Integrand size = 15, antiderivative size = 260 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=\frac {210 d^6 (b c-a d)^4 x}{b^{10}}-\frac {(b c-a d)^{10}}{5 b^{11} (a+b x)^5}-\frac {5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac {15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}-\frac {60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac {210 d^4 (b c-a d)^6}{b^{11} (a+b x)}+\frac {60 d^7 (b c-a d)^3 (a+b x)^2}{b^{11}}+\frac {15 d^8 (b c-a d)^2 (a+b x)^3}{b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^4}{2 b^{11}}+\frac {d^{10} (a+b x)^5}{5 b^{11}}+\frac {252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}} \]
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Time = 0.32 (sec) , antiderivative size = 260, 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} \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=\frac {5 d^9 (a+b x)^4 (b c-a d)}{2 b^{11}}+\frac {15 d^8 (a+b x)^3 (b c-a d)^2}{b^{11}}+\frac {60 d^7 (a+b x)^2 (b c-a d)^3}{b^{11}}+\frac {252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}}-\frac {210 d^4 (b c-a d)^6}{b^{11} (a+b x)}-\frac {60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac {15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}-\frac {5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac {(b c-a d)^{10}}{5 b^{11} (a+b x)^5}+\frac {d^{10} (a+b x)^5}{5 b^{11}}+\frac {210 d^6 x (b c-a d)^4}{b^{10}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {210 d^6 (b c-a d)^4}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^6}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^5}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^4}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^3}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^2}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)}+\frac {120 d^7 (b c-a d)^3 (a+b x)}{b^{10}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^2}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^3}{b^{10}}+\frac {d^{10} (a+b x)^4}{b^{10}}\right ) \, dx \\ & = \frac {210 d^6 (b c-a d)^4 x}{b^{10}}-\frac {(b c-a d)^{10}}{5 b^{11} (a+b x)^5}-\frac {5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac {15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}-\frac {60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac {210 d^4 (b c-a d)^6}{b^{11} (a+b x)}+\frac {60 d^7 (b c-a d)^3 (a+b x)^2}{b^{11}}+\frac {15 d^8 (b c-a d)^2 (a+b x)^3}{b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^4}{2 b^{11}}+\frac {d^{10} (a+b x)^5}{5 b^{11}}+\frac {252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.17 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=\frac {10 b d^6 \left (210 b^4 c^4-720 a b^3 c^3 d+945 a^2 b^2 c^2 d^2-560 a^3 b c d^3+126 a^4 d^4\right ) x+10 b^2 d^7 \left (60 b^3 c^3-135 a b^2 c^2 d+105 a^2 b c d^2-28 a^3 d^3\right ) x^2+10 b^3 d^8 \left (15 b^2 c^2-20 a b c d+7 a^2 d^2\right ) x^3+5 b^4 d^9 (5 b c-3 a d) x^4+2 b^5 d^{10} x^5-\frac {2 (b c-a d)^{10}}{(a+b x)^5}+\frac {25 d (-b c+a d)^9}{(a+b x)^4}-\frac {150 d^2 (b c-a d)^8}{(a+b x)^3}+\frac {600 d^3 (-b c+a d)^7}{(a+b x)^2}-\frac {2100 d^4 (b c-a d)^6}{a+b x}+2520 d^5 (b c-a d)^5 \log (a+b x)}{10 b^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(846\) vs. \(2(252)=504\).
Time = 0.22 (sec) , antiderivative size = 847, normalized size of antiderivative = 3.26
method | result | size |
norman | \(\frac {-\frac {5754 a^{10} d^{10}-28770 a^{9} b c \,d^{9}+57540 a^{8} b^{2} c^{2} d^{8}-57540 a^{7} b^{3} c^{3} d^{7}+28770 a^{6} b^{4} c^{4} d^{6}-5754 a^{5} b^{5} c^{5} d^{5}+420 a^{4} b^{6} c^{6} d^{4}+60 a^{3} b^{7} c^{7} d^{3}+15 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +2 b^{10} c^{10}}{10 b^{11}}+\frac {d^{10} x^{10}}{5 b}-\frac {5 \left (252 a^{6} d^{10}-1260 a^{5} b c \,d^{9}+2520 a^{4} b^{2} c^{2} d^{8}-2520 a^{3} b^{3} c^{3} d^{7}+1260 a^{2} b^{4} c^{4} d^{6}-252 a \,b^{5} c^{5} d^{5}+42 b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}-\frac {10 \left (378 a^{7} d^{10}-1890 a^{6} b c \,d^{9}+3780 a^{5} b^{2} c^{2} d^{8}-3780 a^{4} b^{3} c^{3} d^{7}+1890 a^{3} b^{4} c^{4} d^{6}-378 a^{2} b^{5} c^{5} d^{5}+42 a \,b^{6} c^{6} d^{4}+6 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}-\frac {5 \left (924 a^{8} d^{10}-4620 a^{7} b c \,d^{9}+9240 a^{6} b^{2} c^{2} d^{8}-9240 a^{5} b^{3} c^{3} d^{7}+4620 a^{4} b^{4} c^{4} d^{6}-924 a^{3} b^{5} c^{5} d^{5}+84 a^{2} b^{6} c^{6} d^{4}+12 a \,b^{7} c^{7} d^{3}+3 b^{8} c^{8} d^{2}\right ) x^{2}}{b^{9}}-\frac {5 \left (1050 a^{9} d^{10}-5250 a^{8} b c \,d^{9}+10500 a^{7} b^{2} c^{2} d^{8}-10500 a^{6} b^{3} c^{3} d^{7}+5250 a^{5} b^{4} c^{4} d^{6}-1050 a^{4} b^{5} c^{5} d^{5}+84 a^{3} b^{6} c^{6} d^{4}+12 a^{2} b^{7} c^{7} d^{3}+3 a \,b^{8} c^{8} d^{2}+b^{9} c^{9} d \right ) x}{2 b^{10}}+\frac {42 d^{6} \left (a^{4} d^{4}-5 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}-10 a \,b^{3} c^{3} d +5 b^{4} c^{4}\right ) x^{6}}{b^{5}}-\frac {6 d^{7} \left (a^{3} d^{3}-5 a^{2} b c \,d^{2}+10 a \,b^{2} c^{2} d -10 b^{3} c^{3}\right ) x^{7}}{b^{4}}+\frac {3 d^{8} \left (a^{2} d^{2}-5 a b c d +10 b^{2} c^{2}\right ) x^{8}}{2 b^{3}}-\frac {d^{9} \left (a d -5 b c \right ) x^{9}}{2 b^{2}}}{\left (b x +a \right )^{5}}-\frac {252 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \ln \left (b x +a \right )}{b^{11}}\) | \(847\) |
default | \(\frac {d^{6} \left (\frac {1}{5} d^{4} x^{5} b^{4}-\frac {3}{2} a \,b^{3} d^{4} x^{4}+\frac {5}{2} b^{4} c \,d^{3} x^{4}+7 a^{2} b^{2} d^{4} x^{3}-20 a \,b^{3} c \,d^{3} x^{3}+15 b^{4} c^{2} d^{2} x^{3}-28 a^{3} b \,d^{4} x^{2}+105 a^{2} b^{2} c \,d^{3} x^{2}-135 a \,b^{3} c^{2} d^{2} x^{2}+60 b^{4} c^{3} d \,x^{2}+126 a^{4} d^{4} x -560 a^{3} b c \,d^{3} x +945 a^{2} b^{2} c^{2} d^{2} x -720 a \,b^{3} c^{3} d x +210 b^{4} c^{4} x \right )}{b^{10}}-\frac {15 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}{b^{11} \left (b x +a \right )^{3}}-\frac {252 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \ln \left (b x +a \right )}{b^{11}}+\frac {5 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{2 b^{11} \left (b x +a \right )^{4}}+\frac {60 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{2}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{5 b^{11} \left (b x +a \right )^{5}}-\frac {210 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{b^{11} \left (b x +a \right )}\) | \(870\) |
risch | \(\frac {d^{10} x^{5}}{5 b^{6}}-\frac {3 d^{10} a \,x^{4}}{2 b^{7}}+\frac {5 d^{9} c \,x^{4}}{2 b^{6}}+\frac {7 d^{10} a^{2} x^{3}}{b^{8}}-\frac {20 d^{9} a c \,x^{3}}{b^{7}}+\frac {15 d^{8} c^{2} x^{3}}{b^{6}}-\frac {28 d^{10} a^{3} x^{2}}{b^{9}}+\frac {105 d^{9} a^{2} c \,x^{2}}{b^{8}}-\frac {135 d^{8} a \,c^{2} x^{2}}{b^{7}}+\frac {60 d^{7} c^{3} x^{2}}{b^{6}}+\frac {126 d^{10} a^{4} x}{b^{10}}-\frac {560 d^{9} a^{3} c x}{b^{9}}+\frac {945 d^{8} a^{2} c^{2} x}{b^{8}}-\frac {720 d^{7} a \,c^{3} x}{b^{7}}+\frac {210 d^{6} c^{4} x}{b^{6}}+\frac {\left (-210 a^{6} b^{3} d^{10}+1260 a^{5} c \,d^{9} b^{4}-3150 a^{4} c^{2} d^{8} b^{5}+4200 a^{3} c^{3} d^{7} b^{6}-3150 a^{2} c^{4} d^{6} b^{7}+1260 a \,b^{8} c^{5} d^{5}-210 c^{6} d^{4} b^{9}\right ) x^{4}-60 b^{2} d^{3} \left (13 a^{7} d^{7}-77 a^{6} b c \,d^{6}+189 a^{5} b^{2} c^{2} d^{5}-245 a^{4} b^{3} c^{3} d^{4}+175 a^{3} b^{4} c^{4} d^{3}-63 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +b^{7} c^{7}\right ) x^{3}-15 b \,d^{2} \left (73 a^{8} d^{8}-428 a^{7} b c \,d^{7}+1036 a^{6} b^{2} c^{2} d^{6}-1316 a^{5} b^{3} c^{3} d^{5}+910 a^{4} b^{4} c^{4} d^{4}-308 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}+4 a \,b^{7} c^{7} d +b^{8} c^{8}\right ) x^{2}-\frac {5 d \left (275 a^{9} d^{9}-1599 a^{8} b c \,d^{8}+3828 a^{7} b^{2} c^{2} d^{7}-4788 a^{6} b^{3} c^{3} d^{6}+3234 a^{5} b^{4} c^{4} d^{5}-1050 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}+12 a^{2} b^{7} c^{7} d^{2}+3 a \,b^{8} c^{8} d +b^{9} c^{9}\right ) x}{2}-\frac {1627 a^{10} d^{10}-9395 a^{9} b c \,d^{9}+22290 a^{8} b^{2} c^{2} d^{8}-27540 a^{7} b^{3} c^{3} d^{7}+18270 a^{6} b^{4} c^{4} d^{6}-5754 a^{5} b^{5} c^{5} d^{5}+420 a^{4} b^{6} c^{6} d^{4}+60 a^{3} b^{7} c^{7} d^{3}+15 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +2 b^{10} c^{10}}{10 b}}{b^{10} \left (b x +a \right )^{5}}-\frac {252 d^{10} \ln \left (b x +a \right ) a^{5}}{b^{11}}+\frac {1260 d^{9} \ln \left (b x +a \right ) a^{4} c}{b^{10}}-\frac {2520 d^{8} \ln \left (b x +a \right ) a^{3} c^{2}}{b^{9}}+\frac {2520 d^{7} \ln \left (b x +a \right ) a^{2} c^{3}}{b^{8}}-\frac {1260 d^{6} \ln \left (b x +a \right ) a \,c^{4}}{b^{7}}+\frac {252 d^{5} \ln \left (b x +a \right ) c^{5}}{b^{6}}\) | \(898\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1625\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1395 vs. \(2 (252) = 504\).
Time = 0.23 (sec) , antiderivative size = 1395, normalized size of antiderivative = 5.37 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (252) = 504\).
Time = 0.26 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.51 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=-\frac {2 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 15 \, a^{2} b^{8} c^{8} d^{2} + 60 \, a^{3} b^{7} c^{7} d^{3} + 420 \, a^{4} b^{6} c^{6} d^{4} - 5754 \, a^{5} b^{5} c^{5} d^{5} + 18270 \, a^{6} b^{4} c^{4} d^{6} - 27540 \, a^{7} b^{3} c^{3} d^{7} + 22290 \, a^{8} b^{2} c^{2} d^{8} - 9395 \, a^{9} b c d^{9} + 1627 \, a^{10} d^{10} + 2100 \, {\left (b^{10} c^{6} d^{4} - 6 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 20 \, a^{3} b^{7} c^{3} d^{7} + 15 \, a^{4} b^{6} c^{2} d^{8} - 6 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 600 \, {\left (b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} - 63 \, a^{2} b^{8} c^{5} d^{5} + 175 \, a^{3} b^{7} c^{4} d^{6} - 245 \, a^{4} b^{6} c^{3} d^{7} + 189 \, a^{5} b^{5} c^{2} d^{8} - 77 \, a^{6} b^{4} c d^{9} + 13 \, a^{7} b^{3} d^{10}\right )} x^{3} + 150 \, {\left (b^{10} c^{8} d^{2} + 4 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} - 308 \, a^{3} b^{7} c^{5} d^{5} + 910 \, a^{4} b^{6} c^{4} d^{6} - 1316 \, a^{5} b^{5} c^{3} d^{7} + 1036 \, a^{6} b^{4} c^{2} d^{8} - 428 \, a^{7} b^{3} c d^{9} + 73 \, a^{8} b^{2} d^{10}\right )} x^{2} + 25 \, {\left (b^{10} c^{9} d + 3 \, a b^{9} c^{8} d^{2} + 12 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} - 1050 \, a^{4} b^{6} c^{5} d^{5} + 3234 \, a^{5} b^{5} c^{4} d^{6} - 4788 \, a^{6} b^{4} c^{3} d^{7} + 3828 \, a^{7} b^{3} c^{2} d^{8} - 1599 \, a^{8} b^{2} c d^{9} + 275 \, a^{9} b d^{10}\right )} x}{10 \, {\left (b^{16} x^{5} + 5 \, a b^{15} x^{4} + 10 \, a^{2} b^{14} x^{3} + 10 \, a^{3} b^{13} x^{2} + 5 \, a^{4} b^{12} x + a^{5} b^{11}\right )}} + \frac {2 \, b^{4} d^{10} x^{5} + 5 \, {\left (5 \, b^{4} c d^{9} - 3 \, a b^{3} d^{10}\right )} x^{4} + 10 \, {\left (15 \, b^{4} c^{2} d^{8} - 20 \, a b^{3} c d^{9} + 7 \, a^{2} b^{2} d^{10}\right )} x^{3} + 10 \, {\left (60 \, b^{4} c^{3} d^{7} - 135 \, a b^{3} c^{2} d^{8} + 105 \, a^{2} b^{2} c d^{9} - 28 \, a^{3} b d^{10}\right )} x^{2} + 10 \, {\left (210 \, b^{4} c^{4} d^{6} - 720 \, a b^{3} c^{3} d^{7} + 945 \, a^{2} b^{2} c^{2} d^{8} - 560 \, a^{3} b c d^{9} + 126 \, a^{4} d^{10}\right )} x}{10 \, b^{10}} + \frac {252 \, {\left (b^{5} c^{5} d^{5} - 5 \, a b^{4} c^{4} d^{6} + 10 \, a^{2} b^{3} c^{3} d^{7} - 10 \, a^{3} b^{2} c^{2} d^{8} + 5 \, a^{4} b c d^{9} - a^{5} 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. 883 vs. \(2 (252) = 504\).
Time = 0.37 (sec) , antiderivative size = 883, normalized size of antiderivative = 3.40 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=\frac {252 \, {\left (b^{5} c^{5} d^{5} - 5 \, a b^{4} c^{4} d^{6} + 10 \, a^{2} b^{3} c^{3} d^{7} - 10 \, a^{3} b^{2} c^{2} d^{8} + 5 \, a^{4} b c d^{9} - a^{5} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {2 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 15 \, a^{2} b^{8} c^{8} d^{2} + 60 \, a^{3} b^{7} c^{7} d^{3} + 420 \, a^{4} b^{6} c^{6} d^{4} - 5754 \, a^{5} b^{5} c^{5} d^{5} + 18270 \, a^{6} b^{4} c^{4} d^{6} - 27540 \, a^{7} b^{3} c^{3} d^{7} + 22290 \, a^{8} b^{2} c^{2} d^{8} - 9395 \, a^{9} b c d^{9} + 1627 \, a^{10} d^{10} + 2100 \, {\left (b^{10} c^{6} d^{4} - 6 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 20 \, a^{3} b^{7} c^{3} d^{7} + 15 \, a^{4} b^{6} c^{2} d^{8} - 6 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 600 \, {\left (b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} - 63 \, a^{2} b^{8} c^{5} d^{5} + 175 \, a^{3} b^{7} c^{4} d^{6} - 245 \, a^{4} b^{6} c^{3} d^{7} + 189 \, a^{5} b^{5} c^{2} d^{8} - 77 \, a^{6} b^{4} c d^{9} + 13 \, a^{7} b^{3} d^{10}\right )} x^{3} + 150 \, {\left (b^{10} c^{8} d^{2} + 4 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} - 308 \, a^{3} b^{7} c^{5} d^{5} + 910 \, a^{4} b^{6} c^{4} d^{6} - 1316 \, a^{5} b^{5} c^{3} d^{7} + 1036 \, a^{6} b^{4} c^{2} d^{8} - 428 \, a^{7} b^{3} c d^{9} + 73 \, a^{8} b^{2} d^{10}\right )} x^{2} + 25 \, {\left (b^{10} c^{9} d + 3 \, a b^{9} c^{8} d^{2} + 12 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} - 1050 \, a^{4} b^{6} c^{5} d^{5} + 3234 \, a^{5} b^{5} c^{4} d^{6} - 4788 \, a^{6} b^{4} c^{3} d^{7} + 3828 \, a^{7} b^{3} c^{2} d^{8} - 1599 \, a^{8} b^{2} c d^{9} + 275 \, a^{9} b d^{10}\right )} x}{10 \, {\left (b x + a\right )}^{5} b^{11}} + \frac {2 \, b^{24} d^{10} x^{5} + 25 \, b^{24} c d^{9} x^{4} - 15 \, a b^{23} d^{10} x^{4} + 150 \, b^{24} c^{2} d^{8} x^{3} - 200 \, a b^{23} c d^{9} x^{3} + 70 \, a^{2} b^{22} d^{10} x^{3} + 600 \, b^{24} c^{3} d^{7} x^{2} - 1350 \, a b^{23} c^{2} d^{8} x^{2} + 1050 \, a^{2} b^{22} c d^{9} x^{2} - 280 \, a^{3} b^{21} d^{10} x^{2} + 2100 \, b^{24} c^{4} d^{6} x - 7200 \, a b^{23} c^{3} d^{7} x + 9450 \, a^{2} b^{22} c^{2} d^{8} x - 5600 \, a^{3} b^{21} c d^{9} x + 1260 \, a^{4} b^{20} d^{10} x}{10 \, b^{30}} \]
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Time = 0.50 (sec) , antiderivative size = 1141, normalized size of antiderivative = 4.39 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=x^3\,\left (\frac {2\,a\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b}-\frac {5\,a^2\,d^{10}}{b^8}+\frac {15\,c^2\,d^8}{b^6}\right )-x^2\,\left (\frac {3\,a\,\left (\frac {6\,a\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b}-\frac {15\,a^2\,d^{10}}{b^8}+\frac {45\,c^2\,d^8}{b^6}\right )}{b}+\frac {10\,a^3\,d^{10}}{b^9}-\frac {60\,c^3\,d^7}{b^6}-\frac {15\,a^2\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{2\,b^2}\right )-x^4\,\left (\frac {3\,a\,d^{10}}{2\,b^7}-\frac {5\,c\,d^9}{2\,b^6}\right )-\frac {x^4\,\left (210\,a^6\,b^3\,d^{10}-1260\,a^5\,b^4\,c\,d^9+3150\,a^4\,b^5\,c^2\,d^8-4200\,a^3\,b^6\,c^3\,d^7+3150\,a^2\,b^7\,c^4\,d^6-1260\,a\,b^8\,c^5\,d^5+210\,b^9\,c^6\,d^4\right )+\frac {1627\,a^{10}\,d^{10}-9395\,a^9\,b\,c\,d^9+22290\,a^8\,b^2\,c^2\,d^8-27540\,a^7\,b^3\,c^3\,d^7+18270\,a^6\,b^4\,c^4\,d^6-5754\,a^5\,b^5\,c^5\,d^5+420\,a^4\,b^6\,c^6\,d^4+60\,a^3\,b^7\,c^7\,d^3+15\,a^2\,b^8\,c^8\,d^2+5\,a\,b^9\,c^9\,d+2\,b^{10}\,c^{10}}{10\,b}+x\,\left (\frac {1375\,a^9\,d^{10}}{2}-\frac {7995\,a^8\,b\,c\,d^9}{2}+9570\,a^7\,b^2\,c^2\,d^8-11970\,a^6\,b^3\,c^3\,d^7+8085\,a^5\,b^4\,c^4\,d^6-2625\,a^4\,b^5\,c^5\,d^5+210\,a^3\,b^6\,c^6\,d^4+30\,a^2\,b^7\,c^7\,d^3+\frac {15\,a\,b^8\,c^8\,d^2}{2}+\frac {5\,b^9\,c^9\,d}{2}\right )+x^3\,\left (780\,a^7\,b^2\,d^{10}-4620\,a^6\,b^3\,c\,d^9+11340\,a^5\,b^4\,c^2\,d^8-14700\,a^4\,b^5\,c^3\,d^7+10500\,a^3\,b^6\,c^4\,d^6-3780\,a^2\,b^7\,c^5\,d^5+420\,a\,b^8\,c^6\,d^4+60\,b^9\,c^7\,d^3\right )+x^2\,\left (1095\,a^8\,b\,d^{10}-6420\,a^7\,b^2\,c\,d^9+15540\,a^6\,b^3\,c^2\,d^8-19740\,a^5\,b^4\,c^3\,d^7+13650\,a^4\,b^5\,c^4\,d^6-4620\,a^3\,b^6\,c^5\,d^5+420\,a^2\,b^7\,c^6\,d^4+60\,a\,b^8\,c^7\,d^3+15\,b^9\,c^8\,d^2\right )}{a^5\,b^{10}+5\,a^4\,b^{11}\,x+10\,a^3\,b^{12}\,x^2+10\,a^2\,b^{13}\,x^3+5\,a\,b^{14}\,x^4+b^{15}\,x^5}+x\,\left (\frac {6\,a\,\left (\frac {6\,a\,\left (\frac {6\,a\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b}-\frac {15\,a^2\,d^{10}}{b^8}+\frac {45\,c^2\,d^8}{b^6}\right )}{b}+\frac {20\,a^3\,d^{10}}{b^9}-\frac {120\,c^3\,d^7}{b^6}-\frac {15\,a^2\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b^2}\right )}{b}-\frac {15\,a^4\,d^{10}}{b^{10}}+\frac {210\,c^4\,d^6}{b^6}+\frac {20\,a^3\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b^3}-\frac {15\,a^2\,\left (\frac {6\,a\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b}-\frac {15\,a^2\,d^{10}}{b^8}+\frac {45\,c^2\,d^8}{b^6}\right )}{b^2}\right )+\frac {d^{10}\,x^5}{5\,b^6}-\frac {\ln \left (a+b\,x\right )\,\left (252\,a^5\,d^{10}-1260\,a^4\,b\,c\,d^9+2520\,a^3\,b^2\,c^2\,d^8-2520\,a^2\,b^3\,c^3\,d^7+1260\,a\,b^4\,c^4\,d^6-252\,b^5\,c^5\,d^5\right )}{b^{11}} \]
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