Integrand size = 15, antiderivative size = 209 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\frac {b^8 x}{d^8}-\frac {(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac {4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac {28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}+\frac {14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac {70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac {28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac {28 b^6 (b c-a d)^2}{d^9 (c+d x)}-\frac {8 b^7 (b c-a d) \log (c+d x)}{d^9} \]
[Out]
Time = 0.19 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=-\frac {8 b^7 (b c-a d) \log (c+d x)}{d^9}-\frac {28 b^6 (b c-a d)^2}{d^9 (c+d x)}+\frac {28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac {70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac {14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac {28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}+\frac {4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac {(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac {b^8 x}{d^8} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^8}{d^8}+\frac {(-b c+a d)^8}{d^8 (c+d x)^8}-\frac {8 b (b c-a d)^7}{d^8 (c+d x)^7}+\frac {28 b^2 (b c-a d)^6}{d^8 (c+d x)^6}-\frac {56 b^3 (b c-a d)^5}{d^8 (c+d x)^5}+\frac {70 b^4 (b c-a d)^4}{d^8 (c+d x)^4}-\frac {56 b^5 (b c-a d)^3}{d^8 (c+d x)^3}+\frac {28 b^6 (b c-a d)^2}{d^8 (c+d x)^2}-\frac {8 b^7 (b c-a d)}{d^8 (c+d x)}\right ) \, dx \\ & = \frac {b^8 x}{d^8}-\frac {(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac {4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac {28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}+\frac {14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac {70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac {28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac {28 b^6 (b c-a d)^2}{d^9 (c+d x)}-\frac {8 b^7 (b c-a d) \log (c+d x)}{d^9} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(474\) vs. \(2(209)=418\).
Time = 0.12 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.27 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=-\frac {15 a^8 d^8+20 a^7 b d^7 (c+7 d x)+28 a^6 b^2 d^6 \left (c^2+7 c d x+21 d^2 x^2\right )+42 a^5 b^3 d^5 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+70 a^4 b^4 d^4 \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )+140 a^3 b^5 d^3 \left (c^5+7 c^4 d x+21 c^3 d^2 x^2+35 c^2 d^3 x^3+35 c d^4 x^4+21 d^5 x^5\right )+420 a^2 b^6 d^2 \left (c^6+7 c^5 d x+21 c^4 d^2 x^2+35 c^3 d^3 x^3+35 c^2 d^4 x^4+21 c d^5 x^5+7 d^6 x^6\right )-2 a b^7 c d \left (1089 c^6+7203 c^5 d x+20139 c^4 d^2 x^2+30625 c^3 d^3 x^3+26950 c^2 d^4 x^4+13230 c d^5 x^5+2940 d^6 x^6\right )+b^8 \left (1443 c^8+9261 c^7 d x+24843 c^6 d^2 x^2+35525 c^5 d^3 x^3+28175 c^4 d^4 x^4+11025 c^3 d^5 x^5+735 c^2 d^6 x^6-735 c d^7 x^7-105 d^8 x^8\right )+840 b^7 (b c-a d) (c+d x)^7 \log (c+d x)}{105 d^9 (c+d x)^7} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(569\) vs. \(2(201)=402\).
Time = 0.23 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.73
method | result | size |
risch | \(\frac {b^{8} x}{d^{8}}+\frac {\left (-28 a^{2} b^{6} d^{7}+56 a \,b^{7} c \,d^{6}-28 b^{8} c^{2} d^{5}\right ) x^{6}-28 b^{5} d^{4} \left (a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +5 b^{3} c^{3}\right ) x^{5}-\frac {70 b^{4} d^{3} \left (a^{4} d^{4}+2 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-22 a \,b^{3} c^{3} d +13 b^{4} c^{4}\right ) x^{4}}{3}-\frac {14 b^{3} d^{2} \left (3 a^{5} d^{5}+5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}+30 a^{2} b^{3} c^{3} d^{2}-125 a \,b^{4} c^{4} d +77 b^{5} c^{5}\right ) x^{3}}{3}-\frac {14 b^{2} d \left (2 a^{6} d^{6}+3 a^{5} b c \,d^{5}+5 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+30 a^{2} b^{4} c^{4} d^{2}-137 a \,b^{5} c^{5} d +87 b^{6} c^{6}\right ) x^{2}}{5}-\frac {2 b \left (10 a^{7} d^{7}+14 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+210 a^{2} b^{5} c^{5} d^{2}-1029 a \,b^{6} c^{6} d +669 b^{7} c^{7}\right ) x}{15}-\frac {15 a^{8} d^{8}+20 a^{7} b c \,d^{7}+28 a^{6} b^{2} d^{6} c^{2}+42 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} d^{4} c^{4}+140 a^{3} b^{5} c^{5} d^{3}+420 a^{2} b^{6} d^{2} c^{6}-2178 a \,b^{7} c^{7} d +1443 b^{8} c^{8}}{105 d}}{d^{8} \left (d x +c \right )^{7}}+\frac {8 b^{7} \ln \left (d x +c \right ) a}{d^{8}}-\frac {8 b^{8} \ln \left (d x +c \right ) c}{d^{9}}\) | \(570\) |
default | \(\frac {b^{8} x}{d^{8}}-\frac {14 b^{3} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{d^{9} \left (d x +c \right )^{4}}-\frac {28 b^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{9} \left (d x +c \right )^{2}}+\frac {8 b^{7} \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{9}}-\frac {4 b \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{3 d^{9} \left (d x +c \right )^{6}}-\frac {28 b^{2} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{5 d^{9} \left (d x +c \right )^{5}}-\frac {a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} d^{6} c^{2}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} d^{4} c^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} d^{2} c^{6}-8 a \,b^{7} c^{7} d +b^{8} c^{8}}{7 d^{9} \left (d x +c \right )^{7}}-\frac {70 b^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{3 d^{9} \left (d x +c \right )^{3}}-\frac {28 b^{6} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{9} \left (d x +c \right )}\) | \(576\) |
norman | \(\frac {\frac {b^{8} x^{8}}{d}-\frac {15 a^{8} d^{8}+20 a^{7} b c \,d^{7}+28 a^{6} b^{2} d^{6} c^{2}+42 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} d^{4} c^{4}+140 a^{3} b^{5} c^{5} d^{3}+420 a^{2} b^{6} d^{2} c^{6}-2178 a \,b^{7} c^{7} d +2178 b^{8} c^{8}}{105 d^{9}}-\frac {7 \left (4 a^{2} b^{6} d^{2}-8 a \,b^{7} c d +8 b^{8} c^{2}\right ) x^{6}}{d^{3}}-\frac {7 \left (4 a^{3} b^{5} d^{3}+12 a^{2} b^{6} c \,d^{2}-36 a \,b^{7} c^{2} d +36 b^{8} c^{3}\right ) x^{5}}{d^{4}}-\frac {35 \left (2 a^{4} b^{4} d^{4}+4 a^{3} b^{5} c \,d^{3}+12 a^{2} b^{6} c^{2} d^{2}-44 a \,b^{7} c^{3} d +44 b^{8} c^{4}\right ) x^{4}}{3 d^{5}}-\frac {7 \left (6 a^{5} b^{3} d^{5}+10 a^{4} b^{4} c \,d^{4}+20 a^{3} b^{5} c^{2} d^{3}+60 a^{2} b^{6} c^{3} d^{2}-250 a \,b^{7} c^{4} d +250 b^{8} c^{5}\right ) x^{3}}{3 d^{6}}-\frac {7 \left (4 a^{6} b^{2} d^{6}+6 a^{5} b^{3} c \,d^{5}+10 a^{4} b^{4} c^{2} d^{4}+20 a^{3} b^{5} c^{3} d^{3}+60 a^{2} b^{6} c^{4} d^{2}-274 a \,b^{7} c^{5} d +274 b^{8} c^{6}\right ) x^{2}}{5 d^{7}}-\frac {\left (20 a^{7} b \,d^{7}+28 a^{6} b^{2} c \,d^{6}+42 a^{5} b^{3} c^{2} d^{5}+70 a^{4} b^{4} c^{3} d^{4}+140 a^{3} b^{5} c^{4} d^{3}+420 a^{2} b^{6} c^{5} d^{2}-2058 a \,b^{7} c^{6} d +2058 b^{8} c^{7}\right ) x}{15 d^{8}}}{\left (d x +c \right )^{7}}+\frac {8 b^{7} \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{9}}\) | \(577\) |
parallelrisch | \(\frac {-8820 x^{2} a^{2} b^{6} c^{4} d^{4}-14700 x^{4} a^{2} b^{6} c^{2} d^{6}+53900 x^{4} a \,b^{7} c^{3} d^{5}-8820 x^{5} a^{2} b^{6} c \,d^{7}+26460 x^{5} a \,b^{7} c^{2} d^{6}+5880 x^{6} a \,b^{7} c \,d^{7}+840 \ln \left (d x +c \right ) x^{7} a \,b^{7} d^{8}-840 \ln \left (d x +c \right ) x^{7} b^{8} c \,d^{7}-5880 \ln \left (d x +c \right ) x^{6} b^{8} c^{2} d^{6}-17640 \ln \left (d x +c \right ) x^{5} b^{8} c^{3} d^{5}+40278 x^{2} a \,b^{7} c^{5} d^{3}-2450 x^{3} a^{4} b^{4} c \,d^{7}-4900 x^{3} a^{3} b^{5} c^{2} d^{6}-14700 x^{3} a^{2} b^{6} c^{3} d^{5}+61250 x^{3} a \,b^{7} c^{4} d^{4}-4900 x^{4} a^{3} b^{5} c \,d^{7}-196 x \,a^{6} b^{2} c \,d^{7}-294 x \,a^{5} b^{3} c^{2} d^{6}-490 x \,a^{4} b^{4} c^{3} d^{5}-29400 \ln \left (d x +c \right ) x^{4} b^{8} c^{4} d^{4}-29400 \ln \left (d x +c \right ) x^{3} b^{8} c^{5} d^{3}-17640 \ln \left (d x +c \right ) x^{2} b^{8} c^{6} d^{2}-5880 \ln \left (d x +c \right ) x \,b^{8} c^{7} d +840 \ln \left (d x +c \right ) a \,b^{7} c^{7} d -980 x \,a^{3} b^{5} c^{4} d^{4}-2940 x \,a^{2} b^{6} c^{5} d^{3}+14406 x a \,b^{7} c^{6} d^{2}-882 x^{2} a^{5} b^{3} c \,d^{7}-1470 x^{2} a^{4} b^{4} c^{2} d^{6}-2940 x^{2} a^{3} b^{5} c^{3} d^{5}+29400 \ln \left (d x +c \right ) x^{3} a \,b^{7} c^{4} d^{4}+17640 \ln \left (d x +c \right ) x^{2} a \,b^{7} c^{5} d^{3}+5880 \ln \left (d x +c \right ) x a \,b^{7} c^{6} d^{2}+5880 \ln \left (d x +c \right ) x^{6} a \,b^{7} c \,d^{7}+17640 \ln \left (d x +c \right ) x^{5} a \,b^{7} c^{2} d^{6}+29400 \ln \left (d x +c \right ) x^{4} a \,b^{7} c^{3} d^{5}-14406 b^{8} c^{7} d x -140 a^{7} b \,d^{8} x -40278 b^{8} c^{6} d^{2} x^{2}-588 a^{6} b^{2} d^{8} x^{2}-61250 b^{8} c^{5} d^{3} x^{3}-1470 a^{5} b^{3} d^{8} x^{3}-53900 b^{8} c^{4} d^{4} x^{4}-2450 a^{4} b^{4} d^{8} x^{4}-26460 b^{8} c^{3} d^{5} x^{5}-2940 a^{3} b^{5} d^{8} x^{5}-5880 b^{8} c^{2} d^{6} x^{6}-2940 a^{2} b^{6} d^{8} x^{6}+105 x^{8} b^{8} d^{8}-420 a^{2} b^{6} d^{2} c^{6}-28 a^{6} b^{2} d^{6} c^{2}-70 a^{4} b^{4} d^{4} c^{4}-140 a^{3} b^{5} c^{5} d^{3}-840 \ln \left (d x +c \right ) b^{8} c^{8}-42 a^{5} b^{3} c^{3} d^{5}-15 a^{8} d^{8}-2178 b^{8} c^{8}-20 a^{7} b c \,d^{7}+2178 a \,b^{7} c^{7} d}{105 d^{9} \left (d x +c \right )^{7}}\) | \(916\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (201) = 402\).
Time = 0.22 (sec) , antiderivative size = 852, normalized size of antiderivative = 4.08 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\frac {105 \, b^{8} d^{8} x^{8} + 735 \, b^{8} c d^{7} x^{7} - 1443 \, b^{8} c^{8} + 2178 \, a b^{7} c^{7} d - 420 \, a^{2} b^{6} c^{6} d^{2} - 140 \, a^{3} b^{5} c^{5} d^{3} - 70 \, a^{4} b^{4} c^{4} d^{4} - 42 \, a^{5} b^{3} c^{3} d^{5} - 28 \, a^{6} b^{2} c^{2} d^{6} - 20 \, a^{7} b c d^{7} - 15 \, a^{8} d^{8} - 735 \, {\left (b^{8} c^{2} d^{6} - 8 \, a b^{7} c d^{7} + 4 \, a^{2} b^{6} d^{8}\right )} x^{6} - 735 \, {\left (15 \, b^{8} c^{3} d^{5} - 36 \, a b^{7} c^{2} d^{6} + 12 \, a^{2} b^{6} c d^{7} + 4 \, a^{3} b^{5} d^{8}\right )} x^{5} - 1225 \, {\left (23 \, b^{8} c^{4} d^{4} - 44 \, a b^{7} c^{3} d^{5} + 12 \, a^{2} b^{6} c^{2} d^{6} + 4 \, a^{3} b^{5} c d^{7} + 2 \, a^{4} b^{4} d^{8}\right )} x^{4} - 245 \, {\left (145 \, b^{8} c^{5} d^{3} - 250 \, a b^{7} c^{4} d^{4} + 60 \, a^{2} b^{6} c^{3} d^{5} + 20 \, a^{3} b^{5} c^{2} d^{6} + 10 \, a^{4} b^{4} c d^{7} + 6 \, a^{5} b^{3} d^{8}\right )} x^{3} - 147 \, {\left (169 \, b^{8} c^{6} d^{2} - 274 \, a b^{7} c^{5} d^{3} + 60 \, a^{2} b^{6} c^{4} d^{4} + 20 \, a^{3} b^{5} c^{3} d^{5} + 10 \, a^{4} b^{4} c^{2} d^{6} + 6 \, a^{5} b^{3} c d^{7} + 4 \, a^{6} b^{2} d^{8}\right )} x^{2} - 7 \, {\left (1323 \, b^{8} c^{7} d - 2058 \, a b^{7} c^{6} d^{2} + 420 \, a^{2} b^{6} c^{5} d^{3} + 140 \, a^{3} b^{5} c^{4} d^{4} + 70 \, a^{4} b^{4} c^{3} d^{5} + 42 \, a^{5} b^{3} c^{2} d^{6} + 28 \, a^{6} b^{2} c d^{7} + 20 \, a^{7} b d^{8}\right )} x - 840 \, {\left (b^{8} c^{8} - a b^{7} c^{7} d + {\left (b^{8} c d^{7} - a b^{7} d^{8}\right )} x^{7} + 7 \, {\left (b^{8} c^{2} d^{6} - a b^{7} c d^{7}\right )} x^{6} + 21 \, {\left (b^{8} c^{3} d^{5} - a b^{7} c^{2} d^{6}\right )} x^{5} + 35 \, {\left (b^{8} c^{4} d^{4} - a b^{7} c^{3} d^{5}\right )} x^{4} + 35 \, {\left (b^{8} c^{5} d^{3} - a b^{7} c^{4} d^{4}\right )} x^{3} + 21 \, {\left (b^{8} c^{6} d^{2} - a b^{7} c^{5} d^{3}\right )} x^{2} + 7 \, {\left (b^{8} c^{7} d - a b^{7} c^{6} d^{2}\right )} x\right )} \log \left (d x + c\right )}{105 \, {\left (d^{16} x^{7} + 7 \, c d^{15} x^{6} + 21 \, c^{2} d^{14} x^{5} + 35 \, c^{3} d^{13} x^{4} + 35 \, c^{4} d^{12} x^{3} + 21 \, c^{5} d^{11} x^{2} + 7 \, c^{6} d^{10} x + c^{7} d^{9}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (201) = 402\).
Time = 0.24 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\frac {b^{8} x}{d^{8}} - \frac {1443 \, b^{8} c^{8} - 2178 \, a b^{7} c^{7} d + 420 \, a^{2} b^{6} c^{6} d^{2} + 140 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} + 42 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} + 20 \, a^{7} b c d^{7} + 15 \, a^{8} d^{8} + 2940 \, {\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{6} + 2940 \, {\left (5 \, b^{8} c^{3} d^{5} - 9 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x^{5} + 2450 \, {\left (13 \, b^{8} c^{4} d^{4} - 22 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} + 2 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 490 \, {\left (77 \, b^{8} c^{5} d^{3} - 125 \, a b^{7} c^{4} d^{4} + 30 \, a^{2} b^{6} c^{3} d^{5} + 10 \, a^{3} b^{5} c^{2} d^{6} + 5 \, a^{4} b^{4} c d^{7} + 3 \, a^{5} b^{3} d^{8}\right )} x^{3} + 294 \, {\left (87 \, b^{8} c^{6} d^{2} - 137 \, a b^{7} c^{5} d^{3} + 30 \, a^{2} b^{6} c^{4} d^{4} + 10 \, a^{3} b^{5} c^{3} d^{5} + 5 \, a^{4} b^{4} c^{2} d^{6} + 3 \, a^{5} b^{3} c d^{7} + 2 \, a^{6} b^{2} d^{8}\right )} x^{2} + 14 \, {\left (669 \, b^{8} c^{7} d - 1029 \, a b^{7} c^{6} d^{2} + 210 \, a^{2} b^{6} c^{5} d^{3} + 70 \, a^{3} b^{5} c^{4} d^{4} + 35 \, a^{4} b^{4} c^{3} d^{5} + 21 \, a^{5} b^{3} c^{2} d^{6} + 14 \, a^{6} b^{2} c d^{7} + 10 \, a^{7} b d^{8}\right )} x}{105 \, {\left (d^{16} x^{7} + 7 \, c d^{15} x^{6} + 21 \, c^{2} d^{14} x^{5} + 35 \, c^{3} d^{13} x^{4} + 35 \, c^{4} d^{12} x^{3} + 21 \, c^{5} d^{11} x^{2} + 7 \, c^{6} d^{10} x + c^{7} d^{9}\right )}} - \frac {8 \, {\left (b^{8} c - a b^{7} d\right )} \log \left (d x + c\right )}{d^{9}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (201) = 402\).
Time = 0.29 (sec) , antiderivative size = 581, normalized size of antiderivative = 2.78 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\frac {b^{8} x}{d^{8}} - \frac {8 \, {\left (b^{8} c - a b^{7} d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{9}} - \frac {1443 \, b^{8} c^{8} - 2178 \, a b^{7} c^{7} d + 420 \, a^{2} b^{6} c^{6} d^{2} + 140 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} + 42 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} + 20 \, a^{7} b c d^{7} + 15 \, a^{8} d^{8} + 2940 \, {\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{6} + 2940 \, {\left (5 \, b^{8} c^{3} d^{5} - 9 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x^{5} + 2450 \, {\left (13 \, b^{8} c^{4} d^{4} - 22 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} + 2 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 490 \, {\left (77 \, b^{8} c^{5} d^{3} - 125 \, a b^{7} c^{4} d^{4} + 30 \, a^{2} b^{6} c^{3} d^{5} + 10 \, a^{3} b^{5} c^{2} d^{6} + 5 \, a^{4} b^{4} c d^{7} + 3 \, a^{5} b^{3} d^{8}\right )} x^{3} + 294 \, {\left (87 \, b^{8} c^{6} d^{2} - 137 \, a b^{7} c^{5} d^{3} + 30 \, a^{2} b^{6} c^{4} d^{4} + 10 \, a^{3} b^{5} c^{3} d^{5} + 5 \, a^{4} b^{4} c^{2} d^{6} + 3 \, a^{5} b^{3} c d^{7} + 2 \, a^{6} b^{2} d^{8}\right )} x^{2} + 14 \, {\left (669 \, b^{8} c^{7} d - 1029 \, a b^{7} c^{6} d^{2} + 210 \, a^{2} b^{6} c^{5} d^{3} + 70 \, a^{3} b^{5} c^{4} d^{4} + 35 \, a^{4} b^{4} c^{3} d^{5} + 21 \, a^{5} b^{3} c^{2} d^{6} + 14 \, a^{6} b^{2} c d^{7} + 10 \, a^{7} b d^{8}\right )} x}{105 \, {\left (d x + c\right )}^{7} d^{9}} \]
[In]
[Out]
Time = 0.47 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^8}{(c+d x)^8} \, dx=\frac {b^8\,x}{d^8}-\frac {\ln \left (c+d\,x\right )\,\left (8\,b^8\,c-8\,a\,b^7\,d\right )}{d^9}-\frac {x^4\,\left (\frac {70\,a^4\,b^4\,d^7}{3}+\frac {140\,a^3\,b^5\,c\,d^6}{3}+140\,a^2\,b^6\,c^2\,d^5-\frac {1540\,a\,b^7\,c^3\,d^4}{3}+\frac {910\,b^8\,c^4\,d^3}{3}\right )+x^6\,\left (28\,a^2\,b^6\,d^7-56\,a\,b^7\,c\,d^6+28\,b^8\,c^2\,d^5\right )+\frac {15\,a^8\,d^8+20\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6+42\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4+140\,a^3\,b^5\,c^5\,d^3+420\,a^2\,b^6\,c^6\,d^2-2178\,a\,b^7\,c^7\,d+1443\,b^8\,c^8}{105\,d}+x\,\left (\frac {4\,a^7\,b\,d^7}{3}+\frac {28\,a^6\,b^2\,c\,d^6}{15}+\frac {14\,a^5\,b^3\,c^2\,d^5}{5}+\frac {14\,a^4\,b^4\,c^3\,d^4}{3}+\frac {28\,a^3\,b^5\,c^4\,d^3}{3}+28\,a^2\,b^6\,c^5\,d^2-\frac {686\,a\,b^7\,c^6\,d}{5}+\frac {446\,b^8\,c^7}{5}\right )+x^3\,\left (14\,a^5\,b^3\,d^7+\frac {70\,a^4\,b^4\,c\,d^6}{3}+\frac {140\,a^3\,b^5\,c^2\,d^5}{3}+140\,a^2\,b^6\,c^3\,d^4-\frac {1750\,a\,b^7\,c^4\,d^3}{3}+\frac {1078\,b^8\,c^5\,d^2}{3}\right )+x^2\,\left (\frac {28\,a^6\,b^2\,d^7}{5}+\frac {42\,a^5\,b^3\,c\,d^6}{5}+14\,a^4\,b^4\,c^2\,d^5+28\,a^3\,b^5\,c^3\,d^4+84\,a^2\,b^6\,c^4\,d^3-\frac {1918\,a\,b^7\,c^5\,d^2}{5}+\frac {1218\,b^8\,c^6\,d}{5}\right )+x^5\,\left (28\,a^3\,b^5\,d^7+84\,a^2\,b^6\,c\,d^6-252\,a\,b^7\,c^2\,d^5+140\,b^8\,c^3\,d^4\right )}{c^7\,d^8+7\,c^6\,d^9\,x+21\,c^5\,d^{10}\,x^2+35\,c^4\,d^{11}\,x^3+35\,c^3\,d^{12}\,x^4+21\,c^2\,d^{13}\,x^5+7\,c\,d^{14}\,x^6+d^{15}\,x^7} \]
[In]
[Out]