Integrand size = 15, antiderivative size = 181 \[ \int \frac {(c+d x)^7}{(a+b x)^6} \, dx=\frac {d^6 (7 b c-6 a d) x}{b^7}+\frac {d^7 x^2}{2 b^6}-\frac {(b c-a d)^7}{5 b^8 (a+b x)^5}-\frac {7 d (b c-a d)^6}{4 b^8 (a+b x)^4}-\frac {7 d^2 (b c-a d)^5}{b^8 (a+b x)^3}-\frac {35 d^3 (b c-a d)^4}{2 b^8 (a+b x)^2}-\frac {35 d^4 (b c-a d)^3}{b^8 (a+b x)}+\frac {21 d^5 (b c-a d)^2 \log (a+b x)}{b^8} \] Output:
d^6*(-6*a*d+7*b*c)*x/b^7+1/2*d^7*x^2/b^6-1/5*(-a*d+b*c)^7/b^8/(b*x+a)^5-7/ 4*d*(-a*d+b*c)^6/b^8/(b*x+a)^4-7*d^2*(-a*d+b*c)^5/b^8/(b*x+a)^3-35/2*d^3*( -a*d+b*c)^4/b^8/(b*x+a)^2-35*d^4*(-a*d+b*c)^3/b^8/(b*x+a)+21*d^5*(-a*d+b*c )^2*ln(b*x+a)/b^8
Leaf count is larger than twice the leaf count of optimal. \(389\) vs. \(2(181)=362\).
Time = 0.09 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.15 \[ \int \frac {(c+d x)^7}{(a+b x)^6} \, dx=\frac {459 a^7 d^7+3 a^6 b d^6 (-406 c+625 d x)+a^5 b^2 d^5 \left (959 c^2-5250 c d x+2700 d^2 x^2\right )+5 a^4 b^3 d^4 \left (-28 c^3+875 c^2 d x-1680 c d^2 x^2+260 d^3 x^3\right )-5 a^3 b^4 d^3 \left (7 c^4+140 c^3 d x-1540 c^2 d^2 x^2+1120 c d^3 x^3+80 d^4 x^4\right )-a^2 b^5 d^2 \left (14 c^5+175 c^4 d x+1400 c^3 d^2 x^2-6300 c^2 d^3 x^3+700 c d^4 x^4+500 d^5 x^5\right )-7 a b^6 d \left (c^6+10 c^5 d x+50 c^4 d^2 x^2+200 c^3 d^3 x^3-300 c^2 d^4 x^4-100 c d^5 x^5+10 d^6 x^6\right )-b^7 \left (4 c^7+35 c^6 d x+140 c^5 d^2 x^2+350 c^4 d^3 x^3+700 c^3 d^4 x^4-140 c d^6 x^6-10 d^7 x^7\right )+420 d^5 (b c-a d)^2 (a+b x)^5 \log (a+b x)}{20 b^8 (a+b x)^5} \] Input:
Integrate[(c + d*x)^7/(a + b*x)^6,x]
Output:
(459*a^7*d^7 + 3*a^6*b*d^6*(-406*c + 625*d*x) + a^5*b^2*d^5*(959*c^2 - 525 0*c*d*x + 2700*d^2*x^2) + 5*a^4*b^3*d^4*(-28*c^3 + 875*c^2*d*x - 1680*c*d^ 2*x^2 + 260*d^3*x^3) - 5*a^3*b^4*d^3*(7*c^4 + 140*c^3*d*x - 1540*c^2*d^2*x ^2 + 1120*c*d^3*x^3 + 80*d^4*x^4) - a^2*b^5*d^2*(14*c^5 + 175*c^4*d*x + 14 00*c^3*d^2*x^2 - 6300*c^2*d^3*x^3 + 700*c*d^4*x^4 + 500*d^5*x^5) - 7*a*b^6 *d*(c^6 + 10*c^5*d*x + 50*c^4*d^2*x^2 + 200*c^3*d^3*x^3 - 300*c^2*d^4*x^4 - 100*c*d^5*x^5 + 10*d^6*x^6) - b^7*(4*c^7 + 35*c^6*d*x + 140*c^5*d^2*x^2 + 350*c^4*d^3*x^3 + 700*c^3*d^4*x^4 - 140*c*d^6*x^6 - 10*d^7*x^7) + 420*d^ 5*(b*c - a*d)^2*(a + b*x)^5*Log[a + b*x])/(20*b^8*(a + b*x)^5)
Time = 0.36 (sec) , antiderivative size = 181, 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.
| \(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^6} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {d^6 (7 b c-6 a d)}{b^7}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^2}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^3}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^4}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^5}+\frac {(b c-a d)^7}{b^7 (a+b x)^6}+\frac {d^7 x}{b^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {21 d^5 (b c-a d)^2 \log (a+b x)}{b^8}-\frac {35 d^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {35 d^3 (b c-a d)^4}{2 b^8 (a+b x)^2}-\frac {7 d^2 (b c-a d)^5}{b^8 (a+b x)^3}-\frac {7 d (b c-a d)^6}{4 b^8 (a+b x)^4}-\frac {(b c-a d)^7}{5 b^8 (a+b x)^5}+\frac {d^6 x (7 b c-6 a d)}{b^7}+\frac {d^7 x^2}{2 b^6}\) |
Input:
Int[(c + d*x)^7/(a + b*x)^6,x]
Output:
(d^6*(7*b*c - 6*a*d)*x)/b^7 + (d^7*x^2)/(2*b^6) - (b*c - a*d)^7/(5*b^8*(a + b*x)^5) - (7*d*(b*c - a*d)^6)/(4*b^8*(a + b*x)^4) - (7*d^2*(b*c - a*d)^5 )/(b^8*(a + b*x)^3) - (35*d^3*(b*c - a*d)^4)/(2*b^8*(a + b*x)^2) - (35*d^4 *(b*c - a*d)^3)/(b^8*(a + b*x)) + (21*d^5*(b*c - a*d)^2*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. \(450\) vs. \(2(173)=346\).
Time = 0.12 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.49
| method | result | size |
| default | \(-\frac {d^{6} \left (-\frac {1}{2} b d \,x^{2}+6 a d x -7 b c x \right )}{b^{7}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{5 b^{8} \left (b x +a \right )^{5}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{4 b^{8} \left (b x +a \right )^{4}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{2 b^{8} \left (b x +a \right )^{2}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{8} \left (b x +a \right )}+\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{8}}+\frac {7 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{b^{8} \left (b x +a \right )^{3}}\) | \(451\) |
| norman | \(\frac {\frac {959 a^{7} d^{7}-1918 a^{6} b c \,d^{6}+959 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}-14 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -4 b^{7} c^{7}}{20 b^{8}}+\frac {d^{7} x^{7}}{2 b}+\frac {5 \left (21 a^{3} d^{7}-42 a^{2} b c \,d^{6}+21 a \,b^{2} c^{2} d^{5}-7 b^{3} c^{3} d^{4}\right ) x^{4}}{b^{4}}+\frac {5 \left (126 a^{4} d^{7}-252 a^{3} b c \,d^{6}+126 a^{2} b^{2} c^{2} d^{5}-28 a \,b^{3} c^{3} d^{4}-7 b^{4} c^{4} d^{3}\right ) x^{3}}{2 b^{5}}+\frac {\left (770 a^{5} d^{7}-1540 a^{4} b c \,d^{6}+770 a^{3} b^{2} c^{2} d^{5}-140 a^{2} b^{3} c^{3} d^{4}-35 a \,b^{4} c^{4} d^{3}-14 b^{5} c^{5} d^{2}\right ) x^{2}}{2 b^{6}}+\frac {\left (875 a^{6} d^{7}-1750 a^{5} b c \,d^{6}+875 a^{4} b^{2} c^{2} d^{5}-140 a^{3} b^{3} c^{3} d^{4}-35 a^{2} b^{4} c^{4} d^{3}-14 a \,b^{5} c^{5} d^{2}-7 b^{6} c^{6} d \right ) x}{4 b^{7}}-\frac {7 d^{6} \left (a d -2 b c \right ) x^{6}}{2 b^{2}}}{\left (b x +a \right )^{5}}+\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{8}}\) | \(454\) |
| risch | \(\frac {d^{7} x^{2}}{2 b^{6}}-\frac {6 d^{7} a x}{b^{7}}+\frac {7 d^{6} c x}{b^{6}}+\frac {\left (35 a^{3} b^{3} d^{7}-105 a^{2} b^{4} c \,d^{6}+105 a \,b^{5} c^{2} d^{5}-35 b^{6} c^{3} d^{4}\right ) x^{4}+\frac {35 b^{2} d^{3} \left (7 a^{4} d^{4}-20 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -c^{4} b^{4}\right ) x^{3}}{2}+\frac {7 b \,d^{2} \left (47 a^{5} d^{5}-130 a^{4} b c \,d^{4}+110 a^{3} b^{2} c^{2} d^{3}-20 a^{2} b^{3} c^{3} d^{2}-5 a \,b^{4} c^{4} d -2 c^{5} b^{5}\right ) x^{2}}{2}+\frac {7 d \left (57 a^{6} d^{6}-154 a^{5} b c \,d^{5}+125 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}-5 a^{2} b^{4} c^{4} d^{2}-2 a \,b^{5} c^{5} d -c^{6} b^{6}\right ) x}{4}+\frac {459 a^{7} d^{7}-1218 a^{6} b c \,d^{6}+959 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}-14 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -4 b^{7} c^{7}}{20 b}}{b^{7} \left (b x +a \right )^{5}}+\frac {21 d^{7} \ln \left (b x +a \right ) a^{2}}{b^{8}}-\frac {42 d^{6} \ln \left (b x +a \right ) a c}{b^{7}}+\frac {21 d^{5} \ln \left (b x +a \right ) c^{2}}{b^{6}}\) | \(463\) |
| parallelrisch | \(\frac {-1918 a^{6} b c \,d^{6}+959 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-4200 x^{4} a^{2} b^{5} c \,d^{6}+2100 x^{4} a \,b^{6} c^{2} d^{5}-840 \ln \left (b x +a \right ) a^{6} b c \,d^{6}+420 \ln \left (b x +a \right ) a^{5} b^{2} c^{2} d^{5}+10 x^{7} d^{7} b^{7}+4375 x \,a^{6} b \,d^{7}-35 x \,b^{7} c^{6} d +7700 x^{2} a^{5} b^{2} d^{7}-140 x^{2} b^{7} c^{5} d^{2}+6300 x^{3} a^{4} b^{3} d^{7}-350 x^{3} b^{7} c^{4} d^{3}+2100 x^{4} a^{3} b^{4} d^{7}-700 x^{4} b^{7} c^{3} d^{4}-70 x^{6} a \,b^{6} d^{7}+140 x^{6} b^{7} c \,d^{6}-4 b^{7} c^{7}+959 a^{7} d^{7}+4200 \ln \left (b x +a \right ) x^{3} a^{4} b^{3} d^{7}+4200 \ln \left (b x +a \right ) x^{2} a^{5} b^{2} d^{7}+420 \ln \left (b x +a \right ) x^{5} a^{2} b^{5} d^{7}+420 \ln \left (b x +a \right ) x^{5} b^{7} c^{2} d^{5}-35 a^{3} b^{4} c^{4} d^{3}-14 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -840 \ln \left (b x +a \right ) x^{5} a \,b^{6} c \,d^{6}-8750 x \,a^{5} b^{2} c \,d^{6}+4375 x \,a^{4} b^{3} c^{2} d^{5}-700 x \,a^{3} b^{4} c^{3} d^{4}-175 x \,a^{2} b^{5} c^{4} d^{3}-70 x a \,b^{6} c^{5} d^{2}-15400 x^{2} a^{4} b^{3} c \,d^{6}+7700 x^{2} a^{3} b^{4} c^{2} d^{5}-1400 x^{2} a^{2} b^{5} c^{3} d^{4}-350 x^{2} a \,b^{6} c^{4} d^{3}-12600 x^{3} a^{3} b^{4} c \,d^{6}+6300 x^{3} a^{2} b^{5} c^{2} d^{5}-1400 x^{3} a \,b^{6} c^{3} d^{4}+2100 \ln \left (b x +a \right ) x \,a^{4} b^{3} c^{2} d^{5}-8400 \ln \left (b x +a \right ) x^{2} a^{4} b^{3} c \,d^{6}+4200 \ln \left (b x +a \right ) x^{2} a^{3} b^{4} c^{2} d^{5}+2100 \ln \left (b x +a \right ) x \,a^{6} b \,d^{7}-4200 \ln \left (b x +a \right ) x \,a^{5} b^{2} c \,d^{6}+420 \ln \left (b x +a \right ) a^{7} d^{7}+2100 \ln \left (b x +a \right ) x^{4} a^{3} b^{4} d^{7}-4200 \ln \left (b x +a \right ) x^{4} a^{2} b^{5} c \,d^{6}+2100 \ln \left (b x +a \right ) x^{4} a \,b^{6} c^{2} d^{5}-8400 \ln \left (b x +a \right ) x^{3} a^{3} b^{4} c \,d^{6}+4200 \ln \left (b x +a \right ) x^{3} a^{2} b^{5} c^{2} d^{5}}{20 b^{8} \left (b x +a \right )^{5}}\) | \(812\) |
Input:
int((d*x+c)^7/(b*x+a)^6,x,method=_RETURNVERBOSE)
Output:
-d^6/b^7*(-1/2*b*d*x^2+6*a*d*x-7*b*c*x)-1/5/b^8*(-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)/(b*x+a)^5-7/4*d/b^8*(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+b^6*c^6)/(b *x+a)^4-35/2/b^8*d^3*(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+b^4*c^4)/(b*x+a)^2+35/b^8*d^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c ^3)/(b*x+a)+21/b^8*d^5*(a^2*d^2-2*a*b*c*d+b^2*c^2)*ln(b*x+a)+7/b^8*d^2*(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-b^ 5*c^5)/(b*x+a)^3
Leaf count of result is larger than twice the leaf count of optimal. 732 vs. \(2 (173) = 346\).
Time = 0.08 (sec) , antiderivative size = 732, normalized size of antiderivative = 4.04 \[ \int \frac {(c+d x)^7}{(a+b x)^6} \, dx=\frac {10 \, b^{7} d^{7} x^{7} - 4 \, b^{7} c^{7} - 7 \, a b^{6} c^{6} d - 14 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} - 140 \, a^{4} b^{3} c^{3} d^{4} + 959 \, a^{5} b^{2} c^{2} d^{5} - 1218 \, a^{6} b c d^{6} + 459 \, a^{7} d^{7} + 70 \, {\left (2 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 100 \, {\left (7 \, a b^{6} c d^{6} - 5 \, a^{2} b^{5} d^{7}\right )} x^{5} - 100 \, {\left (7 \, b^{7} c^{3} d^{4} - 21 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} + 4 \, a^{3} b^{4} d^{7}\right )} x^{4} - 50 \, {\left (7 \, b^{7} c^{4} d^{3} + 28 \, a b^{6} c^{3} d^{4} - 126 \, a^{2} b^{5} c^{2} d^{5} + 112 \, a^{3} b^{4} c d^{6} - 26 \, a^{4} b^{3} d^{7}\right )} x^{3} - 10 \, {\left (14 \, b^{7} c^{5} d^{2} + 35 \, a b^{6} c^{4} d^{3} + 140 \, a^{2} b^{5} c^{3} d^{4} - 770 \, a^{3} b^{4} c^{2} d^{5} + 840 \, a^{4} b^{3} c d^{6} - 270 \, a^{5} b^{2} d^{7}\right )} x^{2} - 5 \, {\left (7 \, b^{7} c^{6} d + 14 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} + 140 \, a^{3} b^{4} c^{3} d^{4} - 875 \, a^{4} b^{3} c^{2} d^{5} + 1050 \, a^{5} b^{2} c d^{6} - 375 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{5} b^{2} c^{2} d^{5} - 2 \, a^{6} b c d^{6} + a^{7} d^{7} + {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 5 \, {\left (a b^{6} c^{2} d^{5} - 2 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 10 \, {\left (a^{2} b^{5} c^{2} d^{5} - 2 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 10 \, {\left (a^{3} b^{4} c^{2} d^{5} - 2 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 5 \, {\left (a^{4} b^{3} c^{2} d^{5} - 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} \] Input:
integrate((d*x+c)^7/(b*x+a)^6,x, algorithm="fricas")
Output:
1/20*(10*b^7*d^7*x^7 - 4*b^7*c^7 - 7*a*b^6*c^6*d - 14*a^2*b^5*c^5*d^2 - 35 *a^3*b^4*c^4*d^3 - 140*a^4*b^3*c^3*d^4 + 959*a^5*b^2*c^2*d^5 - 1218*a^6*b* c*d^6 + 459*a^7*d^7 + 70*(2*b^7*c*d^6 - a*b^6*d^7)*x^6 + 100*(7*a*b^6*c*d^ 6 - 5*a^2*b^5*d^7)*x^5 - 100*(7*b^7*c^3*d^4 - 21*a*b^6*c^2*d^5 + 7*a^2*b^5 *c*d^6 + 4*a^3*b^4*d^7)*x^4 - 50*(7*b^7*c^4*d^3 + 28*a*b^6*c^3*d^4 - 126*a ^2*b^5*c^2*d^5 + 112*a^3*b^4*c*d^6 - 26*a^4*b^3*d^7)*x^3 - 10*(14*b^7*c^5* d^2 + 35*a*b^6*c^4*d^3 + 140*a^2*b^5*c^3*d^4 - 770*a^3*b^4*c^2*d^5 + 840*a ^4*b^3*c*d^6 - 270*a^5*b^2*d^7)*x^2 - 5*(7*b^7*c^6*d + 14*a*b^6*c^5*d^2 + 35*a^2*b^5*c^4*d^3 + 140*a^3*b^4*c^3*d^4 - 875*a^4*b^3*c^2*d^5 + 1050*a^5* b^2*c*d^6 - 375*a^6*b*d^7)*x + 420*(a^5*b^2*c^2*d^5 - 2*a^6*b*c*d^6 + a^7* d^7 + (b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 5*(a*b^6*c^2*d^5 - 2*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 10*(a^2*b^5*c^2*d^5 - 2*a^3*b^4*c*d^ 6 + a^4*b^3*d^7)*x^3 + 10*(a^3*b^4*c^2*d^5 - 2*a^4*b^3*c*d^6 + a^5*b^2*d^7 )*x^2 + 5*(a^4*b^3*c^2*d^5 - 2*a^5*b^2*c*d^6 + a^6*b*d^7)*x)*log(b*x + a)) /(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9* x + a^5*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (168) = 336\).
Time = 62.28 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.90 \[ \int \frac {(c+d x)^7}{(a+b x)^6} \, dx=x \left (- \frac {6 a d^{7}}{b^{7}} + \frac {7 c d^{6}}{b^{6}}\right ) + \frac {459 a^{7} d^{7} - 1218 a^{6} b c d^{6} + 959 a^{5} b^{2} c^{2} d^{5} - 140 a^{4} b^{3} c^{3} d^{4} - 35 a^{3} b^{4} c^{4} d^{3} - 14 a^{2} b^{5} c^{5} d^{2} - 7 a b^{6} c^{6} d - 4 b^{7} c^{7} + x^{4} \cdot \left (700 a^{3} b^{4} d^{7} - 2100 a^{2} b^{5} c d^{6} + 2100 a b^{6} c^{2} d^{5} - 700 b^{7} c^{3} d^{4}\right ) + x^{3} \cdot \left (2450 a^{4} b^{3} d^{7} - 7000 a^{3} b^{4} c d^{6} + 6300 a^{2} b^{5} c^{2} d^{5} - 1400 a b^{6} c^{3} d^{4} - 350 b^{7} c^{4} d^{3}\right ) + x^{2} \cdot \left (3290 a^{5} b^{2} d^{7} - 9100 a^{4} b^{3} c d^{6} + 7700 a^{3} b^{4} c^{2} d^{5} - 1400 a^{2} b^{5} c^{3} d^{4} - 350 a b^{6} c^{4} d^{3} - 140 b^{7} c^{5} d^{2}\right ) + x \left (1995 a^{6} b d^{7} - 5390 a^{5} b^{2} c d^{6} + 4375 a^{4} b^{3} c^{2} d^{5} - 700 a^{3} b^{4} c^{3} d^{4} - 175 a^{2} b^{5} c^{4} d^{3} - 70 a b^{6} c^{5} d^{2} - 35 b^{7} c^{6} d\right )}{20 a^{5} b^{8} + 100 a^{4} b^{9} x + 200 a^{3} b^{10} x^{2} + 200 a^{2} b^{11} x^{3} + 100 a b^{12} x^{4} + 20 b^{13} x^{5}} + \frac {d^{7} x^{2}}{2 b^{6}} + \frac {21 d^{5} \left (a d - b c\right )^{2} \log {\left (a + b x \right )}}{b^{8}} \] Input:
integrate((d*x+c)**7/(b*x+a)**6,x)
Output:
x*(-6*a*d**7/b**7 + 7*c*d**6/b**6) + (459*a**7*d**7 - 1218*a**6*b*c*d**6 + 959*a**5*b**2*c**2*d**5 - 140*a**4*b**3*c**3*d**4 - 35*a**3*b**4*c**4*d** 3 - 14*a**2*b**5*c**5*d**2 - 7*a*b**6*c**6*d - 4*b**7*c**7 + x**4*(700*a** 3*b**4*d**7 - 2100*a**2*b**5*c*d**6 + 2100*a*b**6*c**2*d**5 - 700*b**7*c** 3*d**4) + x**3*(2450*a**4*b**3*d**7 - 7000*a**3*b**4*c*d**6 + 6300*a**2*b* *5*c**2*d**5 - 1400*a*b**6*c**3*d**4 - 350*b**7*c**4*d**3) + x**2*(3290*a* *5*b**2*d**7 - 9100*a**4*b**3*c*d**6 + 7700*a**3*b**4*c**2*d**5 - 1400*a** 2*b**5*c**3*d**4 - 350*a*b**6*c**4*d**3 - 140*b**7*c**5*d**2) + x*(1995*a* *6*b*d**7 - 5390*a**5*b**2*c*d**6 + 4375*a**4*b**3*c**2*d**5 - 700*a**3*b* *4*c**3*d**4 - 175*a**2*b**5*c**4*d**3 - 70*a*b**6*c**5*d**2 - 35*b**7*c** 6*d))/(20*a**5*b**8 + 100*a**4*b**9*x + 200*a**3*b**10*x**2 + 200*a**2*b** 11*x**3 + 100*a*b**12*x**4 + 20*b**13*x**5) + d**7*x**2/(2*b**6) + 21*d**5 *(a*d - b*c)**2*log(a + b*x)/b**8
Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (173) = 346\).
Time = 0.08 (sec) , antiderivative size = 504, normalized size of antiderivative = 2.78 \[ \int \frac {(c+d x)^7}{(a+b x)^6} \, dx=-\frac {4 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 14 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} - 959 \, a^{5} b^{2} c^{2} d^{5} + 1218 \, a^{6} b c d^{6} - 459 \, a^{7} d^{7} + 700 \, {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 350 \, {\left (b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} - 18 \, a^{2} b^{5} c^{2} d^{5} + 20 \, a^{3} b^{4} c d^{6} - 7 \, a^{4} b^{3} d^{7}\right )} x^{3} + 70 \, {\left (2 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} - 110 \, a^{3} b^{4} c^{2} d^{5} + 130 \, a^{4} b^{3} c d^{6} - 47 \, a^{5} b^{2} d^{7}\right )} x^{2} + 35 \, {\left (b^{7} c^{6} d + 2 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} - 125 \, a^{4} b^{3} c^{2} d^{5} + 154 \, a^{5} b^{2} c d^{6} - 57 \, a^{6} b d^{7}\right )} x}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} + \frac {b d^{7} x^{2} + 2 \, {\left (7 \, b c d^{6} - 6 \, a d^{7}\right )} x}{2 \, b^{7}} + \frac {21 \, {\left (b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \] Input:
integrate((d*x+c)^7/(b*x+a)^6,x, algorithm="maxima")
Output:
-1/20*(4*b^7*c^7 + 7*a*b^6*c^6*d + 14*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 - 959*a^5*b^2*c^2*d^5 + 1218*a^6*b*c*d^6 - 459*a^7* d^7 + 700*(b^7*c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 - a^3*b^4*d^7)* x^4 + 350*(b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 - 18*a^2*b^5*c^2*d^5 + 20*a^3*b^4 *c*d^6 - 7*a^4*b^3*d^7)*x^3 + 70*(2*b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 + 20*a^2 *b^5*c^3*d^4 - 110*a^3*b^4*c^2*d^5 + 130*a^4*b^3*c*d^6 - 47*a^5*b^2*d^7)*x ^2 + 35*(b^7*c^6*d + 2*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3* d^4 - 125*a^4*b^3*c^2*d^5 + 154*a^5*b^2*c*d^6 - 57*a^6*b*d^7)*x)/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b^ 8) + 1/2*(b*d^7*x^2 + 2*(7*b*c*d^6 - 6*a*d^7)*x)/b^7 + 21*(b^2*c^2*d^5 - 2 *a*b*c*d^6 + a^2*d^7)*log(b*x + a)/b^8
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (173) = 346\).
Time = 0.12 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.56 \[ \int \frac {(c+d x)^7}{(a+b x)^6} \, dx=\frac {21 \, {\left (b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} + \frac {b^{6} d^{7} x^{2} + 14 \, b^{6} c d^{6} x - 12 \, a b^{5} d^{7} x}{2 \, b^{12}} - \frac {4 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 14 \, a^{2} b^{5} c^{5} d^{2} + 35 \, a^{3} b^{4} c^{4} d^{3} + 140 \, a^{4} b^{3} c^{3} d^{4} - 959 \, a^{5} b^{2} c^{2} d^{5} + 1218 \, a^{6} b c d^{6} - 459 \, a^{7} d^{7} + 700 \, {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 350 \, {\left (b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} - 18 \, a^{2} b^{5} c^{2} d^{5} + 20 \, a^{3} b^{4} c d^{6} - 7 \, a^{4} b^{3} d^{7}\right )} x^{3} + 70 \, {\left (2 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 20 \, a^{2} b^{5} c^{3} d^{4} - 110 \, a^{3} b^{4} c^{2} d^{5} + 130 \, a^{4} b^{3} c d^{6} - 47 \, a^{5} b^{2} d^{7}\right )} x^{2} + 35 \, {\left (b^{7} c^{6} d + 2 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 20 \, a^{3} b^{4} c^{3} d^{4} - 125 \, a^{4} b^{3} c^{2} d^{5} + 154 \, a^{5} b^{2} c d^{6} - 57 \, a^{6} b d^{7}\right )} x}{20 \, {\left (b x + a\right )}^{5} b^{8}} \] Input:
integrate((d*x+c)^7/(b*x+a)^6,x, algorithm="giac")
Output:
21*(b^2*c^2*d^5 - 2*a*b*c*d^6 + a^2*d^7)*log(abs(b*x + a))/b^8 + 1/2*(b^6* d^7*x^2 + 14*b^6*c*d^6*x - 12*a*b^5*d^7*x)/b^12 - 1/20*(4*b^7*c^7 + 7*a*b^ 6*c^6*d + 14*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 - 959*a^5*b^2*c^2*d^5 + 1218*a^6*b*c*d^6 - 459*a^7*d^7 + 700*(b^7*c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 - a^3*b^4*d^7)*x^4 + 350*(b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 - 18*a^2*b^5*c^2*d^5 + 20*a^3*b^4*c*d^6 - 7*a^4*b^3*d^7)*x ^3 + 70*(2*b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 + 20*a^2*b^5*c^3*d^4 - 110*a^3*b^ 4*c^2*d^5 + 130*a^4*b^3*c*d^6 - 47*a^5*b^2*d^7)*x^2 + 35*(b^7*c^6*d + 2*a* b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 20*a^3*b^4*c^3*d^4 - 125*a^4*b^3*c^2*d^5 + 154*a^5*b^2*c*d^6 - 57*a^6*b*d^7)*x)/((b*x + a)^5*b^8)
Time = 0.11 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.81 \[ \int \frac {(c+d x)^7}{(a+b x)^6} \, dx=\frac {\ln \left (a+b\,x\right )\,\left (21\,a^2\,d^7-42\,a\,b\,c\,d^6+21\,b^2\,c^2\,d^5\right )}{b^8}-x\,\left (\frac {6\,a\,d^7}{b^7}-\frac {7\,c\,d^6}{b^6}\right )-\frac {\frac {-459\,a^7\,d^7+1218\,a^6\,b\,c\,d^6-959\,a^5\,b^2\,c^2\,d^5+140\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3+14\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d+4\,b^7\,c^7}{20\,b}+x\,\left (-\frac {399\,a^6\,d^7}{4}+\frac {539\,a^5\,b\,c\,d^6}{2}-\frac {875\,a^4\,b^2\,c^2\,d^5}{4}+35\,a^3\,b^3\,c^3\,d^4+\frac {35\,a^2\,b^4\,c^4\,d^3}{4}+\frac {7\,a\,b^5\,c^5\,d^2}{2}+\frac {7\,b^6\,c^6\,d}{4}\right )+x^3\,\left (-\frac {245\,a^4\,b^2\,d^7}{2}+350\,a^3\,b^3\,c\,d^6-315\,a^2\,b^4\,c^2\,d^5+70\,a\,b^5\,c^3\,d^4+\frac {35\,b^6\,c^4\,d^3}{2}\right )+x^2\,\left (-\frac {329\,a^5\,b\,d^7}{2}+455\,a^4\,b^2\,c\,d^6-385\,a^3\,b^3\,c^2\,d^5+70\,a^2\,b^4\,c^3\,d^4+\frac {35\,a\,b^5\,c^4\,d^3}{2}+7\,b^6\,c^5\,d^2\right )-x^4\,\left (35\,a^3\,b^3\,d^7-105\,a^2\,b^4\,c\,d^6+105\,a\,b^5\,c^2\,d^5-35\,b^6\,c^3\,d^4\right )}{a^5\,b^7+5\,a^4\,b^8\,x+10\,a^3\,b^9\,x^2+10\,a^2\,b^{10}\,x^3+5\,a\,b^{11}\,x^4+b^{12}\,x^5}+\frac {d^7\,x^2}{2\,b^6} \] Input:
int((c + d*x)^7/(a + b*x)^6,x)
Output:
(log(a + b*x)*(21*a^2*d^7 + 21*b^2*c^2*d^5 - 42*a*b*c*d^6))/b^8 - x*((6*a* d^7)/b^7 - (7*c*d^6)/b^6) - ((4*b^7*c^7 - 459*a^7*d^7 + 14*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 + 140*a^4*b^3*c^3*d^4 - 959*a^5*b^2*c^2*d^5 + 7*a*b^ 6*c^6*d + 1218*a^6*b*c*d^6)/(20*b) + x*((7*b^6*c^6*d)/4 - (399*a^6*d^7)/4 + (7*a*b^5*c^5*d^2)/2 + (35*a^2*b^4*c^4*d^3)/4 + 35*a^3*b^3*c^3*d^4 - (875 *a^4*b^2*c^2*d^5)/4 + (539*a^5*b*c*d^6)/2) + x^3*((35*b^6*c^4*d^3)/2 - (24 5*a^4*b^2*d^7)/2 + 70*a*b^5*c^3*d^4 + 350*a^3*b^3*c*d^6 - 315*a^2*b^4*c^2* d^5) + x^2*(7*b^6*c^5*d^2 - (329*a^5*b*d^7)/2 + (35*a*b^5*c^4*d^3)/2 + 455 *a^4*b^2*c*d^6 + 70*a^2*b^4*c^3*d^4 - 385*a^3*b^3*c^2*d^5) - x^4*(35*a^3*b ^3*d^7 - 35*b^6*c^3*d^4 + 105*a*b^5*c^2*d^5 - 105*a^2*b^4*c*d^6))/(a^5*b^7 + b^12*x^5 + 5*a^4*b^8*x + 5*a*b^11*x^4 + 10*a^3*b^9*x^2 + 10*a^2*b^10*x^ 3) + (d^7*x^2)/(2*b^6)
Time = 0.17 (sec) , antiderivative size = 816, normalized size of antiderivative = 4.51 \[ \int \frac {(c+d x)^7}{(a+b x)^6} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^7/(b*x+a)^6,x)
Output:
(420*log(a + b*x)*a**8*d**7 - 840*log(a + b*x)*a**7*b*c*d**6 + 2100*log(a + b*x)*a**7*b*d**7*x + 420*log(a + b*x)*a**6*b**2*c**2*d**5 - 4200*log(a + b*x)*a**6*b**2*c*d**6*x + 4200*log(a + b*x)*a**6*b**2*d**7*x**2 + 2100*lo g(a + b*x)*a**5*b**3*c**2*d**5*x - 8400*log(a + b*x)*a**5*b**3*c*d**6*x**2 + 4200*log(a + b*x)*a**5*b**3*d**7*x**3 + 4200*log(a + b*x)*a**4*b**4*c** 2*d**5*x**2 - 8400*log(a + b*x)*a**4*b**4*c*d**6*x**3 + 2100*log(a + b*x)* a**4*b**4*d**7*x**4 + 4200*log(a + b*x)*a**3*b**5*c**2*d**5*x**3 - 4200*lo g(a + b*x)*a**3*b**5*c*d**6*x**4 + 420*log(a + b*x)*a**3*b**5*d**7*x**5 + 2100*log(a + b*x)*a**2*b**6*c**2*d**5*x**4 - 840*log(a + b*x)*a**2*b**6*c* d**6*x**5 + 420*log(a + b*x)*a*b**7*c**2*d**5*x**5 + 539*a**8*d**7 - 1078* a**7*b*c*d**6 + 2275*a**7*b*d**7*x + 539*a**6*b**2*c**2*d**5 - 4550*a**6*b **2*c*d**6*x + 3500*a**6*b**2*d**7*x**2 + 2275*a**5*b**3*c**2*d**5*x - 700 0*a**5*b**3*c*d**6*x**2 + 2100*a**5*b**3*d**7*x**3 - 35*a**4*b**4*c**4*d** 3 + 3500*a**4*b**4*c**2*d**5*x**2 - 4200*a**4*b**4*c*d**6*x**3 - 14*a**3*b **5*c**5*d**2 - 175*a**3*b**5*c**4*d**3*x + 2100*a**3*b**5*c**2*d**5*x**3 - 420*a**3*b**5*d**7*x**5 - 7*a**2*b**6*c**6*d - 70*a**2*b**6*c**5*d**2*x - 350*a**2*b**6*c**4*d**3*x**2 + 840*a**2*b**6*c*d**6*x**5 - 70*a**2*b**6* d**7*x**6 - 4*a*b**7*c**7 - 35*a*b**7*c**6*d*x - 140*a*b**7*c**5*d**2*x**2 - 350*a*b**7*c**4*d**3*x**3 - 420*a*b**7*c**2*d**5*x**5 + 140*a*b**7*c*d* *6*x**6 + 10*a*b**7*d**7*x**7 + 140*b**8*c**3*d**4*x**5)/(20*a*b**8*(a*...