Integrand size = 15, antiderivative size = 258 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=\frac {45 d^8 (b c-a d)^2 x}{b^{10}}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}+\frac {5 d^9 (b c-a d) (a+b x)^2}{b^{11}}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}} \]
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Time = 0.25 (sec) , antiderivative size = 258, 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} \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=\frac {5 d^9 (a+b x)^2 (b c-a d)}{b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {45 d^8 x (b c-a d)^2}{b^{10}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {45 d^8 (b c-a d)^2}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^8}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^7}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^6}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^5}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^4}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^3}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^2}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)}+\frac {10 d^9 (b c-a d) (a+b x)}{b^{10}}+\frac {d^{10} (a+b x)^2}{b^{10}}\right ) \, dx \\ & = \frac {45 d^8 (b c-a d)^2 x}{b^{10}}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}+\frac {5 d^9 (b c-a d) (a+b x)^2}{b^{11}}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=\frac {21 b d^8 \left (45 b^2 c^2-80 a b c d+36 a^2 d^2\right ) x+21 b^2 d^9 (5 b c-4 a d) x^2+7 b^3 d^{10} x^3-\frac {3 (b c-a d)^{10}}{(a+b x)^7}+\frac {35 d (-b c+a d)^9}{(a+b x)^6}-\frac {189 d^2 (b c-a d)^8}{(a+b x)^5}+\frac {630 d^3 (-b c+a d)^7}{(a+b x)^4}-\frac {1470 d^4 (b c-a d)^6}{(a+b x)^3}+\frac {2646 d^5 (-b c+a d)^5}{(a+b x)^2}-\frac {4410 d^6 (b c-a d)^4}{a+b x}+2520 d^7 (b c-a d)^3 \log (a+b x)}{21 b^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(853\) vs. \(2(252)=504\).
Time = 0.24 (sec) , antiderivative size = 854, normalized size of antiderivative = 3.31
method | result | size |
norman | \(\frac {-\frac {6534 a^{10} d^{10}-19602 a^{9} b c \,d^{9}+19602 a^{8} b^{2} c^{2} d^{8}-6534 a^{7} b^{3} c^{3} d^{7}+630 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+42 a^{4} b^{6} c^{6} d^{4}+18 a^{3} b^{7} c^{7} d^{3}+9 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +3 b^{10} c^{10}}{21 b^{11}}+\frac {d^{10} x^{10}}{3 b}-\frac {7 \left (120 a^{4} d^{10}-360 a^{3} b c \,d^{9}+360 a^{2} b^{2} c^{2} d^{8}-120 a \,b^{3} c^{3} d^{7}+30 b^{4} c^{4} d^{6}\right ) x^{6}}{b^{5}}-\frac {21 \left (180 a^{5} d^{10}-540 a^{4} b c \,d^{9}+540 a^{3} b^{2} c^{2} d^{8}-180 a^{2} b^{3} c^{3} d^{7}+30 a \,b^{4} c^{4} d^{6}+6 b^{5} c^{5} d^{5}\right ) x^{5}}{b^{6}}-\frac {35 \left (220 a^{6} d^{10}-660 a^{5} b c \,d^{9}+660 a^{4} b^{2} c^{2} d^{8}-220 a^{3} b^{3} c^{3} d^{7}+30 a^{2} b^{4} c^{4} d^{6}+6 a \,b^{5} c^{5} d^{5}+2 b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}-\frac {5 \left (1750 a^{7} d^{10}-5250 a^{6} b c \,d^{9}+5250 a^{5} b^{2} c^{2} d^{8}-1750 a^{4} b^{3} c^{3} d^{7}+210 a^{3} b^{4} c^{4} d^{6}+42 a^{2} b^{5} c^{5} d^{5}+14 a \,b^{6} c^{6} d^{4}+6 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}-\frac {3 \left (1918 a^{8} d^{10}-5754 a^{7} b c \,d^{9}+5754 a^{6} b^{2} c^{2} d^{8}-1918 a^{5} b^{3} c^{3} d^{7}+210 a^{4} b^{4} c^{4} d^{6}+42 a^{3} b^{5} c^{5} d^{5}+14 a^{2} b^{6} c^{6} d^{4}+6 a \,b^{7} c^{7} d^{3}+3 b^{8} c^{8} d^{2}\right ) x^{2}}{b^{9}}-\frac {\left (6174 a^{9} d^{10}-18522 a^{8} b c \,d^{9}+18522 a^{7} b^{2} c^{2} d^{8}-6174 a^{6} b^{3} c^{3} d^{7}+630 a^{5} b^{4} c^{4} d^{6}+126 a^{4} b^{5} c^{5} d^{5}+42 a^{3} b^{6} c^{6} d^{4}+18 a^{2} b^{7} c^{7} d^{3}+9 a \,b^{8} c^{8} d^{2}+5 b^{9} c^{9} d \right ) x}{3 b^{10}}+\frac {15 d^{8} \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{8}}{b^{3}}-\frac {5 d^{9} \left (a d -3 b c \right ) x^{9}}{3 b^{2}}}{\left (b x +a \right )^{7}}-\frac {120 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{11}}\) | \(854\) |
default | \(\frac {d^{8} \left (\frac {1}{3} d^{2} x^{3} b^{2}-4 x^{2} a b \,d^{2}+5 x^{2} b^{2} c d +36 a^{2} d^{2} x -80 a b c d x +45 b^{2} c^{2} x \right )}{b^{10}}-\frac {70 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{b^{11} \left (b x +a \right )^{3}}-\frac {120 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{11}}+\frac {5 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{3 b^{11} \left (b x +a \right )^{6}}+\frac {30 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{4}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{7 b^{11} \left (b x +a \right )^{7}}+\frac {126 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{b^{11} \left (b x +a \right )^{2}}-\frac {9 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}{b^{11} \left (b x +a \right )^{5}}-\frac {210 d^{6} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{b^{11} \left (b x +a \right )}\) | \(856\) |
risch | \(\frac {d^{10} x^{3}}{3 b^{8}}-\frac {4 d^{10} x^{2} a}{b^{9}}+\frac {5 d^{9} x^{2} c}{b^{8}}+\frac {36 d^{10} a^{2} x}{b^{10}}-\frac {80 d^{9} a c x}{b^{9}}+\frac {45 d^{8} c^{2} x}{b^{8}}+\frac {\left (-210 a^{4} b^{5} d^{10}+840 a^{3} b^{6} c \,d^{9}-1260 a^{2} b^{7} c^{2} d^{8}+840 a \,b^{8} c^{3} d^{7}-210 b^{9} c^{4} d^{6}\right ) x^{6}-126 b^{4} d^{5} \left (9 a^{5} d^{5}-35 a^{4} b c \,d^{4}+50 a^{3} b^{2} c^{2} d^{3}-30 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x^{5}-70 b^{3} d^{4} \left (37 a^{6} d^{6}-141 a^{5} b c \,d^{5}+195 a^{4} b^{2} c^{2} d^{4}-110 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) x^{4}-10 b^{2} d^{3} \left (319 a^{7} d^{7}-1197 a^{6} b c \,d^{6}+1617 a^{5} b^{2} c^{2} d^{5}-875 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +3 b^{7} c^{7}\right ) x^{3}-3 b \,d^{2} \left (743 a^{8} d^{8}-2754 a^{7} b c \,d^{7}+3654 a^{6} b^{2} c^{2} d^{6}-1918 a^{5} b^{3} c^{3} d^{5}+210 a^{4} b^{4} c^{4} d^{4}+42 a^{3} b^{5} c^{5} d^{3}+14 a^{2} b^{6} c^{6} d^{2}+6 a \,b^{7} c^{7} d +3 b^{8} c^{8}\right ) x^{2}-\frac {d \left (2509 a^{9} d^{9}-9207 a^{8} b c \,d^{8}+12042 a^{7} b^{2} c^{2} d^{7}-6174 a^{6} b^{3} c^{3} d^{6}+630 a^{5} b^{4} c^{4} d^{5}+126 a^{4} b^{5} c^{5} d^{4}+42 a^{3} b^{6} c^{6} d^{3}+18 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d +5 b^{9} c^{9}\right ) x}{3}-\frac {2761 a^{10} d^{10}-10047 a^{9} b c \,d^{9}+12987 a^{8} b^{2} c^{2} d^{8}-6534 a^{7} b^{3} c^{3} d^{7}+630 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+42 a^{4} b^{6} c^{6} d^{4}+18 a^{3} b^{7} c^{7} d^{3}+9 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +3 b^{10} c^{10}}{21 b}}{b^{10} \left (b x +a \right )^{7}}-\frac {120 d^{10} \ln \left (b x +a \right ) a^{3}}{b^{11}}+\frac {360 d^{9} \ln \left (b x +a \right ) a^{2} c}{b^{10}}-\frac {360 d^{8} \ln \left (b x +a \right ) a \,c^{2}}{b^{9}}+\frac {120 d^{7} \ln \left (b x +a \right ) c^{3}}{b^{8}}\) | \(868\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1567\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1362 vs. \(2 (252) = 504\).
Time = 0.25 (sec) , antiderivative size = 1362, normalized size of antiderivative = 5.28 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 934 vs. \(2 (252) = 504\).
Time = 0.27 (sec) , antiderivative size = 934, normalized size of antiderivative = 3.62 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=-\frac {3 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 18 \, a^{3} b^{7} c^{7} d^{3} + 42 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} - 6534 \, a^{7} b^{3} c^{3} d^{7} + 12987 \, a^{8} b^{2} c^{2} d^{8} - 10047 \, a^{9} b c d^{9} + 2761 \, a^{10} d^{10} + 4410 \, {\left (b^{10} c^{4} d^{6} - 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 4 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 2646 \, {\left (b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} - 30 \, a^{2} b^{8} c^{3} d^{7} + 50 \, a^{3} b^{7} c^{2} d^{8} - 35 \, a^{4} b^{6} c d^{9} + 9 \, a^{5} b^{5} d^{10}\right )} x^{5} + 1470 \, {\left (b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 110 \, a^{3} b^{7} c^{3} d^{7} + 195 \, a^{4} b^{6} c^{2} d^{8} - 141 \, a^{5} b^{5} c d^{9} + 37 \, a^{6} b^{4} d^{10}\right )} x^{4} + 210 \, {\left (3 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} - 875 \, a^{4} b^{6} c^{3} d^{7} + 1617 \, a^{5} b^{5} c^{2} d^{8} - 1197 \, a^{6} b^{4} c d^{9} + 319 \, a^{7} b^{3} d^{10}\right )} x^{3} + 63 \, {\left (3 \, b^{10} c^{8} d^{2} + 6 \, a b^{9} c^{7} d^{3} + 14 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} - 1918 \, a^{5} b^{5} c^{3} d^{7} + 3654 \, a^{6} b^{4} c^{2} d^{8} - 2754 \, a^{7} b^{3} c d^{9} + 743 \, a^{8} b^{2} d^{10}\right )} x^{2} + 7 \, {\left (5 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 18 \, a^{2} b^{8} c^{7} d^{3} + 42 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} - 6174 \, a^{6} b^{4} c^{3} d^{7} + 12042 \, a^{7} b^{3} c^{2} d^{8} - 9207 \, a^{8} b^{2} c d^{9} + 2509 \, a^{9} b d^{10}\right )} x}{21 \, {\left (b^{18} x^{7} + 7 \, a b^{17} x^{6} + 21 \, a^{2} b^{16} x^{5} + 35 \, a^{3} b^{15} x^{4} + 35 \, a^{4} b^{14} x^{3} + 21 \, a^{5} b^{13} x^{2} + 7 \, a^{6} b^{12} x + a^{7} b^{11}\right )}} + \frac {b^{2} d^{10} x^{3} + 3 \, {\left (5 \, b^{2} c d^{9} - 4 \, a b d^{10}\right )} x^{2} + 3 \, {\left (45 \, b^{2} c^{2} d^{8} - 80 \, a b c d^{9} + 36 \, a^{2} d^{10}\right )} x}{3 \, b^{10}} + \frac {120 \, {\left (b^{3} c^{3} d^{7} - 3 \, a b^{2} c^{2} d^{8} + 3 \, a^{2} b c d^{9} - a^{3} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (252) = 504\).
Time = 0.37 (sec) , antiderivative size = 872, normalized size of antiderivative = 3.38 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=\frac {120 \, {\left (b^{3} c^{3} d^{7} - 3 \, a b^{2} c^{2} d^{8} + 3 \, a^{2} b c d^{9} - a^{3} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {3 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 18 \, a^{3} b^{7} c^{7} d^{3} + 42 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} - 6534 \, a^{7} b^{3} c^{3} d^{7} + 12987 \, a^{8} b^{2} c^{2} d^{8} - 10047 \, a^{9} b c d^{9} + 2761 \, a^{10} d^{10} + 4410 \, {\left (b^{10} c^{4} d^{6} - 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 4 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 2646 \, {\left (b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} - 30 \, a^{2} b^{8} c^{3} d^{7} + 50 \, a^{3} b^{7} c^{2} d^{8} - 35 \, a^{4} b^{6} c d^{9} + 9 \, a^{5} b^{5} d^{10}\right )} x^{5} + 1470 \, {\left (b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 110 \, a^{3} b^{7} c^{3} d^{7} + 195 \, a^{4} b^{6} c^{2} d^{8} - 141 \, a^{5} b^{5} c d^{9} + 37 \, a^{6} b^{4} d^{10}\right )} x^{4} + 210 \, {\left (3 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} - 875 \, a^{4} b^{6} c^{3} d^{7} + 1617 \, a^{5} b^{5} c^{2} d^{8} - 1197 \, a^{6} b^{4} c d^{9} + 319 \, a^{7} b^{3} d^{10}\right )} x^{3} + 63 \, {\left (3 \, b^{10} c^{8} d^{2} + 6 \, a b^{9} c^{7} d^{3} + 14 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} - 1918 \, a^{5} b^{5} c^{3} d^{7} + 3654 \, a^{6} b^{4} c^{2} d^{8} - 2754 \, a^{7} b^{3} c d^{9} + 743 \, a^{8} b^{2} d^{10}\right )} x^{2} + 7 \, {\left (5 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 18 \, a^{2} b^{8} c^{7} d^{3} + 42 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} - 6174 \, a^{6} b^{4} c^{3} d^{7} + 12042 \, a^{7} b^{3} c^{2} d^{8} - 9207 \, a^{8} b^{2} c d^{9} + 2509 \, a^{9} b d^{10}\right )} x}{21 \, {\left (b x + a\right )}^{7} b^{11}} + \frac {b^{16} d^{10} x^{3} + 15 \, b^{16} c d^{9} x^{2} - 12 \, a b^{15} d^{10} x^{2} + 135 \, b^{16} c^{2} d^{8} x - 240 \, a b^{15} c d^{9} x + 108 \, a^{2} b^{14} d^{10} x}{3 \, b^{24}} \]
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Time = 0.47 (sec) , antiderivative size = 950, normalized size of antiderivative = 3.68 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=x\,\left (\frac {8\,a\,\left (\frac {8\,a\,d^{10}}{b^9}-\frac {10\,c\,d^9}{b^8}\right )}{b}-\frac {28\,a^2\,d^{10}}{b^{10}}+\frac {45\,c^2\,d^8}{b^8}\right )-\frac {x^4\,\left (2590\,a^6\,b^3\,d^{10}-9870\,a^5\,b^4\,c\,d^9+13650\,a^4\,b^5\,c^2\,d^8-7700\,a^3\,b^6\,c^3\,d^7+1050\,a^2\,b^7\,c^4\,d^6+210\,a\,b^8\,c^5\,d^5+70\,b^9\,c^6\,d^4\right )+x^6\,\left (210\,a^4\,b^5\,d^{10}-840\,a^3\,b^6\,c\,d^9+1260\,a^2\,b^7\,c^2\,d^8-840\,a\,b^8\,c^3\,d^7+210\,b^9\,c^4\,d^6\right )+\frac {2761\,a^{10}\,d^{10}-10047\,a^9\,b\,c\,d^9+12987\,a^8\,b^2\,c^2\,d^8-6534\,a^7\,b^3\,c^3\,d^7+630\,a^6\,b^4\,c^4\,d^6+126\,a^5\,b^5\,c^5\,d^5+42\,a^4\,b^6\,c^6\,d^4+18\,a^3\,b^7\,c^7\,d^3+9\,a^2\,b^8\,c^8\,d^2+5\,a\,b^9\,c^9\,d+3\,b^{10}\,c^{10}}{21\,b}+x\,\left (\frac {2509\,a^9\,d^{10}}{3}-3069\,a^8\,b\,c\,d^9+4014\,a^7\,b^2\,c^2\,d^8-2058\,a^6\,b^3\,c^3\,d^7+210\,a^5\,b^4\,c^4\,d^6+42\,a^4\,b^5\,c^5\,d^5+14\,a^3\,b^6\,c^6\,d^4+6\,a^2\,b^7\,c^7\,d^3+3\,a\,b^8\,c^8\,d^2+\frac {5\,b^9\,c^9\,d}{3}\right )+x^3\,\left (3190\,a^7\,b^2\,d^{10}-11970\,a^6\,b^3\,c\,d^9+16170\,a^5\,b^4\,c^2\,d^8-8750\,a^4\,b^5\,c^3\,d^7+1050\,a^3\,b^6\,c^4\,d^6+210\,a^2\,b^7\,c^5\,d^5+70\,a\,b^8\,c^6\,d^4+30\,b^9\,c^7\,d^3\right )+x^2\,\left (2229\,a^8\,b\,d^{10}-8262\,a^7\,b^2\,c\,d^9+10962\,a^6\,b^3\,c^2\,d^8-5754\,a^5\,b^4\,c^3\,d^7+630\,a^4\,b^5\,c^4\,d^6+126\,a^3\,b^6\,c^5\,d^5+42\,a^2\,b^7\,c^6\,d^4+18\,a\,b^8\,c^7\,d^3+9\,b^9\,c^8\,d^2\right )+x^5\,\left (1134\,a^5\,b^4\,d^{10}-4410\,a^4\,b^5\,c\,d^9+6300\,a^3\,b^6\,c^2\,d^8-3780\,a^2\,b^7\,c^3\,d^7+630\,a\,b^8\,c^4\,d^6+126\,b^9\,c^5\,d^5\right )}{a^7\,b^{10}+7\,a^6\,b^{11}\,x+21\,a^5\,b^{12}\,x^2+35\,a^4\,b^{13}\,x^3+35\,a^3\,b^{14}\,x^4+21\,a^2\,b^{15}\,x^5+7\,a\,b^{16}\,x^6+b^{17}\,x^7}-x^2\,\left (\frac {4\,a\,d^{10}}{b^9}-\frac {5\,c\,d^9}{b^8}\right )-\frac {\ln \left (a+b\,x\right )\,\left (120\,a^3\,d^{10}-360\,a^2\,b\,c\,d^9+360\,a\,b^2\,c^2\,d^8-120\,b^3\,c^3\,d^7\right )}{b^{11}}+\frac {d^{10}\,x^3}{3\,b^8} \]
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