Integrand size = 15, antiderivative size = 169 \[ \int \frac {(c+d x)^7}{a+b x} \, dx=\frac {d (b c-a d)^6 x}{b^7}+\frac {(b c-a d)^5 (c+d x)^2}{2 b^6}+\frac {(b c-a d)^4 (c+d x)^3}{3 b^5}+\frac {(b c-a d)^3 (c+d x)^4}{4 b^4}+\frac {(b c-a d)^2 (c+d x)^5}{5 b^3}+\frac {(b c-a d) (c+d x)^6}{6 b^2}+\frac {(c+d x)^7}{7 b}+\frac {(b c-a d)^7 \log (a+b x)}{b^8} \]
d*(-a*d+b*c)^6*x/b^7+1/2*(-a*d+b*c)^5*(d*x+c)^2/b^6+1/3*(-a*d+b*c)^4*(d*x+ c)^3/b^5+1/4*(-a*d+b*c)^3*(d*x+c)^4/b^4+1/5*(-a*d+b*c)^2*(d*x+c)^5/b^3+1/6 *(-a*d+b*c)*(d*x+c)^6/b^2+1/7*(d*x+c)^7/b+(-a*d+b*c)^7*ln(b*x+a)/b^8
Time = 0.09 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.80 \[ \int \frac {(c+d x)^7}{a+b x} \, dx=\frac {d x \left (420 a^6 d^6-210 a^5 b d^5 (14 c+d x)+70 a^4 b^2 d^4 \left (126 c^2+21 c d x+2 d^2 x^2\right )-35 a^3 b^3 d^3 \left (420 c^3+126 c^2 d x+28 c d^2 x^2+3 d^3 x^3\right )+21 a^2 b^4 d^2 \left (700 c^4+350 c^3 d x+140 c^2 d^2 x^2+35 c d^3 x^3+4 d^4 x^4\right )-7 a b^5 d \left (1260 c^5+1050 c^4 d x+700 c^3 d^2 x^2+315 c^2 d^3 x^3+84 c d^4 x^4+10 d^5 x^5\right )+b^6 \left (2940 c^6+4410 c^5 d x+4900 c^4 d^2 x^2+3675 c^3 d^3 x^3+1764 c^2 d^4 x^4+490 c d^5 x^5+60 d^6 x^6\right )\right )}{420 b^7}+\frac {(b c-a d)^7 \log (a+b x)}{b^8} \]
(d*x*(420*a^6*d^6 - 210*a^5*b*d^5*(14*c + d*x) + 70*a^4*b^2*d^4*(126*c^2 + 21*c*d*x + 2*d^2*x^2) - 35*a^3*b^3*d^3*(420*c^3 + 126*c^2*d*x + 28*c*d^2* x^2 + 3*d^3*x^3) + 21*a^2*b^4*d^2*(700*c^4 + 350*c^3*d*x + 140*c^2*d^2*x^2 + 35*c*d^3*x^3 + 4*d^4*x^4) - 7*a*b^5*d*(1260*c^5 + 1050*c^4*d*x + 700*c^ 3*d^2*x^2 + 315*c^2*d^3*x^3 + 84*c*d^4*x^4 + 10*d^5*x^5) + b^6*(2940*c^6 + 4410*c^5*d*x + 4900*c^4*d^2*x^2 + 3675*c^3*d^3*x^3 + 1764*c^2*d^4*x^4 + 4 90*c*d^5*x^5 + 60*d^6*x^6)))/(420*b^7) + ((b*c - a*d)^7*Log[a + b*x])/b^8
Time = 0.30 (sec) , antiderivative size = 169, 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.
| \(\displaystyle \int \frac {(c+d x)^7}{a+b x} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {(b c-a d)^7}{b^7 (a+b x)}+\frac {d (b c-a d)^6}{b^7}+\frac {d (c+d x) (b c-a d)^5}{b^6}+\frac {d (c+d x)^2 (b c-a d)^4}{b^5}+\frac {d (c+d x)^3 (b c-a d)^3}{b^4}+\frac {d (c+d x)^4 (b c-a d)^2}{b^3}+\frac {d (c+d x)^5 (b c-a d)}{b^2}+\frac {d (c+d x)^6}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(b c-a d)^7 \log (a+b x)}{b^8}+\frac {d x (b c-a d)^6}{b^7}+\frac {(c+d x)^2 (b c-a d)^5}{2 b^6}+\frac {(c+d x)^3 (b c-a d)^4}{3 b^5}+\frac {(c+d x)^4 (b c-a d)^3}{4 b^4}+\frac {(c+d x)^5 (b c-a d)^2}{5 b^3}+\frac {(c+d x)^6 (b c-a d)}{6 b^2}+\frac {(c+d x)^7}{7 b}\) |
(d*(b*c - a*d)^6*x)/b^7 + ((b*c - a*d)^5*(c + d*x)^2)/(2*b^6) + ((b*c - a* d)^4*(c + d*x)^3)/(3*b^5) + ((b*c - a*d)^3*(c + d*x)^4)/(4*b^4) + ((b*c - a*d)^2*(c + d*x)^5)/(5*b^3) + ((b*c - a*d)*(c + d*x)^6)/(6*b^2) + (c + d*x )^7/(7*b) + ((b*c - a*d)^7*Log[a + b*x])/b^8
3.13.83.3.1 Defintions of rubi rules used
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. \(434\) vs. \(2(157)=314\).
Time = 0.61 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.57
| method | result | size |
| norman | \(\frac {d \left (a^{6} d^{6}-7 a^{5} b c \,d^{5}+21 a^{4} b^{2} c^{2} d^{4}-35 a^{3} b^{3} c^{3} d^{3}+35 a^{2} b^{4} c^{4} d^{2}-21 a \,b^{5} c^{5} d +7 b^{6} c^{6}\right ) x}{b^{7}}+\frac {d^{7} x^{7}}{7 b}-\frac {d^{2} \left (a^{5} d^{5}-7 a^{4} b c \,d^{4}+21 a^{3} b^{2} c^{2} d^{3}-35 a^{2} b^{3} c^{3} d^{2}+35 a \,b^{4} c^{4} d -21 b^{5} c^{5}\right ) x^{2}}{2 b^{6}}+\frac {d^{3} \left (a^{4} d^{4}-7 a^{3} b c \,d^{3}+21 a^{2} b^{2} c^{2} d^{2}-35 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right ) x^{3}}{3 b^{5}}-\frac {d^{4} \left (a^{3} d^{3}-7 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d -35 b^{3} c^{3}\right ) x^{4}}{4 b^{4}}+\frac {d^{5} \left (a^{2} d^{2}-7 a b c d +21 b^{2} c^{2}\right ) x^{5}}{5 b^{3}}-\frac {d^{6} \left (a d -7 b c \right ) x^{6}}{6 b^{2}}-\frac {\left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right ) \ln \left (b x +a \right )}{b^{8}}\) | \(435\) |
| default | \(\frac {d \left (\frac {21}{5} b^{6} c^{2} d^{4} x^{5}-21 a \,b^{5} c^{5} d x +7 b^{6} c^{6} x +a^{6} d^{6} x +\frac {1}{7} d^{6} x^{7} b^{6}-\frac {35}{3} a \,b^{5} c^{3} d^{3} x^{3}+\frac {7}{2} a^{4} b^{2} c \,d^{5} x^{2}-\frac {21}{2} a^{3} b^{3} c^{2} d^{4} x^{2}+\frac {35}{2} a^{2} b^{4} c^{3} d^{3} x^{2}-\frac {35}{2} a \,b^{5} c^{4} d^{2} x^{2}-7 a^{5} b c \,d^{5} x +21 a^{4} b^{2} c^{2} d^{4} x -35 a^{3} b^{3} c^{3} d^{3} x +35 a^{2} b^{4} c^{4} d^{2} x +\frac {35}{4} b^{6} c^{3} d^{3} x^{4}+\frac {1}{3} a^{4} b^{2} d^{6} x^{3}+\frac {35}{3} b^{6} c^{4} d^{2} x^{3}-\frac {1}{2} a^{5} b \,d^{6} x^{2}+\frac {21}{2} b^{6} c^{5} d \,x^{2}-\frac {1}{6} a \,b^{5} d^{6} x^{6}+\frac {7}{6} b^{6} c \,d^{5} x^{6}+\frac {1}{5} a^{2} b^{4} d^{6} x^{5}-\frac {1}{4} a^{3} b^{3} d^{6} x^{4}+\frac {7}{4} a^{2} b^{4} c \,d^{5} x^{4}-\frac {21}{4} a \,b^{5} c^{2} d^{4} x^{4}-\frac {7}{3} a^{3} b^{3} c \,d^{5} x^{3}+7 a^{2} b^{4} c^{2} d^{4} x^{3}-\frac {7}{5} a \,b^{5} c \,d^{5} x^{5}\right )}{b^{7}}+\frac {\left (-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}\right ) \ln \left (b x +a \right )}{b^{8}}\) | \(491\) |
| risch | \(\frac {\ln \left (b x +a \right ) c^{7}}{b}+\frac {d^{7} x^{7}}{7 b}-\frac {21 d^{2} a \,c^{5} x}{b^{2}}-\frac {35 d^{4} a \,c^{3} x^{3}}{3 b^{2}}+\frac {7 d^{6} a^{4} c \,x^{2}}{2 b^{5}}-\frac {21 d^{5} a^{3} c^{2} x^{2}}{2 b^{4}}+\frac {35 d^{4} a^{2} c^{3} x^{2}}{2 b^{3}}-\frac {35 d^{3} a \,c^{4} x^{2}}{2 b^{2}}-\frac {7 d^{6} a^{5} c x}{b^{6}}+\frac {21 d^{5} a^{4} c^{2} x}{b^{5}}-\frac {35 d^{4} a^{3} c^{3} x}{b^{4}}+\frac {35 d^{3} a^{2} c^{4} x}{b^{3}}+\frac {7 d^{6} a^{2} c \,x^{4}}{4 b^{3}}-\frac {21 d^{5} a \,c^{2} x^{4}}{4 b^{2}}-\frac {7 d^{6} a^{3} c \,x^{3}}{3 b^{4}}+\frac {7 d^{5} a^{2} c^{2} x^{3}}{b^{3}}-\frac {7 d^{6} a c \,x^{5}}{5 b^{2}}+\frac {7 \ln \left (b x +a \right ) a^{6} c \,d^{6}}{b^{7}}-\frac {21 \ln \left (b x +a \right ) a^{5} c^{2} d^{5}}{b^{6}}+\frac {35 \ln \left (b x +a \right ) a^{4} c^{3} d^{4}}{b^{5}}-\frac {35 \ln \left (b x +a \right ) a^{3} c^{4} d^{3}}{b^{4}}+\frac {21 \ln \left (b x +a \right ) a^{2} c^{5} d^{2}}{b^{3}}-\frac {7 \ln \left (b x +a \right ) a \,c^{6} d}{b^{2}}-\frac {\ln \left (b x +a \right ) a^{7} d^{7}}{b^{8}}+\frac {21 d^{5} c^{2} x^{5}}{5 b}+\frac {7 d \,c^{6} x}{b}+\frac {d^{7} a^{6} x}{b^{7}}+\frac {35 d^{4} c^{3} x^{4}}{4 b}+\frac {d^{7} a^{4} x^{3}}{3 b^{5}}+\frac {35 d^{3} c^{4} x^{3}}{3 b}-\frac {d^{7} a^{5} x^{2}}{2 b^{6}}+\frac {21 d^{2} c^{5} x^{2}}{2 b}-\frac {d^{7} a \,x^{6}}{6 b^{2}}+\frac {7 d^{6} c \,x^{6}}{6 b}+\frac {d^{7} a^{2} x^{5}}{5 b^{3}}-\frac {d^{7} a^{3} x^{4}}{4 b^{4}}\) | \(539\) |
| parallelrisch | \(-\frac {-8820 \ln \left (b x +a \right ) a^{2} b^{5} c^{5} d^{2}+2940 \ln \left (b x +a \right ) a \,b^{6} c^{6} d -7350 x^{2} a^{2} b^{5} c^{3} d^{4}+7350 x^{2} a \,b^{6} c^{4} d^{3}+980 x^{3} a^{3} b^{4} c \,d^{6}-2940 x^{3} a^{2} b^{5} c^{2} d^{5}+4900 x^{3} a \,b^{6} c^{3} d^{4}-735 x^{4} a^{2} b^{5} c \,d^{6}+14700 \ln \left (b x +a \right ) a^{3} b^{4} c^{4} d^{3}-2940 \ln \left (b x +a \right ) a^{6} b c \,d^{6}+8820 \ln \left (b x +a \right ) a^{5} b^{2} c^{2} d^{5}-14700 \ln \left (b x +a \right ) a^{4} b^{3} c^{3} d^{4}+2205 x^{4} a \,b^{6} c^{2} d^{5}+588 x^{5} a \,b^{6} c \,d^{6}+2940 x \,a^{5} b^{2} c \,d^{6}-8820 x \,a^{4} b^{3} c^{2} d^{5}+14700 x \,a^{3} b^{4} c^{3} d^{4}-14700 x \,a^{2} b^{5} c^{4} d^{3}+8820 x a \,b^{6} c^{5} d^{2}-1470 x^{2} a^{4} b^{3} c \,d^{6}+4410 x^{2} a^{3} b^{4} c^{2} d^{5}+420 \ln \left (b x +a \right ) a^{7} d^{7}-420 \ln \left (b x +a \right ) b^{7} c^{7}-60 x^{7} d^{7} b^{7}-84 x^{5} a^{2} b^{5} d^{7}-1764 x^{5} b^{7} c^{2} d^{5}+70 x^{6} a \,b^{6} d^{7}-420 x \,a^{6} b \,d^{7}-2940 x \,b^{7} c^{6} d +210 x^{2} a^{5} b^{2} d^{7}-4410 x^{2} b^{7} c^{5} d^{2}-140 x^{3} a^{4} b^{3} d^{7}-4900 x^{3} b^{7} c^{4} d^{3}+105 x^{4} a^{3} b^{4} d^{7}-3675 x^{4} b^{7} c^{3} d^{4}-490 x^{6} b^{7} c \,d^{6}}{420 b^{8}}\) | \(539\) |
d*(a^6*d^6-7*a^5*b*c*d^5+21*a^4*b^2*c^2*d^4-35*a^3*b^3*c^3*d^3+35*a^2*b^4* c^4*d^2-21*a*b^5*c^5*d+7*b^6*c^6)/b^7*x+1/7/b*d^7*x^7-1/2/b^6*d^2*(a^5*d^5 -7*a^4*b*c*d^4+21*a^3*b^2*c^2*d^3-35*a^2*b^3*c^3*d^2+35*a*b^4*c^4*d-21*b^5 *c^5)*x^2+1/3/b^5*d^3*(a^4*d^4-7*a^3*b*c*d^3+21*a^2*b^2*c^2*d^2-35*a*b^3*c ^3*d+35*b^4*c^4)*x^3-1/4/b^4*d^4*(a^3*d^3-7*a^2*b*c*d^2+21*a*b^2*c^2*d-35* b^3*c^3)*x^4+1/5/b^3*d^5*(a^2*d^2-7*a*b*c*d+21*b^2*c^2)*x^5-1/6/b^2*d^6*(a *d-7*b*c)*x^6-(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^8*ln(b*x+a )
Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (157) = 314\).
Time = 0.23 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.73 \[ \int \frac {(c+d x)^7}{a+b x} \, dx=\frac {60 \, b^{7} d^{7} x^{7} + 70 \, {\left (7 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 84 \, {\left (21 \, b^{7} c^{2} d^{5} - 7 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 105 \, {\left (35 \, b^{7} c^{3} d^{4} - 21 \, a b^{6} c^{2} d^{5} + 7 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 140 \, {\left (35 \, b^{7} c^{4} d^{3} - 35 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} - 7 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 210 \, {\left (21 \, b^{7} c^{5} d^{2} - 35 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} - 21 \, a^{3} b^{4} c^{2} d^{5} + 7 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 420 \, {\left (7 \, b^{7} c^{6} d - 21 \, a b^{6} c^{5} d^{2} + 35 \, a^{2} b^{5} c^{4} d^{3} - 35 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} - 7 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x + 420 \, {\left (b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} \log \left (b x + a\right )}{420 \, b^{8}} \]
1/420*(60*b^7*d^7*x^7 + 70*(7*b^7*c*d^6 - a*b^6*d^7)*x^6 + 84*(21*b^7*c^2* d^5 - 7*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 105*(35*b^7*c^3*d^4 - 21*a*b^6*c^ 2*d^5 + 7*a^2*b^5*c*d^6 - a^3*b^4*d^7)*x^4 + 140*(35*b^7*c^4*d^3 - 35*a*b^ 6*c^3*d^4 + 21*a^2*b^5*c^2*d^5 - 7*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 210* (21*b^7*c^5*d^2 - 35*a*b^6*c^4*d^3 + 35*a^2*b^5*c^3*d^4 - 21*a^3*b^4*c^2*d ^5 + 7*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^2 + 420*(7*b^7*c^6*d - 21*a*b^6*c^5* d^2 + 35*a^2*b^5*c^4*d^3 - 35*a^3*b^4*c^3*d^4 + 21*a^4*b^3*c^2*d^5 - 7*a^5 *b^2*c*d^6 + a^6*b*d^7)*x + 420*(b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5*c^5* d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a^6 *b*c*d^6 - a^7*d^7)*log(b*x + a))/b^8
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (144) = 288\).
Time = 0.48 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.41 \[ \int \frac {(c+d x)^7}{a+b x} \, dx=x^{6} \left (- \frac {a d^{7}}{6 b^{2}} + \frac {7 c d^{6}}{6 b}\right ) + x^{5} \left (\frac {a^{2} d^{7}}{5 b^{3}} - \frac {7 a c d^{6}}{5 b^{2}} + \frac {21 c^{2} d^{5}}{5 b}\right ) + x^{4} \left (- \frac {a^{3} d^{7}}{4 b^{4}} + \frac {7 a^{2} c d^{6}}{4 b^{3}} - \frac {21 a c^{2} d^{5}}{4 b^{2}} + \frac {35 c^{3} d^{4}}{4 b}\right ) + x^{3} \left (\frac {a^{4} d^{7}}{3 b^{5}} - \frac {7 a^{3} c d^{6}}{3 b^{4}} + \frac {7 a^{2} c^{2} d^{5}}{b^{3}} - \frac {35 a c^{3} d^{4}}{3 b^{2}} + \frac {35 c^{4} d^{3}}{3 b}\right ) + x^{2} \left (- \frac {a^{5} d^{7}}{2 b^{6}} + \frac {7 a^{4} c d^{6}}{2 b^{5}} - \frac {21 a^{3} c^{2} d^{5}}{2 b^{4}} + \frac {35 a^{2} c^{3} d^{4}}{2 b^{3}} - \frac {35 a c^{4} d^{3}}{2 b^{2}} + \frac {21 c^{5} d^{2}}{2 b}\right ) + x \left (\frac {a^{6} d^{7}}{b^{7}} - \frac {7 a^{5} c d^{6}}{b^{6}} + \frac {21 a^{4} c^{2} d^{5}}{b^{5}} - \frac {35 a^{3} c^{3} d^{4}}{b^{4}} + \frac {35 a^{2} c^{4} d^{3}}{b^{3}} - \frac {21 a c^{5} d^{2}}{b^{2}} + \frac {7 c^{6} d}{b}\right ) + \frac {d^{7} x^{7}}{7 b} - \frac {\left (a d - b c\right )^{7} \log {\left (a + b x \right )}}{b^{8}} \]
x**6*(-a*d**7/(6*b**2) + 7*c*d**6/(6*b)) + x**5*(a**2*d**7/(5*b**3) - 7*a* c*d**6/(5*b**2) + 21*c**2*d**5/(5*b)) + x**4*(-a**3*d**7/(4*b**4) + 7*a**2 *c*d**6/(4*b**3) - 21*a*c**2*d**5/(4*b**2) + 35*c**3*d**4/(4*b)) + x**3*(a **4*d**7/(3*b**5) - 7*a**3*c*d**6/(3*b**4) + 7*a**2*c**2*d**5/b**3 - 35*a* c**3*d**4/(3*b**2) + 35*c**4*d**3/(3*b)) + x**2*(-a**5*d**7/(2*b**6) + 7*a **4*c*d**6/(2*b**5) - 21*a**3*c**2*d**5/(2*b**4) + 35*a**2*c**3*d**4/(2*b* *3) - 35*a*c**4*d**3/(2*b**2) + 21*c**5*d**2/(2*b)) + x*(a**6*d**7/b**7 - 7*a**5*c*d**6/b**6 + 21*a**4*c**2*d**5/b**5 - 35*a**3*c**3*d**4/b**4 + 35* a**2*c**4*d**3/b**3 - 21*a*c**5*d**2/b**2 + 7*c**6*d/b) + d**7*x**7/(7*b) - (a*d - b*c)**7*log(a + b*x)/b**8
Leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (157) = 314\).
Time = 0.22 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.72 \[ \int \frac {(c+d x)^7}{a+b x} \, dx=\frac {60 \, b^{6} d^{7} x^{7} + 70 \, {\left (7 \, b^{6} c d^{6} - a b^{5} d^{7}\right )} x^{6} + 84 \, {\left (21 \, b^{6} c^{2} d^{5} - 7 \, a b^{5} c d^{6} + a^{2} b^{4} d^{7}\right )} x^{5} + 105 \, {\left (35 \, b^{6} c^{3} d^{4} - 21 \, a b^{5} c^{2} d^{5} + 7 \, a^{2} b^{4} c d^{6} - a^{3} b^{3} d^{7}\right )} x^{4} + 140 \, {\left (35 \, b^{6} c^{4} d^{3} - 35 \, a b^{5} c^{3} d^{4} + 21 \, a^{2} b^{4} c^{2} d^{5} - 7 \, a^{3} b^{3} c d^{6} + a^{4} b^{2} d^{7}\right )} x^{3} + 210 \, {\left (21 \, b^{6} c^{5} d^{2} - 35 \, a b^{5} c^{4} d^{3} + 35 \, a^{2} b^{4} c^{3} d^{4} - 21 \, a^{3} b^{3} c^{2} d^{5} + 7 \, a^{4} b^{2} c d^{6} - a^{5} b d^{7}\right )} x^{2} + 420 \, {\left (7 \, b^{6} c^{6} d - 21 \, a b^{5} c^{5} d^{2} + 35 \, a^{2} b^{4} c^{4} d^{3} - 35 \, a^{3} b^{3} c^{3} d^{4} + 21 \, a^{4} b^{2} c^{2} d^{5} - 7 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} x}{420 \, b^{7}} + \frac {{\left (b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \]
1/420*(60*b^6*d^7*x^7 + 70*(7*b^6*c*d^6 - a*b^5*d^7)*x^6 + 84*(21*b^6*c^2* d^5 - 7*a*b^5*c*d^6 + a^2*b^4*d^7)*x^5 + 105*(35*b^6*c^3*d^4 - 21*a*b^5*c^ 2*d^5 + 7*a^2*b^4*c*d^6 - a^3*b^3*d^7)*x^4 + 140*(35*b^6*c^4*d^3 - 35*a*b^ 5*c^3*d^4 + 21*a^2*b^4*c^2*d^5 - 7*a^3*b^3*c*d^6 + a^4*b^2*d^7)*x^3 + 210* (21*b^6*c^5*d^2 - 35*a*b^5*c^4*d^3 + 35*a^2*b^4*c^3*d^4 - 21*a^3*b^3*c^2*d ^5 + 7*a^4*b^2*c*d^6 - a^5*b*d^7)*x^2 + 420*(7*b^6*c^6*d - 21*a*b^5*c^5*d^ 2 + 35*a^2*b^4*c^4*d^3 - 35*a^3*b^3*c^3*d^4 + 21*a^4*b^2*c^2*d^5 - 7*a^5*b *c*d^6 + a^6*d^7)*x)/b^7 + (b^7*c^7 - 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a^6*b*c* d^6 - a^7*d^7)*log(b*x + a)/b^8
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (157) = 314\).
Time = 0.29 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.94 \[ \int \frac {(c+d x)^7}{a+b x} \, dx=\frac {60 \, b^{6} d^{7} x^{7} + 490 \, b^{6} c d^{6} x^{6} - 70 \, a b^{5} d^{7} x^{6} + 1764 \, b^{6} c^{2} d^{5} x^{5} - 588 \, a b^{5} c d^{6} x^{5} + 84 \, a^{2} b^{4} d^{7} x^{5} + 3675 \, b^{6} c^{3} d^{4} x^{4} - 2205 \, a b^{5} c^{2} d^{5} x^{4} + 735 \, a^{2} b^{4} c d^{6} x^{4} - 105 \, a^{3} b^{3} d^{7} x^{4} + 4900 \, b^{6} c^{4} d^{3} x^{3} - 4900 \, a b^{5} c^{3} d^{4} x^{3} + 2940 \, a^{2} b^{4} c^{2} d^{5} x^{3} - 980 \, a^{3} b^{3} c d^{6} x^{3} + 140 \, a^{4} b^{2} d^{7} x^{3} + 4410 \, b^{6} c^{5} d^{2} x^{2} - 7350 \, a b^{5} c^{4} d^{3} x^{2} + 7350 \, a^{2} b^{4} c^{3} d^{4} x^{2} - 4410 \, a^{3} b^{3} c^{2} d^{5} x^{2} + 1470 \, a^{4} b^{2} c d^{6} x^{2} - 210 \, a^{5} b d^{7} x^{2} + 2940 \, b^{6} c^{6} d x - 8820 \, a b^{5} c^{5} d^{2} x + 14700 \, a^{2} b^{4} c^{4} d^{3} x - 14700 \, a^{3} b^{3} c^{3} d^{4} x + 8820 \, a^{4} b^{2} c^{2} d^{5} x - 2940 \, a^{5} b c d^{6} x + 420 \, a^{6} d^{7} x}{420 \, b^{7}} + \frac {{\left (b^{7} c^{7} - 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} - 35 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} - 21 \, a^{5} b^{2} c^{2} d^{5} + 7 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} \]
1/420*(60*b^6*d^7*x^7 + 490*b^6*c*d^6*x^6 - 70*a*b^5*d^7*x^6 + 1764*b^6*c^ 2*d^5*x^5 - 588*a*b^5*c*d^6*x^5 + 84*a^2*b^4*d^7*x^5 + 3675*b^6*c^3*d^4*x^ 4 - 2205*a*b^5*c^2*d^5*x^4 + 735*a^2*b^4*c*d^6*x^4 - 105*a^3*b^3*d^7*x^4 + 4900*b^6*c^4*d^3*x^3 - 4900*a*b^5*c^3*d^4*x^3 + 2940*a^2*b^4*c^2*d^5*x^3 - 980*a^3*b^3*c*d^6*x^3 + 140*a^4*b^2*d^7*x^3 + 4410*b^6*c^5*d^2*x^2 - 735 0*a*b^5*c^4*d^3*x^2 + 7350*a^2*b^4*c^3*d^4*x^2 - 4410*a^3*b^3*c^2*d^5*x^2 + 1470*a^4*b^2*c*d^6*x^2 - 210*a^5*b*d^7*x^2 + 2940*b^6*c^6*d*x - 8820*a*b ^5*c^5*d^2*x + 14700*a^2*b^4*c^4*d^3*x - 14700*a^3*b^3*c^3*d^4*x + 8820*a^ 4*b^2*c^2*d^5*x - 2940*a^5*b*c*d^6*x + 420*a^6*d^7*x)/b^7 + (b^7*c^7 - 7*a *b^6*c^6*d + 21*a^2*b^5*c^5*d^2 - 35*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 21*a^5*b^2*c^2*d^5 + 7*a^6*b*c*d^6 - a^7*d^7)*log(abs(b*x + a))/b^8
Time = 0.23 (sec) , antiderivative size = 509, normalized size of antiderivative = 3.01 \[ \int \frac {(c+d x)^7}{a+b x} \, dx=x\,\left (\frac {7\,c^6\,d}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {35\,c^3\,d^4}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{b}\right )}{b}-\frac {35\,c^4\,d^3}{b}\right )}{b}+\frac {21\,c^5\,d^2}{b}\right )}{b}\right )-x^6\,\left (\frac {a\,d^7}{6\,b^2}-\frac {7\,c\,d^6}{6\,b}\right )+x^4\,\left (\frac {35\,c^3\,d^4}{4\,b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{4\,b}\right )+x^2\,\left (\frac {a\,\left (\frac {a\,\left (\frac {35\,c^3\,d^4}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{b}\right )}{b}-\frac {35\,c^4\,d^3}{b}\right )}{2\,b}+\frac {21\,c^5\,d^2}{2\,b}\right )+x^5\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{5\,b}+\frac {21\,c^2\,d^5}{5\,b}\right )-x^3\,\left (\frac {a\,\left (\frac {35\,c^3\,d^4}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^7}{b^2}-\frac {7\,c\,d^6}{b}\right )}{b}+\frac {21\,c^2\,d^5}{b}\right )}{b}\right )}{3\,b}-\frac {35\,c^4\,d^3}{3\,b}\right )-\frac {\ln \left (a+b\,x\right )\,\left (a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7\right )}{b^8}+\frac {d^7\,x^7}{7\,b} \]
x*((7*c^6*d)/b - (a*((a*((a*((35*c^3*d^4)/b - (a*((a*((a*d^7)/b^2 - (7*c*d ^6)/b))/b + (21*c^2*d^5)/b))/b))/b - (35*c^4*d^3)/b))/b + (21*c^5*d^2)/b)) /b) - x^6*((a*d^7)/(6*b^2) - (7*c*d^6)/(6*b)) + x^4*((35*c^3*d^4)/(4*b) - (a*((a*((a*d^7)/b^2 - (7*c*d^6)/b))/b + (21*c^2*d^5)/b))/(4*b)) + x^2*((a* ((a*((35*c^3*d^4)/b - (a*((a*((a*d^7)/b^2 - (7*c*d^6)/b))/b + (21*c^2*d^5) /b))/b))/b - (35*c^4*d^3)/b))/(2*b) + (21*c^5*d^2)/(2*b)) + x^5*((a*((a*d^ 7)/b^2 - (7*c*d^6)/b))/(5*b) + (21*c^2*d^5)/(5*b)) - x^3*((a*((35*c^3*d^4) /b - (a*((a*((a*d^7)/b^2 - (7*c*d^6)/b))/b + (21*c^2*d^5)/b))/b))/(3*b) - (35*c^4*d^3)/(3*b)) - (log(a + b*x)*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^ 2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6 *c^6*d - 7*a^6*b*c*d^6))/b^8 + (d^7*x^7)/(7*b)