Integrand size = 15, antiderivative size = 271 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=-\frac {(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}-\frac {10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac {45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac {120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac {35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac {252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac {105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac {40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac {45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac {10 d^9 (b c-a d)}{b^{11} (a+b x)}+\frac {d^{10} \log (a+b x)}{b^{11}} \]
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Time = 0.20 (sec) , antiderivative size = 271, 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} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=-\frac {10 d^9 (b c-a d)}{b^{11} (a+b x)}-\frac {45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac {40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac {105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac {252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac {35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac {120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac {45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac {10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac {(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}+\frac {d^{10} \log (a+b x)}{b^{11}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^{10}}{b^{10} (a+b x)^{11}}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^{10}}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^9}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^8}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^7}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^6}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^5}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)^4}+\frac {45 d^8 (b c-a d)^2}{b^{10} (a+b x)^3}+\frac {10 d^9 (b c-a d)}{b^{10} (a+b x)^2}+\frac {d^{10}}{b^{10} (a+b x)}\right ) \, dx \\ & = -\frac {(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}-\frac {10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac {45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac {120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac {35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac {252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac {105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac {40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac {45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac {10 d^9 (b c-a d)}{b^{11} (a+b x)}+\frac {d^{10} \log (a+b x)}{b^{11}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(591\) vs. \(2(271)=542\).
Time = 0.22 (sec) , antiderivative size = 591, normalized size of antiderivative = 2.18 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=-\frac {(b c-a d) \left (7381 a^9 d^9+a^8 b d^8 (4861 c+71290 d x)+a^7 b^2 d^7 \left (3601 c^2+46090 c d x+308205 d^2 x^2\right )+a^6 b^3 d^6 \left (2761 c^3+33490 c^2 d x+194805 c d^2 x^2+784080 d^3 x^3\right )+a^5 b^4 d^5 \left (2131 c^4+25090 c^3 d x+138105 c^2 d^2 x^2+481680 c d^3 x^3+1296540 d^4 x^4\right )+a^4 b^5 d^4 \left (1627 c^5+18790 c^4 d x+100305 c^3 d^2 x^2+330480 c^2 d^3 x^3+767340 c d^4 x^4+1450008 d^5 x^5\right )+a^3 b^6 d^3 \left (1207 c^6+13750 c^5 d x+71955 c^4 d^2 x^2+229680 c^3 d^3 x^3+502740 c^2 d^4 x^4+814968 c d^5 x^5+1102500 d^6 x^6\right )+a^2 b^7 d^2 \left (847 c^7+9550 c^6 d x+49275 c^5 d^2 x^2+154080 c^4 d^3 x^3+326340 c^3 d^4 x^4+497448 c^2 d^5 x^5+573300 c d^6 x^6+554400 d^7 x^7\right )+a b^8 d \left (532 c^8+5950 c^7 d x+30375 c^6 d^2 x^2+93600 c^5 d^3 x^3+194040 c^4 d^4 x^4+285768 c^3 d^5 x^5+308700 c^2 d^6 x^6+252000 c d^7 x^7+170100 d^8 x^8\right )+b^9 \left (252 c^9+2800 c^8 d x+14175 c^7 d^2 x^2+43200 c^6 d^3 x^3+88200 c^5 d^4 x^4+127008 c^4 d^5 x^5+132300 c^3 d^6 x^6+100800 c^2 d^7 x^7+56700 c d^8 x^8+25200 d^9 x^9\right )\right )}{2520 b^{11} (a+b x)^{10}}+\frac {d^{10} \log (a+b x)}{b^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(842\) vs. \(2(257)=514\).
Time = 0.23 (sec) , antiderivative size = 843, normalized size of antiderivative = 3.11
method | result | size |
risch | \(\frac {\frac {10 d^{9} \left (a d -b c \right ) x^{9}}{b^{2}}+\frac {45 d^{8} \left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x^{8}}{2 b^{3}}+\frac {20 d^{7} \left (11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{7}}{b^{4}}+\frac {35 d^{6} \left (25 a^{4} d^{4}-12 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x^{6}}{2 b^{5}}+\frac {21 d^{5} \left (137 a^{5} d^{5}-60 a^{4} b c \,d^{4}-30 a^{3} b^{2} c^{2} d^{3}-20 a^{2} b^{3} c^{3} d^{2}-15 a \,b^{4} c^{4} d -12 b^{5} c^{5}\right ) x^{5}}{5 b^{6}}+\frac {7 d^{4} \left (147 a^{6} d^{6}-60 a^{5} b c \,d^{5}-30 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}-15 a^{2} b^{4} c^{4} d^{2}-12 a \,b^{5} c^{5} d -10 b^{6} c^{6}\right ) x^{4}}{2 b^{7}}+\frac {2 d^{3} \left (1089 a^{7} d^{7}-420 a^{6} b c \,d^{6}-210 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-105 a^{3} b^{4} c^{4} d^{3}-84 a^{2} b^{5} c^{5} d^{2}-70 a \,b^{6} c^{6} d -60 b^{7} c^{7}\right ) x^{3}}{7 b^{8}}+\frac {3 d^{2} \left (2283 a^{8} d^{8}-840 a^{7} b c \,d^{7}-420 a^{6} b^{2} c^{2} d^{6}-280 a^{5} b^{3} c^{3} d^{5}-210 a^{4} b^{4} c^{4} d^{4}-168 a^{3} b^{5} c^{5} d^{3}-140 a^{2} b^{6} c^{6} d^{2}-120 a \,b^{7} c^{7} d -105 b^{8} c^{8}\right ) x^{2}}{56 b^{9}}+\frac {d \left (7129 a^{9} d^{9}-2520 a^{8} b c \,d^{8}-1260 a^{7} b^{2} c^{2} d^{7}-840 a^{6} b^{3} c^{3} d^{6}-630 a^{5} b^{4} c^{4} d^{5}-504 a^{4} b^{5} c^{5} d^{4}-420 a^{3} b^{6} c^{6} d^{3}-360 a^{2} b^{7} c^{7} d^{2}-315 a \,b^{8} c^{8} d -280 b^{9} c^{9}\right ) x}{252 b^{10}}+\frac {7381 a^{10} d^{10}-2520 a^{9} b c \,d^{9}-1260 a^{8} b^{2} c^{2} d^{8}-840 a^{7} b^{3} c^{3} d^{7}-630 a^{6} b^{4} c^{4} d^{6}-504 a^{5} b^{5} c^{5} d^{5}-420 a^{4} b^{6} c^{6} d^{4}-360 a^{3} b^{7} c^{7} d^{3}-315 a^{2} b^{8} c^{8} d^{2}-280 a \,b^{9} c^{9} d -252 b^{10} c^{10}}{2520 b^{11}}}{\left (b x +a \right )^{10}}+\frac {d^{10} \ln \left (b x +a \right )}{b^{11}}\) | \(843\) |
norman | \(\frac {\frac {7381 a^{10} d^{10}-2520 a^{9} b c \,d^{9}-1260 a^{8} b^{2} c^{2} d^{8}-840 a^{7} b^{3} c^{3} d^{7}-630 a^{6} b^{4} c^{4} d^{6}-504 a^{5} b^{5} c^{5} d^{5}-420 a^{4} b^{6} c^{6} d^{4}-360 a^{3} b^{7} c^{7} d^{3}-315 a^{2} b^{8} c^{8} d^{2}-280 a \,b^{9} c^{9} d -252 b^{10} c^{10}}{2520 b^{11}}+\frac {10 \left (a \,d^{10}-b c \,d^{9}\right ) x^{9}}{b^{2}}+\frac {45 \left (3 a^{2} d^{10}-2 a b c \,d^{9}-b^{2} c^{2} d^{8}\right ) x^{8}}{2 b^{3}}+\frac {20 \left (11 a^{3} d^{10}-6 a^{2} b c \,d^{9}-3 a \,b^{2} c^{2} d^{8}-2 b^{3} c^{3} d^{7}\right ) x^{7}}{b^{4}}+\frac {35 \left (25 a^{4} d^{10}-12 a^{3} b c \,d^{9}-6 a^{2} b^{2} c^{2} d^{8}-4 a \,b^{3} c^{3} d^{7}-3 b^{4} c^{4} d^{6}\right ) x^{6}}{2 b^{5}}+\frac {21 \left (137 a^{5} d^{10}-60 a^{4} b c \,d^{9}-30 a^{3} b^{2} c^{2} d^{8}-20 a^{2} b^{3} c^{3} d^{7}-15 a \,b^{4} c^{4} d^{6}-12 b^{5} c^{5} d^{5}\right ) x^{5}}{5 b^{6}}+\frac {7 \left (147 a^{6} d^{10}-60 a^{5} b c \,d^{9}-30 a^{4} b^{2} c^{2} d^{8}-20 a^{3} b^{3} c^{3} d^{7}-15 a^{2} b^{4} c^{4} d^{6}-12 a \,b^{5} c^{5} d^{5}-10 b^{6} c^{6} d^{4}\right ) x^{4}}{2 b^{7}}+\frac {2 \left (1089 a^{7} d^{10}-420 a^{6} b c \,d^{9}-210 a^{5} b^{2} c^{2} d^{8}-140 a^{4} b^{3} c^{3} d^{7}-105 a^{3} b^{4} c^{4} d^{6}-84 a^{2} b^{5} c^{5} d^{5}-70 a \,b^{6} c^{6} d^{4}-60 b^{7} c^{7} d^{3}\right ) x^{3}}{7 b^{8}}+\frac {3 \left (2283 a^{8} d^{10}-840 a^{7} b c \,d^{9}-420 a^{6} b^{2} c^{2} d^{8}-280 a^{5} b^{3} c^{3} d^{7}-210 a^{4} b^{4} c^{4} d^{6}-168 a^{3} b^{5} c^{5} d^{5}-140 a^{2} b^{6} c^{6} d^{4}-120 a \,b^{7} c^{7} d^{3}-105 b^{8} c^{8} d^{2}\right ) x^{2}}{56 b^{9}}+\frac {\left (7129 a^{9} d^{10}-2520 a^{8} b c \,d^{9}-1260 a^{7} b^{2} c^{2} d^{8}-840 a^{6} b^{3} c^{3} d^{7}-630 a^{5} b^{4} c^{4} d^{6}-504 a^{4} b^{5} c^{5} d^{5}-420 a^{3} b^{6} c^{6} d^{4}-360 a^{2} b^{7} c^{7} d^{3}-315 a \,b^{8} c^{8} d^{2}-280 b^{9} c^{9} d \right ) x}{252 b^{10}}}{\left (b x +a \right )^{10}}+\frac {d^{10} \ln \left (b x +a \right )}{b^{11}}\) | \(861\) |
default | \(\frac {40 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 )^{3}}+\frac {10 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 )}{9 b^{11} \left (b x +a \right )^{9}}+\frac {d^{10} \ln \left (b x +a \right )}{b^{11}}-\frac {35 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 )^{6}}-\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 )}{8 b^{11} \left (b x +a \right )^{8}}-\frac {105 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 )}{2 b^{11} \left (b x +a \right )^{4}}+\frac {120 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 )}{7 b^{11} \left (b x +a \right )^{7}}-\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}}{10 b^{11} \left (b x +a \right )^{10}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{11} \left (b x +a \right )^{2}}+\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 )}{5 b^{11} \left (b x +a \right )^{5}}+\frac {10 d^{9} \left (a d -b c \right )}{b^{11} \left (b x +a \right )}\) | \(865\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1157\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1107 vs. \(2 (257) = 514\).
Time = 0.23 (sec) , antiderivative size = 1107, normalized size of antiderivative = 4.08 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=-\frac {252 \, b^{10} c^{10} + 280 \, a b^{9} c^{9} d + 315 \, a^{2} b^{8} c^{8} d^{2} + 360 \, a^{3} b^{7} c^{7} d^{3} + 420 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} + 840 \, a^{7} b^{3} c^{3} d^{7} + 1260 \, a^{8} b^{2} c^{2} d^{8} + 2520 \, a^{9} b c d^{9} - 7381 \, a^{10} d^{10} + 25200 \, {\left (b^{10} c d^{9} - a b^{9} d^{10}\right )} x^{9} + 56700 \, {\left (b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} - 3 \, a^{2} b^{8} d^{10}\right )} x^{8} + 50400 \, {\left (2 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 6 \, a^{2} b^{8} c d^{9} - 11 \, a^{3} b^{7} d^{10}\right )} x^{7} + 44100 \, {\left (3 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} + 12 \, a^{3} b^{7} c d^{9} - 25 \, a^{4} b^{6} d^{10}\right )} x^{6} + 10584 \, {\left (12 \, b^{10} c^{5} d^{5} + 15 \, a b^{9} c^{4} d^{6} + 20 \, a^{2} b^{8} c^{3} d^{7} + 30 \, a^{3} b^{7} c^{2} d^{8} + 60 \, a^{4} b^{6} c d^{9} - 137 \, a^{5} b^{5} d^{10}\right )} x^{5} + 8820 \, {\left (10 \, b^{10} c^{6} d^{4} + 12 \, 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} + 30 \, a^{4} b^{6} c^{2} d^{8} + 60 \, a^{5} b^{5} c d^{9} - 147 \, a^{6} b^{4} d^{10}\right )} x^{4} + 720 \, {\left (60 \, b^{10} c^{7} d^{3} + 70 \, a b^{9} c^{6} d^{4} + 84 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} + 140 \, a^{4} b^{6} c^{3} d^{7} + 210 \, a^{5} b^{5} c^{2} d^{8} + 420 \, a^{6} b^{4} c d^{9} - 1089 \, a^{7} b^{3} d^{10}\right )} x^{3} + 135 \, {\left (105 \, b^{10} c^{8} d^{2} + 120 \, a b^{9} c^{7} d^{3} + 140 \, a^{2} b^{8} c^{6} d^{4} + 168 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} + 280 \, a^{5} b^{5} c^{3} d^{7} + 420 \, a^{6} b^{4} c^{2} d^{8} + 840 \, a^{7} b^{3} c d^{9} - 2283 \, a^{8} b^{2} d^{10}\right )} x^{2} + 10 \, {\left (280 \, b^{10} c^{9} d + 315 \, a b^{9} c^{8} d^{2} + 360 \, a^{2} b^{8} c^{7} d^{3} + 420 \, a^{3} b^{7} c^{6} d^{4} + 504 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} + 840 \, a^{6} b^{4} c^{3} d^{7} + 1260 \, a^{7} b^{3} c^{2} d^{8} + 2520 \, a^{8} b^{2} c d^{9} - 7129 \, a^{9} b d^{10}\right )} x - 2520 \, {\left (b^{10} d^{10} x^{10} + 10 \, a b^{9} d^{10} x^{9} + 45 \, a^{2} b^{8} d^{10} x^{8} + 120 \, a^{3} b^{7} d^{10} x^{7} + 210 \, a^{4} b^{6} d^{10} x^{6} + 252 \, a^{5} b^{5} d^{10} x^{5} + 210 \, a^{6} b^{4} d^{10} x^{4} + 120 \, a^{7} b^{3} d^{10} x^{3} + 45 \, a^{8} b^{2} d^{10} x^{2} + 10 \, a^{9} b d^{10} x + a^{10} d^{10}\right )} \log \left (b x + a\right )}{2520 \, {\left (b^{21} x^{10} + 10 \, a b^{20} x^{9} + 45 \, a^{2} b^{19} x^{8} + 120 \, a^{3} b^{18} x^{7} + 210 \, a^{4} b^{17} x^{6} + 252 \, a^{5} b^{16} x^{5} + 210 \, a^{6} b^{15} x^{4} + 120 \, a^{7} b^{14} x^{3} + 45 \, a^{8} b^{13} x^{2} + 10 \, a^{9} b^{12} x + a^{10} b^{11}\right )}} \]
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Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 975 vs. \(2 (257) = 514\).
Time = 0.25 (sec) , antiderivative size = 975, normalized size of antiderivative = 3.60 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=-\frac {252 \, b^{10} c^{10} + 280 \, a b^{9} c^{9} d + 315 \, a^{2} b^{8} c^{8} d^{2} + 360 \, a^{3} b^{7} c^{7} d^{3} + 420 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} + 840 \, a^{7} b^{3} c^{3} d^{7} + 1260 \, a^{8} b^{2} c^{2} d^{8} + 2520 \, a^{9} b c d^{9} - 7381 \, a^{10} d^{10} + 25200 \, {\left (b^{10} c d^{9} - a b^{9} d^{10}\right )} x^{9} + 56700 \, {\left (b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} - 3 \, a^{2} b^{8} d^{10}\right )} x^{8} + 50400 \, {\left (2 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 6 \, a^{2} b^{8} c d^{9} - 11 \, a^{3} b^{7} d^{10}\right )} x^{7} + 44100 \, {\left (3 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} + 12 \, a^{3} b^{7} c d^{9} - 25 \, a^{4} b^{6} d^{10}\right )} x^{6} + 10584 \, {\left (12 \, b^{10} c^{5} d^{5} + 15 \, a b^{9} c^{4} d^{6} + 20 \, a^{2} b^{8} c^{3} d^{7} + 30 \, a^{3} b^{7} c^{2} d^{8} + 60 \, a^{4} b^{6} c d^{9} - 137 \, a^{5} b^{5} d^{10}\right )} x^{5} + 8820 \, {\left (10 \, b^{10} c^{6} d^{4} + 12 \, 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} + 30 \, a^{4} b^{6} c^{2} d^{8} + 60 \, a^{5} b^{5} c d^{9} - 147 \, a^{6} b^{4} d^{10}\right )} x^{4} + 720 \, {\left (60 \, b^{10} c^{7} d^{3} + 70 \, a b^{9} c^{6} d^{4} + 84 \, a^{2} b^{8} c^{5} d^{5} + 105 \, a^{3} b^{7} c^{4} d^{6} + 140 \, a^{4} b^{6} c^{3} d^{7} + 210 \, a^{5} b^{5} c^{2} d^{8} + 420 \, a^{6} b^{4} c d^{9} - 1089 \, a^{7} b^{3} d^{10}\right )} x^{3} + 135 \, {\left (105 \, b^{10} c^{8} d^{2} + 120 \, a b^{9} c^{7} d^{3} + 140 \, a^{2} b^{8} c^{6} d^{4} + 168 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} + 280 \, a^{5} b^{5} c^{3} d^{7} + 420 \, a^{6} b^{4} c^{2} d^{8} + 840 \, a^{7} b^{3} c d^{9} - 2283 \, a^{8} b^{2} d^{10}\right )} x^{2} + 10 \, {\left (280 \, b^{10} c^{9} d + 315 \, a b^{9} c^{8} d^{2} + 360 \, a^{2} b^{8} c^{7} d^{3} + 420 \, a^{3} b^{7} c^{6} d^{4} + 504 \, a^{4} b^{6} c^{5} d^{5} + 630 \, a^{5} b^{5} c^{4} d^{6} + 840 \, a^{6} b^{4} c^{3} d^{7} + 1260 \, a^{7} b^{3} c^{2} d^{8} + 2520 \, a^{8} b^{2} c d^{9} - 7129 \, a^{9} b d^{10}\right )} x}{2520 \, {\left (b^{21} x^{10} + 10 \, a b^{20} x^{9} + 45 \, a^{2} b^{19} x^{8} + 120 \, a^{3} b^{18} x^{7} + 210 \, a^{4} b^{17} x^{6} + 252 \, a^{5} b^{16} x^{5} + 210 \, a^{6} b^{15} x^{4} + 120 \, a^{7} b^{14} x^{3} + 45 \, a^{8} b^{13} x^{2} + 10 \, a^{9} b^{12} x + a^{10} b^{11}\right )}} + \frac {d^{10} \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. 874 vs. \(2 (257) = 514\).
Time = 0.30 (sec) , antiderivative size = 874, normalized size of antiderivative = 3.23 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=\frac {d^{10} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {25200 \, {\left (b^{9} c d^{9} - a b^{8} d^{10}\right )} x^{9} + 56700 \, {\left (b^{9} c^{2} d^{8} + 2 \, a b^{8} c d^{9} - 3 \, a^{2} b^{7} d^{10}\right )} x^{8} + 50400 \, {\left (2 \, b^{9} c^{3} d^{7} + 3 \, a b^{8} c^{2} d^{8} + 6 \, a^{2} b^{7} c d^{9} - 11 \, a^{3} b^{6} d^{10}\right )} x^{7} + 44100 \, {\left (3 \, b^{9} c^{4} d^{6} + 4 \, a b^{8} c^{3} d^{7} + 6 \, a^{2} b^{7} c^{2} d^{8} + 12 \, a^{3} b^{6} c d^{9} - 25 \, a^{4} b^{5} d^{10}\right )} x^{6} + 10584 \, {\left (12 \, b^{9} c^{5} d^{5} + 15 \, a b^{8} c^{4} d^{6} + 20 \, a^{2} b^{7} c^{3} d^{7} + 30 \, a^{3} b^{6} c^{2} d^{8} + 60 \, a^{4} b^{5} c d^{9} - 137 \, a^{5} b^{4} d^{10}\right )} x^{5} + 8820 \, {\left (10 \, b^{9} c^{6} d^{4} + 12 \, a b^{8} c^{5} d^{5} + 15 \, a^{2} b^{7} c^{4} d^{6} + 20 \, a^{3} b^{6} c^{3} d^{7} + 30 \, a^{4} b^{5} c^{2} d^{8} + 60 \, a^{5} b^{4} c d^{9} - 147 \, a^{6} b^{3} d^{10}\right )} x^{4} + 720 \, {\left (60 \, b^{9} c^{7} d^{3} + 70 \, a b^{8} c^{6} d^{4} + 84 \, a^{2} b^{7} c^{5} d^{5} + 105 \, a^{3} b^{6} c^{4} d^{6} + 140 \, a^{4} b^{5} c^{3} d^{7} + 210 \, a^{5} b^{4} c^{2} d^{8} + 420 \, a^{6} b^{3} c d^{9} - 1089 \, a^{7} b^{2} d^{10}\right )} x^{3} + 135 \, {\left (105 \, b^{9} c^{8} d^{2} + 120 \, a b^{8} c^{7} d^{3} + 140 \, a^{2} b^{7} c^{6} d^{4} + 168 \, a^{3} b^{6} c^{5} d^{5} + 210 \, a^{4} b^{5} c^{4} d^{6} + 280 \, a^{5} b^{4} c^{3} d^{7} + 420 \, a^{6} b^{3} c^{2} d^{8} + 840 \, a^{7} b^{2} c d^{9} - 2283 \, a^{8} b d^{10}\right )} x^{2} + 10 \, {\left (280 \, b^{9} c^{9} d + 315 \, a b^{8} c^{8} d^{2} + 360 \, a^{2} b^{7} c^{7} d^{3} + 420 \, a^{3} b^{6} c^{6} d^{4} + 504 \, a^{4} b^{5} c^{5} d^{5} + 630 \, a^{5} b^{4} c^{4} d^{6} + 840 \, a^{6} b^{3} c^{3} d^{7} + 1260 \, a^{7} b^{2} c^{2} d^{8} + 2520 \, a^{8} b c d^{9} - 7129 \, a^{9} d^{10}\right )} x + \frac {252 \, b^{10} c^{10} + 280 \, a b^{9} c^{9} d + 315 \, a^{2} b^{8} c^{8} d^{2} + 360 \, a^{3} b^{7} c^{7} d^{3} + 420 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} + 630 \, a^{6} b^{4} c^{4} d^{6} + 840 \, a^{7} b^{3} c^{3} d^{7} + 1260 \, a^{8} b^{2} c^{2} d^{8} + 2520 \, a^{9} b c d^{9} - 7381 \, a^{10} d^{10}}{b}}{2520 \, {\left (b x + a\right )}^{10} b^{10}} \]
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Time = 0.70 (sec) , antiderivative size = 866, normalized size of antiderivative = 3.20 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=\frac {d^{10}\,\ln \left (a+b\,x\right )}{b^{11}}-\frac {x^4\,\left (-\frac {1029\,a^6\,b^4\,d^{10}}{2}+210\,a^5\,b^5\,c\,d^9+105\,a^4\,b^6\,c^2\,d^8+70\,a^3\,b^7\,c^3\,d^7+\frac {105\,a^2\,b^8\,c^4\,d^6}{2}+42\,a\,b^9\,c^5\,d^5+35\,b^{10}\,c^6\,d^4\right )-x^9\,\left (10\,a\,b^9\,d^{10}-10\,b^{10}\,c\,d^9\right )+x\,\left (-\frac {7129\,a^9\,b\,d^{10}}{252}+10\,a^8\,b^2\,c\,d^9+5\,a^7\,b^3\,c^2\,d^8+\frac {10\,a^6\,b^4\,c^3\,d^7}{3}+\frac {5\,a^5\,b^5\,c^4\,d^6}{2}+2\,a^4\,b^6\,c^5\,d^5+\frac {5\,a^3\,b^7\,c^6\,d^4}{3}+\frac {10\,a^2\,b^8\,c^7\,d^3}{7}+\frac {5\,a\,b^9\,c^8\,d^2}{4}+\frac {10\,b^{10}\,c^9\,d}{9}\right )+x^6\,\left (-\frac {875\,a^4\,b^6\,d^{10}}{2}+210\,a^3\,b^7\,c\,d^9+105\,a^2\,b^8\,c^2\,d^8+70\,a\,b^9\,c^3\,d^7+\frac {105\,b^{10}\,c^4\,d^6}{2}\right )+x^8\,\left (-\frac {135\,a^2\,b^8\,d^{10}}{2}+45\,a\,b^9\,c\,d^9+\frac {45\,b^{10}\,c^2\,d^8}{2}\right )+x^3\,\left (-\frac {2178\,a^7\,b^3\,d^{10}}{7}+120\,a^6\,b^4\,c\,d^9+60\,a^5\,b^5\,c^2\,d^8+40\,a^4\,b^6\,c^3\,d^7+30\,a^3\,b^7\,c^4\,d^6+24\,a^2\,b^8\,c^5\,d^5+20\,a\,b^9\,c^6\,d^4+\frac {120\,b^{10}\,c^7\,d^3}{7}\right )+x^5\,\left (-\frac {2877\,a^5\,b^5\,d^{10}}{5}+252\,a^4\,b^6\,c\,d^9+126\,a^3\,b^7\,c^2\,d^8+84\,a^2\,b^8\,c^3\,d^7+63\,a\,b^9\,c^4\,d^6+\frac {252\,b^{10}\,c^5\,d^5}{5}\right )-\frac {7381\,a^{10}\,d^{10}}{2520}+\frac {b^{10}\,c^{10}}{10}+x^7\,\left (-220\,a^3\,b^7\,d^{10}+120\,a^2\,b^8\,c\,d^9+60\,a\,b^9\,c^2\,d^8+40\,b^{10}\,c^3\,d^7\right )+x^2\,\left (-\frac {6849\,a^8\,b^2\,d^{10}}{56}+45\,a^7\,b^3\,c\,d^9+\frac {45\,a^6\,b^4\,c^2\,d^8}{2}+15\,a^5\,b^5\,c^3\,d^7+\frac {45\,a^4\,b^6\,c^4\,d^6}{4}+9\,a^3\,b^7\,c^5\,d^5+\frac {15\,a^2\,b^8\,c^6\,d^4}{2}+\frac {45\,a\,b^9\,c^7\,d^3}{7}+\frac {45\,b^{10}\,c^8\,d^2}{8}\right )+\frac {a^2\,b^8\,c^8\,d^2}{8}+\frac {a^3\,b^7\,c^7\,d^3}{7}+\frac {a^4\,b^6\,c^6\,d^4}{6}+\frac {a^5\,b^5\,c^5\,d^5}{5}+\frac {a^6\,b^4\,c^4\,d^6}{4}+\frac {a^7\,b^3\,c^3\,d^7}{3}+\frac {a^8\,b^2\,c^2\,d^8}{2}+\frac {a\,b^9\,c^9\,d}{9}+a^9\,b\,c\,d^9}{b^{11}\,{\left (a+b\,x\right )}^{10}} \]
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