Integrand size = 15, antiderivative size = 257 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\frac {d^{10} x}{b^{10}}-\frac {(b c-a d)^{10}}{9 b^{11} (a+b x)^9}-\frac {5 d (b c-a d)^9}{4 b^{11} (a+b x)^8}-\frac {45 d^2 (b c-a d)^8}{7 b^{11} (a+b x)^7}-\frac {20 d^3 (b c-a d)^7}{b^{11} (a+b x)^6}-\frac {42 d^4 (b c-a d)^6}{b^{11} (a+b x)^5}-\frac {63 d^5 (b c-a d)^5}{b^{11} (a+b x)^4}-\frac {70 d^6 (b c-a d)^4}{b^{11} (a+b x)^3}-\frac {60 d^7 (b c-a d)^3}{b^{11} (a+b x)^2}-\frac {45 d^8 (b c-a d)^2}{b^{11} (a+b x)}+\frac {10 d^9 (b c-a d) \log (a+b x)}{b^{11}} \]
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Time = 0.22 (sec) , antiderivative size = 257, 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} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\frac {10 d^9 (b c-a d) \log (a+b x)}{b^{11}}-\frac {45 d^8 (b c-a d)^2}{b^{11} (a+b x)}-\frac {60 d^7 (b c-a d)^3}{b^{11} (a+b x)^2}-\frac {70 d^6 (b c-a d)^4}{b^{11} (a+b x)^3}-\frac {63 d^5 (b c-a d)^5}{b^{11} (a+b x)^4}-\frac {42 d^4 (b c-a d)^6}{b^{11} (a+b x)^5}-\frac {20 d^3 (b c-a d)^7}{b^{11} (a+b x)^6}-\frac {45 d^2 (b c-a d)^8}{7 b^{11} (a+b x)^7}-\frac {5 d (b c-a d)^9}{4 b^{11} (a+b x)^8}-\frac {(b c-a d)^{10}}{9 b^{11} (a+b x)^9}+\frac {d^{10} x}{b^{10}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^{10}}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^{10}}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^9}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^8}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^7}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^6}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^5}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^4}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)^3}+\frac {45 d^8 (b c-a d)^2}{b^{10} (a+b x)^2}+\frac {10 d^9 (b c-a d)}{b^{10} (a+b x)}\right ) \, dx \\ & = \frac {d^{10} x}{b^{10}}-\frac {(b c-a d)^{10}}{9 b^{11} (a+b x)^9}-\frac {5 d (b c-a d)^9}{4 b^{11} (a+b x)^8}-\frac {45 d^2 (b c-a d)^8}{7 b^{11} (a+b x)^7}-\frac {20 d^3 (b c-a d)^7}{b^{11} (a+b x)^6}-\frac {42 d^4 (b c-a d)^6}{b^{11} (a+b x)^5}-\frac {63 d^5 (b c-a d)^5}{b^{11} (a+b x)^4}-\frac {70 d^6 (b c-a d)^4}{b^{11} (a+b x)^3}-\frac {60 d^7 (b c-a d)^3}{b^{11} (a+b x)^2}-\frac {45 d^8 (b c-a d)^2}{b^{11} (a+b x)}+\frac {10 d^9 (b c-a d) \log (a+b x)}{b^{11}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(708\) vs. \(2(257)=514\).
Time = 0.25 (sec) , antiderivative size = 708, normalized size of antiderivative = 2.75 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=-\frac {4861 a^{10} d^{10}+a^9 b d^9 (-7129 c+41229 d x)+9 a^8 b^2 d^8 \left (140 c^2-6849 c d x+17064 d^2 x^2\right )+12 a^7 b^3 d^7 \left (35 c^3+945 c^2 d x-19602 c d^2 x^2+27342 d^3 x^3\right )+42 a^6 b^4 d^6 \left (5 c^4+90 c^3 d x+1080 c^2 d^2 x^2-12348 c d^3 x^3+10458 d^4 x^4\right )+126 a^5 b^5 d^5 \left (c^5+15 c^4 d x+120 c^3 d^2 x^2+840 c^2 d^3 x^3-5754 c d^4 x^4+2982 d^5 x^5\right )+42 a^4 b^6 d^4 \left (2 c^6+27 c^5 d x+180 c^4 d^2 x^2+840 c^3 d^3 x^3+3780 c^2 d^4 x^4-15750 c d^5 x^5+4704 d^6 x^6\right )+12 a^3 b^7 d^3 \left (5 c^7+63 c^6 d x+378 c^5 d^2 x^2+1470 c^4 d^3 x^3+4410 c^3 d^4 x^4+13230 c^2 d^5 x^5-32340 c d^6 x^6+4536 d^7 x^7\right )+9 a^2 b^8 d^2 \left (5 c^8+60 c^7 d x+336 c^6 d^2 x^2+1176 c^5 d^3 x^3+2940 c^4 d^4 x^4+5880 c^3 d^5 x^5+11760 c^2 d^6 x^6-15120 c d^7 x^7+252 d^8 x^8\right )+a b^9 d \left (35 c^9+405 c^8 d x+2160 c^7 d^2 x^2+7056 c^6 d^3 x^3+15876 c^5 d^4 x^4+26460 c^4 d^5 x^5+35280 c^3 d^6 x^6+45360 c^2 d^7 x^7-22680 c d^8 x^8-2268 d^9 x^9\right )+b^{10} \left (28 c^{10}+315 c^9 d x+1620 c^8 d^2 x^2+5040 c^7 d^3 x^3+10584 c^6 d^4 x^4+15876 c^5 d^5 x^5+17640 c^4 d^6 x^6+15120 c^3 d^7 x^7+11340 c^2 d^8 x^8-252 d^{10} x^{10}\right )+2520 d^9 (-b c+a d) (a+b x)^9 \log (a+b x)}{252 b^{11} (a+b x)^9} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(847\) vs. \(2(251)=502\).
Time = 0.23 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.30
method | result | size |
risch | \(\frac {d^{10} x}{b^{10}}+\frac {\left (-45 a^{2} b^{7} d^{10}+90 a \,b^{8} c \,d^{9}-45 b^{9} c^{2} d^{8}\right ) x^{8}-60 b^{6} d^{7} \left (5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{7}-70 b^{5} d^{6} \left (13 a^{4} d^{4}-22 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x^{6}-21 b^{4} d^{5} \left (77 a^{5} d^{5}-125 a^{4} b c \,d^{4}+30 a^{3} b^{2} c^{2} d^{3}+10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right ) x^{5}-21 b^{3} d^{4} \left (87 a^{6} d^{6}-137 a^{5} b c \,d^{5}+30 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+5 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +2 b^{6} c^{6}\right ) x^{4}-2 b^{2} d^{3} \left (669 a^{7} d^{7}-1029 a^{6} b c \,d^{6}+210 a^{5} b^{2} c^{2} d^{5}+70 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+14 a \,b^{6} c^{6} d +10 b^{7} c^{7}\right ) x^{3}-\frac {3 b \,d^{2} \left (1443 a^{8} d^{8}-2178 a^{7} b c \,d^{7}+420 a^{6} b^{2} c^{2} d^{6}+140 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}+42 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}+20 a \,b^{7} c^{7} d +15 b^{8} c^{8}\right ) x^{2}}{7}-\frac {d \left (4609 a^{9} d^{9}-6849 a^{8} b c \,d^{8}+1260 a^{7} b^{2} c^{2} d^{7}+420 a^{6} b^{3} c^{3} d^{6}+210 a^{5} b^{4} c^{4} d^{5}+126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}+60 a^{2} b^{7} c^{7} d^{2}+45 a \,b^{8} c^{8} d +35 b^{9} c^{9}\right ) x}{28}-\frac {4861 a^{10} d^{10}-7129 a^{9} b c \,d^{9}+1260 a^{8} b^{2} c^{2} d^{8}+420 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+84 a^{4} b^{6} c^{6} d^{4}+60 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+35 a \,b^{9} c^{9} d +28 b^{10} c^{10}}{252 b}}{b^{10} \left (b x +a \right )^{9}}-\frac {10 d^{10} \ln \left (b x +a \right ) a}{b^{11}}+\frac {10 d^{9} \ln \left (b x +a \right ) c}{b^{10}}\) | \(848\) |
default | \(\frac {d^{10} x}{b^{10}}-\frac {70 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 )^{3}}-\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}}{9 b^{11} \left (b x +a \right )^{9}}-\frac {10 d^{9} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{11}}+\frac {20 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 )^{6}}+\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 )}{4 b^{11} \left (b x +a \right )^{8}}+\frac {63 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 )^{4}}-\frac {45 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 )}{7 b^{11} \left (b x +a \right )^{7}}+\frac {60 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 )}{b^{11} \left (b x +a \right )^{2}}-\frac {42 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 )^{5}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{11} \left (b x +a \right )}\) | \(859\) |
norman | \(\frac {\frac {d^{10} x^{10}}{b}-\frac {7129 a^{10} d^{10}-7129 a^{9} b c \,d^{9}+1260 a^{8} b^{2} c^{2} d^{8}+420 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+84 a^{4} b^{6} c^{6} d^{4}+60 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+35 a \,b^{9} c^{9} d +28 b^{10} c^{10}}{252 b^{11}}-\frac {9 \left (10 a^{2} d^{10}-10 a b c \,d^{9}+5 b^{2} c^{2} d^{8}\right ) x^{8}}{b^{3}}-\frac {12 \left (45 a^{3} d^{10}-45 a^{2} b c \,d^{9}+15 a \,b^{2} c^{2} d^{8}+5 b^{3} c^{3} d^{7}\right ) x^{7}}{b^{4}}-\frac {14 \left (110 a^{4} d^{10}-110 a^{3} b c \,d^{9}+30 a^{2} b^{2} c^{2} d^{8}+10 a \,b^{3} c^{3} d^{7}+5 b^{4} c^{4} d^{6}\right ) x^{6}}{b^{5}}-\frac {21 \left (125 a^{5} d^{10}-125 a^{4} b c \,d^{9}+30 a^{3} b^{2} c^{2} d^{8}+10 a^{2} b^{3} c^{3} d^{7}+5 a \,b^{4} c^{4} d^{6}+3 b^{5} c^{5} d^{5}\right ) x^{5}}{b^{6}}-\frac {21 \left (137 a^{6} d^{10}-137 a^{5} b c \,d^{9}+30 a^{4} b^{2} c^{2} d^{8}+10 a^{3} b^{3} c^{3} d^{7}+5 a^{2} b^{4} c^{4} d^{6}+3 a \,b^{5} c^{5} d^{5}+2 b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}-\frac {2 \left (1029 a^{7} d^{10}-1029 a^{6} b c \,d^{9}+210 a^{5} b^{2} c^{2} d^{8}+70 a^{4} b^{3} c^{3} d^{7}+35 a^{3} b^{4} c^{4} d^{6}+21 a^{2} b^{5} c^{5} d^{5}+14 a \,b^{6} c^{6} d^{4}+10 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}-\frac {3 \left (2178 a^{8} d^{10}-2178 a^{7} b c \,d^{9}+420 a^{6} b^{2} c^{2} d^{8}+140 a^{5} b^{3} c^{3} d^{7}+70 a^{4} b^{4} c^{4} d^{6}+42 a^{3} b^{5} c^{5} d^{5}+28 a^{2} b^{6} c^{6} d^{4}+20 a \,b^{7} c^{7} d^{3}+15 b^{8} c^{8} d^{2}\right ) x^{2}}{7 b^{9}}-\frac {\left (6849 a^{9} d^{10}-6849 a^{8} b c \,d^{9}+1260 a^{7} b^{2} c^{2} d^{8}+420 a^{6} b^{3} c^{3} d^{7}+210 a^{5} b^{4} c^{4} d^{6}+126 a^{4} b^{5} c^{5} d^{5}+84 a^{3} b^{6} c^{6} d^{4}+60 a^{2} b^{7} c^{7} d^{3}+45 a \,b^{8} c^{8} d^{2}+35 b^{9} c^{9} d \right ) x}{28 b^{10}}}{\left (b x +a \right )^{9}}-\frac {10 d^{9} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{11}}\) | \(859\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1325\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1216 vs. \(2 (251) = 502\).
Time = 0.24 (sec) , antiderivative size = 1216, normalized size of antiderivative = 4.73 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 957 vs. \(2 (251) = 502\).
Time = 0.27 (sec) , antiderivative size = 957, normalized size of antiderivative = 3.72 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\frac {d^{10} x}{b^{10}} - \frac {28 \, b^{10} c^{10} + 35 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 60 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 420 \, a^{7} b^{3} c^{3} d^{7} + 1260 \, a^{8} b^{2} c^{2} d^{8} - 7129 \, a^{9} b c d^{9} + 4861 \, a^{10} d^{10} + 11340 \, {\left (b^{10} c^{2} d^{8} - 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 15120 \, {\left (b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} - 9 \, a^{2} b^{8} c d^{9} + 5 \, a^{3} b^{7} d^{10}\right )} x^{7} + 17640 \, {\left (b^{10} c^{4} d^{6} + 2 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 22 \, a^{3} b^{7} c d^{9} + 13 \, a^{4} b^{6} d^{10}\right )} x^{6} + 5292 \, {\left (3 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} + 30 \, a^{3} b^{7} c^{2} d^{8} - 125 \, a^{4} b^{6} c d^{9} + 77 \, a^{5} b^{5} d^{10}\right )} x^{5} + 5292 \, {\left (2 \, b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 10 \, a^{3} b^{7} c^{3} d^{7} + 30 \, a^{4} b^{6} c^{2} d^{8} - 137 \, a^{5} b^{5} c d^{9} + 87 \, a^{6} b^{4} d^{10}\right )} x^{4} + 504 \, {\left (10 \, b^{10} c^{7} d^{3} + 14 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 35 \, a^{3} b^{7} c^{4} d^{6} + 70 \, a^{4} b^{6} c^{3} d^{7} + 210 \, a^{5} b^{5} c^{2} d^{8} - 1029 \, a^{6} b^{4} c d^{9} + 669 \, a^{7} b^{3} d^{10}\right )} x^{3} + 108 \, {\left (15 \, b^{10} c^{8} d^{2} + 20 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} + 140 \, a^{5} b^{5} c^{3} d^{7} + 420 \, a^{6} b^{4} c^{2} d^{8} - 2178 \, a^{7} b^{3} c d^{9} + 1443 \, a^{8} b^{2} d^{10}\right )} x^{2} + 9 \, {\left (35 \, b^{10} c^{9} d + 45 \, a b^{9} c^{8} d^{2} + 60 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 210 \, a^{5} b^{5} c^{4} d^{6} + 420 \, a^{6} b^{4} c^{3} d^{7} + 1260 \, a^{7} b^{3} c^{2} d^{8} - 6849 \, a^{8} b^{2} c d^{9} + 4609 \, a^{9} b d^{10}\right )} x}{252 \, {\left (b^{20} x^{9} + 9 \, a b^{19} x^{8} + 36 \, a^{2} b^{18} x^{7} + 84 \, a^{3} b^{17} x^{6} + 126 \, a^{4} b^{16} x^{5} + 126 \, a^{5} b^{15} x^{4} + 84 \, a^{6} b^{14} x^{3} + 36 \, a^{7} b^{13} x^{2} + 9 \, a^{8} b^{12} x + a^{9} b^{11}\right )}} + \frac {10 \, {\left (b c d^{9} - a 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. 867 vs. \(2 (251) = 502\).
Time = 0.30 (sec) , antiderivative size = 867, normalized size of antiderivative = 3.37 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\frac {d^{10} x}{b^{10}} + \frac {10 \, {\left (b c d^{9} - a d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {28 \, b^{10} c^{10} + 35 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 60 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 420 \, a^{7} b^{3} c^{3} d^{7} + 1260 \, a^{8} b^{2} c^{2} d^{8} - 7129 \, a^{9} b c d^{9} + 4861 \, a^{10} d^{10} + 11340 \, {\left (b^{10} c^{2} d^{8} - 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 15120 \, {\left (b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} - 9 \, a^{2} b^{8} c d^{9} + 5 \, a^{3} b^{7} d^{10}\right )} x^{7} + 17640 \, {\left (b^{10} c^{4} d^{6} + 2 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 22 \, a^{3} b^{7} c d^{9} + 13 \, a^{4} b^{6} d^{10}\right )} x^{6} + 5292 \, {\left (3 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} + 30 \, a^{3} b^{7} c^{2} d^{8} - 125 \, a^{4} b^{6} c d^{9} + 77 \, a^{5} b^{5} d^{10}\right )} x^{5} + 5292 \, {\left (2 \, b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 10 \, a^{3} b^{7} c^{3} d^{7} + 30 \, a^{4} b^{6} c^{2} d^{8} - 137 \, a^{5} b^{5} c d^{9} + 87 \, a^{6} b^{4} d^{10}\right )} x^{4} + 504 \, {\left (10 \, b^{10} c^{7} d^{3} + 14 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 35 \, a^{3} b^{7} c^{4} d^{6} + 70 \, a^{4} b^{6} c^{3} d^{7} + 210 \, a^{5} b^{5} c^{2} d^{8} - 1029 \, a^{6} b^{4} c d^{9} + 669 \, a^{7} b^{3} d^{10}\right )} x^{3} + 108 \, {\left (15 \, b^{10} c^{8} d^{2} + 20 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} + 140 \, a^{5} b^{5} c^{3} d^{7} + 420 \, a^{6} b^{4} c^{2} d^{8} - 2178 \, a^{7} b^{3} c d^{9} + 1443 \, a^{8} b^{2} d^{10}\right )} x^{2} + 9 \, {\left (35 \, b^{10} c^{9} d + 45 \, a b^{9} c^{8} d^{2} + 60 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 210 \, a^{5} b^{5} c^{4} d^{6} + 420 \, a^{6} b^{4} c^{3} d^{7} + 1260 \, a^{7} b^{3} c^{2} d^{8} - 6849 \, a^{8} b^{2} c d^{9} + 4609 \, a^{9} b d^{10}\right )} x}{252 \, {\left (b x + a\right )}^{9} b^{11}} \]
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Time = 0.54 (sec) , antiderivative size = 955, normalized size of antiderivative = 3.72 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\frac {d^{10}\,x}{b^{10}}-\frac {\ln \left (a+b\,x\right )\,\left (10\,a\,d^{10}-10\,b\,c\,d^9\right )}{b^{11}}-\frac {x^4\,\left (1827\,a^6\,b^3\,d^{10}-2877\,a^5\,b^4\,c\,d^9+630\,a^4\,b^5\,c^2\,d^8+210\,a^3\,b^6\,c^3\,d^7+105\,a^2\,b^7\,c^4\,d^6+63\,a\,b^8\,c^5\,d^5+42\,b^9\,c^6\,d^4\right )+x^6\,\left (910\,a^4\,b^5\,d^{10}-1540\,a^3\,b^6\,c\,d^9+420\,a^2\,b^7\,c^2\,d^8+140\,a\,b^8\,c^3\,d^7+70\,b^9\,c^4\,d^6\right )+\frac {4861\,a^{10}\,d^{10}-7129\,a^9\,b\,c\,d^9+1260\,a^8\,b^2\,c^2\,d^8+420\,a^7\,b^3\,c^3\,d^7+210\,a^6\,b^4\,c^4\,d^6+126\,a^5\,b^5\,c^5\,d^5+84\,a^4\,b^6\,c^6\,d^4+60\,a^3\,b^7\,c^7\,d^3+45\,a^2\,b^8\,c^8\,d^2+35\,a\,b^9\,c^9\,d+28\,b^{10}\,c^{10}}{252\,b}+x\,\left (\frac {4609\,a^9\,d^{10}}{28}-\frac {6849\,a^8\,b\,c\,d^9}{28}+45\,a^7\,b^2\,c^2\,d^8+15\,a^6\,b^3\,c^3\,d^7+\frac {15\,a^5\,b^4\,c^4\,d^6}{2}+\frac {9\,a^4\,b^5\,c^5\,d^5}{2}+3\,a^3\,b^6\,c^6\,d^4+\frac {15\,a^2\,b^7\,c^7\,d^3}{7}+\frac {45\,a\,b^8\,c^8\,d^2}{28}+\frac {5\,b^9\,c^9\,d}{4}\right )+x^8\,\left (45\,a^2\,b^7\,d^{10}-90\,a\,b^8\,c\,d^9+45\,b^9\,c^2\,d^8\right )+x^3\,\left (1338\,a^7\,b^2\,d^{10}-2058\,a^6\,b^3\,c\,d^9+420\,a^5\,b^4\,c^2\,d^8+140\,a^4\,b^5\,c^3\,d^7+70\,a^3\,b^6\,c^4\,d^6+42\,a^2\,b^7\,c^5\,d^5+28\,a\,b^8\,c^6\,d^4+20\,b^9\,c^7\,d^3\right )+x^2\,\left (\frac {4329\,a^8\,b\,d^{10}}{7}-\frac {6534\,a^7\,b^2\,c\,d^9}{7}+180\,a^6\,b^3\,c^2\,d^8+60\,a^5\,b^4\,c^3\,d^7+30\,a^4\,b^5\,c^4\,d^6+18\,a^3\,b^6\,c^5\,d^5+12\,a^2\,b^7\,c^6\,d^4+\frac {60\,a\,b^8\,c^7\,d^3}{7}+\frac {45\,b^9\,c^8\,d^2}{7}\right )+x^5\,\left (1617\,a^5\,b^4\,d^{10}-2625\,a^4\,b^5\,c\,d^9+630\,a^3\,b^6\,c^2\,d^8+210\,a^2\,b^7\,c^3\,d^7+105\,a\,b^8\,c^4\,d^6+63\,b^9\,c^5\,d^5\right )+x^7\,\left (300\,a^3\,b^6\,d^{10}-540\,a^2\,b^7\,c\,d^9+180\,a\,b^8\,c^2\,d^8+60\,b^9\,c^3\,d^7\right )}{a^9\,b^{10}+9\,a^8\,b^{11}\,x+36\,a^7\,b^{12}\,x^2+84\,a^6\,b^{13}\,x^3+126\,a^5\,b^{14}\,x^4+126\,a^4\,b^{15}\,x^5+84\,a^3\,b^{16}\,x^6+36\,a^2\,b^{17}\,x^7+9\,a\,b^{18}\,x^8+b^{19}\,x^9} \]
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