*b^3*c*d^6 + 515*a^5*b^2*d^7)*x^2 - 6*(14*b^7*c^6*d + 21*a*b^6*c^5*d ^2 + 35*a^2*b^5*c^4*d^3 + 70*a^3*b^4*c^3*d^4 + 210*a^4*b^3*c^2*d^5 - 959*a ^5*b^2*c*d^6 + 599*a^6*b*d^7)*x + 420*(a^6*b*c*d^6 - a^7*d^7 + (b^7*c*d^6 - a*b^6*d^7)*x^6 + 6*(a*b^6*c*d^6 - a^2*b^5*d^7)*x^5 + 15*(a^2*b^5*c*d^6 - a^3*b^4*d^7)*x^4 + 20*(a^3*b^4*c*d^6 - a^4*b^3*d^7)*x^3 + 15*(a^4*b^3*c*d ^6 - a^5*b^2*d^7)*x^2 + 6*(a^5*b^2*c*d^6 - a^6*b*d^7)*x)*log(b*x + a))/(b^ 14*x^6 + 6*a*b^13*x^5 + 15*a^2*b^12*x^4 + 20*a^3*b^11*x^3 + 15*a^4*b^10*x^ 2 + 6*a^5*b^9*x + a^6*b^8)
Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**7/(b*x+a)**7,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (176) = 352\).
Time = 0.06 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.77 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {d^{7} x}{b^{7}} - \frac {10 \, b^{7} c^{7} + 14 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 70 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} - 1029 \, a^{6} b c d^{6} + 669 \, a^{7} d^{7} + 1260 \, {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1050 \, {\left (b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} - 9 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{4} + 700 \, {\left (b^{7} c^{4} d^{3} + 2 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 22 \, a^{3} b^{4} c d^{6} + 13 \, a^{4} b^{3} d^{7}\right )} x^{3} + 105 \, {\left (3 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} - 125 \, a^{4} b^{3} c d^{6} + 77 \, a^{5} b^{2} d^{7}\right )} x^{2} + 42 \, {\left (2 \, b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} - 137 \, a^{5} b^{2} c d^{6} + 87 \, a^{6} b d^{7}\right )} x}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} + \frac {7 \, {\left (b c d^{6} - a d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \] Input:
integrate((d*x+c)^7/(b*x+a)^7,x, algorithm="maxima")
Output:
d^7*x/b^7 - 1/60*(10*b^7*c^7 + 14*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 35*a^ 3*b^4*c^4*d^3 + 70*a^4*b^3*c^3*d^4 + 210*a^5*b^2*c^2*d^5 - 1029*a^6*b*c*d^ 6 + 669*a^7*d^7 + 1260*(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1 050*(b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 - 9*a^2*b^5*c*d^6 + 5*a^3*b^4*d^7)*x^4 + 700*(b^7*c^4*d^3 + 2*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 22*a^3*b^4*c*d^ 6 + 13*a^4*b^3*d^7)*x^3 + 105*(3*b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 + 10*a^2*b^ 5*c^3*d^4 + 30*a^3*b^4*c^2*d^5 - 125*a^4*b^3*c*d^6 + 77*a^5*b^2*d^7)*x^2 + 42*(2*b^7*c^6*d + 3*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^ 4 + 30*a^4*b^3*c^2*d^5 - 137*a^5*b^2*c*d^6 + 87*a^6*b*d^7)*x)/(b^14*x^6 + 6*a*b^13*x^5 + 15*a^2*b^12*x^4 + 20*a^3*b^11*x^3 + 15*a^4*b^10*x^2 + 6*a^5 *b^9*x + a^6*b^8) + 7*(b*c*d^6 - a*d^7)*log(b*x + a)/b^8
Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (176) = 352\).
Time = 0.13 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.47 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {d^{7} x}{b^{7}} + \frac {7 \, {\left (b c d^{6} - a d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {10 \, b^{7} c^{7} + 14 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 70 \, a^{4} b^{3} c^{3} d^{4} + 210 \, a^{5} b^{2} c^{2} d^{5} - 1029 \, a^{6} b c d^{6} + 669 \, a^{7} d^{7} + 1260 \, {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 1050 \, {\left (b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} - 9 \, a^{2} b^{5} c d^{6} + 5 \, a^{3} b^{4} d^{7}\right )} x^{4} + 700 \, {\left (b^{7} c^{4} d^{3} + 2 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 22 \, a^{3} b^{4} c d^{6} + 13 \, a^{4} b^{3} d^{7}\right )} x^{3} + 105 \, {\left (3 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 30 \, a^{3} b^{4} c^{2} d^{5} - 125 \, a^{4} b^{3} c d^{6} + 77 \, a^{5} b^{2} d^{7}\right )} x^{2} + 42 \, {\left (2 \, b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 30 \, a^{4} b^{3} c^{2} d^{5} - 137 \, a^{5} b^{2} c d^{6} + 87 \, a^{6} b d^{7}\right )} x}{60 \, {\left (b x + a\right )}^{6} b^{8}} \] Input:
integrate((d*x+c)^7/(b*x+a)^7,x, algorithm="giac")
Output:
d^7*x/b^7 + 7*(b*c*d^6 - a*d^7)*log(abs(b*x + a))/b^8 - 1/60*(10*b^7*c^7 + 14*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 70*a^4*b^3*c^3 *d^4 + 210*a^5*b^2*c^2*d^5 - 1029*a^6*b*c*d^6 + 669*a^7*d^7 + 1260*(b^7*c^ 2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 1050*(b^7*c^3*d^4 + 3*a*b^6*c^2 *d^5 - 9*a^2*b^5*c*d^6 + 5*a^3*b^4*d^7)*x^4 + 700*(b^7*c^4*d^3 + 2*a*b^6*c ^3*d^4 + 6*a^2*b^5*c^2*d^5 - 22*a^3*b^4*c*d^6 + 13*a^4*b^3*d^7)*x^3 + 105* (3*b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 + 30*a^3*b^4*c^2*d^5 - 125*a^4*b^3*c*d^6 + 77*a^5*b^2*d^7)*x^2 + 42*(2*b^7*c^6*d + 3*a*b^6*c^5 *d^2 + 5*a^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4 + 30*a^4*b^3*c^2*d^5 - 137*a ^5*b^2*c*d^6 + 87*a^6*b*d^7)*x)/((b*x + a)^6*b^8)
Time = 0.22 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.78 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {d^7\,x}{b^7}-\frac {\ln \left (a+b\,x\right )\,\left (7\,a\,d^7-7\,b\,c\,d^6\right )}{b^8}-\frac {\frac {669\,a^7\,d^7-1029\,a^6\,b\,c\,d^6+210\,a^5\,b^2\,c^2\,d^5+70\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3+21\,a^2\,b^5\,c^5\,d^2+14\,a\,b^6\,c^6\,d+10\,b^7\,c^7}{60\,b}+x\,\left (\frac {609\,a^6\,d^7}{10}-\frac {959\,a^5\,b\,c\,d^6}{10}+21\,a^4\,b^2\,c^2\,d^5+7\,a^3\,b^3\,c^3\,d^4+\frac {7\,a^2\,b^4\,c^4\,d^3}{2}+\frac {21\,a\,b^5\,c^5\,d^2}{10}+\frac {7\,b^6\,c^6\,d}{5}\right )+x^3\,\left (\frac {455\,a^4\,b^2\,d^7}{3}-\frac {770\,a^3\,b^3\,c\,d^6}{3}+70\,a^2\,b^4\,c^2\,d^5+\frac {70\,a\,b^5\,c^3\,d^4}{3}+\frac {35\,b^6\,c^4\,d^3}{3}\right )+x^2\,\left (\frac {539\,a^5\,b\,d^7}{4}-\frac {875\,a^4\,b^2\,c\,d^6}{4}+\frac {105\,a^3\,b^3\,c^2\,d^5}{2}+\frac {35\,a^2\,b^4\,c^3\,d^4}{2}+\frac {35\,a\,b^5\,c^4\,d^3}{4}+\frac {21\,b^6\,c^5\,d^2}{4}\right )+x^5\,\left (21\,a^2\,b^4\,d^7-42\,a\,b^5\,c\,d^6+21\,b^6\,c^2\,d^5\right )+x^4\,\left (\frac {175\,a^3\,b^3\,d^7}{2}-\frac {315\,a^2\,b^4\,c\,d^6}{2}+\frac {105\,a\,b^5\,c^2\,d^5}{2}+\frac {35\,b^6\,c^3\,d^4}{2}\right )}{a^6\,b^7+6\,a^5\,b^8\,x+15\,a^4\,b^9\,x^2+20\,a^3\,b^{10}\,x^3+15\,a^2\,b^{11}\,x^4+6\,a\,b^{12}\,x^5+b^{13}\,x^6} \] Input:
int((c + d*x)^7/(a + b*x)^7,x)
Output:
(d^7*x)/b^7 - (log(a + b*x)*(7*a*d^7 - 7*b*c*d^6))/b^8 - ((669*a^7*d^7 + 1 0*b^7*c^7 + 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 70*a^4*b^3*c^3*d^4 + 210*a^5*b^2*c^2*d^5 + 14*a*b^6*c^6*d - 1029*a^6*b*c*d^6)/(60*b) + x*((609 *a^6*d^7)/10 + (7*b^6*c^6*d)/5 + (21*a*b^5*c^5*d^2)/10 + (7*a^2*b^4*c^4*d^ 3)/2 + 7*a^3*b^3*c^3*d^4 + 21*a^4*b^2*c^2*d^5 - (959*a^5*b*c*d^6)/10) + x^ 3*((455*a^4*b^2*d^7)/3 + (35*b^6*c^4*d^3)/3 + (70*a*b^5*c^3*d^4)/3 - (770* a^3*b^3*c*d^6)/3 + 70*a^2*b^4*c^2*d^5) + x^2*((539*a^5*b*d^7)/4 + (21*b^6* c^5*d^2)/4 + (35*a*b^5*c^4*d^3)/4 - (875*a^4*b^2*c*d^6)/4 + (35*a^2*b^4*c^ 3*d^4)/2 + (105*a^3*b^3*c^2*d^5)/2) + x^5*(21*a^2*b^4*d^7 + 21*b^6*c^2*d^5 - 42*a*b^5*c*d^6) + x^4*((175*a^3*b^3*d^7)/2 + (35*b^6*c^3*d^4)/2 + (105* a*b^5*c^2*d^5)/2 - (315*a^2*b^4*c*d^6)/2))/(a^6*b^7 + b^13*x^6 + 6*a^5*b^8 *x + 6*a*b^12*x^5 + 15*a^4*b^9*x^2 + 20*a^3*b^10*x^3 + 15*a^2*b^11*x^4)
Time = 0.17 (sec) , antiderivative size = 735, normalized size of antiderivative = 3.95 \[ \int \frac {(c+d x)^7}{(a+b x)^7} \, dx=\frac {-8400 \,\mathrm {log}\left (b x +a \right ) a^{5} b^{3} d^{7} x^{3}+8400 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{4} c \,d^{6} x^{3}-609 a^{8} d^{7}+2520 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{6} c \,d^{6} x^{5}+210 b^{8} c^{2} d^{5} x^{6}+6300 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{5} c \,d^{6} x^{4}+2520 \,\mathrm {log}\left (b x +a \right ) a^{6} b^{2} c \,d^{6} x -420 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{6} d^{7} x^{6}-3234 a^{7} b \,d^{7} x -6825 a^{6} b^{2} d^{7} x^{2}-7000 a^{5} b^{3} d^{7} x^{3}-3150 a^{4} b^{4} d^{7} x^{4}+420 a^{2} b^{6} d^{7} x^{6}+60 a \,b^{7} d^{7} x^{7}+609 a^{7} b c \,d^{6}-70 a^{5} b^{3} c^{3} d^{4}-35 a^{4} b^{4} c^{4} d^{3}-21 a^{3} b^{5} c^{5} d^{2}-14 a^{2} b^{6} c^{6} d -2520 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{5} d^{7} x^{5}-6300 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{4} d^{7} x^{4}+420 \,\mathrm {log}\left (b x +a \right ) a \,b^{7} c \,d^{6} x^{6}-420 \,\mathrm {log}\left (b x +a \right ) a^{8} d^{7}-10 a \,b^{7} c^{7}+6300 \,\mathrm {log}\left (b x +a \right ) a^{5} b^{3} c \,d^{6} x^{2}+420 \,\mathrm {log}\left (b x +a \right ) a^{7} b c \,d^{6}-2520 \,\mathrm {log}\left (b x +a \right ) a^{7} b \,d^{7} x +3234 a^{6} b^{2} c \,d^{6} x +6825 a^{5} b^{3} c \,d^{6} x^{2}-420 a^{4} b^{4} c^{3} d^{4} x +7000 a^{4} b^{4} c \,d^{6} x^{3}-210 a^{3} b^{5} c^{4} d^{3} x -1050 a^{3} b^{5} c^{3} d^{4} x^{2}+3150 a^{3} b^{5} c \,d^{6} x^{4}-126 a^{2} b^{6} c^{5} d^{2} x -525 a^{2} b^{6} c^{4} d^{3} x^{2}-1400 a^{2} b^{6} c^{3} d^{4} x^{3}-84 a \,b^{7} c^{6} d x -315 a \,b^{7} c^{5} d^{2} x^{2}-700 a \,b^{7} c^{4} d^{3} x^{3}-1050 a \,b^{7} c^{3} d^{4} x^{4}-420 a \,b^{7} c \,d^{6} x^{6}-6300 \,\mathrm {log}\left (b x +a \right ) a^{6} b^{2} d^{7} x^{2}}{60 a \,b^{8} \left (b^{6} x^{6}+6 a \,b^{5} x^{5}+15 a^{2} b^{4} x^{4}+20 a^{3} b^{3} x^{3}+15 a^{4} b^{2} x^{2}+6 a^{5} b x +a^{6}\right )} \] Input:
int((d*x+c)^7/(b*x+a)^7,x)
Output:
( - 420*log(a + b*x)*a**8*d**7 + 420*log(a + b*x)*a**7*b*c*d**6 - 2520*log (a + b*x)*a**7*b*d**7*x + 2520*log(a + b*x)*a**6*b**2*c*d**6*x - 6300*log( a + b*x)*a**6*b**2*d**7*x**2 + 6300*log(a + b*x)*a**5*b**3*c*d**6*x**2 - 8 400*log(a + b*x)*a**5*b**3*d**7*x**3 + 8400*log(a + b*x)*a**4*b**4*c*d**6* x**3 - 6300*log(a + b*x)*a**4*b**4*d**7*x**4 + 6300*log(a + b*x)*a**3*b**5 *c*d**6*x**4 - 2520*log(a + b*x)*a**3*b**5*d**7*x**5 + 2520*log(a + b*x)*a **2*b**6*c*d**6*x**5 - 420*log(a + b*x)*a**2*b**6*d**7*x**6 + 420*log(a + b*x)*a*b**7*c*d**6*x**6 - 609*a**8*d**7 + 609*a**7*b*c*d**6 - 3234*a**7*b* d**7*x + 3234*a**6*b**2*c*d**6*x - 6825*a**6*b**2*d**7*x**2 - 70*a**5*b**3 *c**3*d**4 + 6825*a**5*b**3*c*d**6*x**2 - 7000*a**5*b**3*d**7*x**3 - 35*a* *4*b**4*c**4*d**3 - 420*a**4*b**4*c**3*d**4*x + 7000*a**4*b**4*c*d**6*x**3 - 3150*a**4*b**4*d**7*x**4 - 21*a**3*b**5*c**5*d**2 - 210*a**3*b**5*c**4* d**3*x - 1050*a**3*b**5*c**3*d**4*x**2 + 3150*a**3*b**5*c*d**6*x**4 - 14*a **2*b**6*c**6*d - 126*a**2*b**6*c**5*d**2*x - 525*a**2*b**6*c**4*d**3*x**2 - 1400*a**2*b**6*c**3*d**4*x**3 + 420*a**2*b**6*d**7*x**6 - 10*a*b**7*c** 7 - 84*a*b**7*c**6*d*x - 315*a*b**7*c**5*d**2*x**2 - 700*a*b**7*c**4*d**3* x**3 - 1050*a*b**7*c**3*d**4*x**4 - 420*a*b**7*c*d**6*x**6 + 60*a*b**7*d** 7*x**7 + 210*b**8*c**2*d**5*x**6)/(60*a*b**8*(a**6 + 6*a**5*b*x + 15*a**4* b**2*x**2 + 20*a**3*b**3*x**3 + 15*a**2*b**4*x**4 + 6*a*b**5*x**5 + b**6*x **6))