Integrand size = 15, antiderivative size = 58 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=-\frac {(c+d x)^{11}}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^{11}}{132 (b c-a d)^2 (a+b x)^{11}} \]
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Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=\frac {d (c+d x)^{11}}{132 (a+b x)^{11} (b c-a d)^2}-\frac {(c+d x)^{11}}{12 (a+b x)^{12} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{11}}{12 (b c-a d) (a+b x)^{12}}-\frac {d \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx}{12 (b c-a d)} \\ & = -\frac {(c+d x)^{11}}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^{11}}{132 (b c-a d)^2 (a+b x)^{11}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(684\) vs. \(2(58)=116\).
Time = 0.17 (sec) , antiderivative size = 684, normalized size of antiderivative = 11.79 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=-\frac {a^{10} d^{10}+2 a^9 b d^9 (c+6 d x)+3 a^8 b^2 d^8 \left (c^2+8 c d x+22 d^2 x^2\right )+4 a^7 b^3 d^7 \left (c^3+9 c^2 d x+33 c d^2 x^2+55 d^3 x^3\right )+a^6 b^4 d^6 \left (5 c^4+48 c^3 d x+198 c^2 d^2 x^2+440 c d^3 x^3+495 d^4 x^4\right )+6 a^5 b^5 d^5 \left (c^5+10 c^4 d x+44 c^3 d^2 x^2+110 c^2 d^3 x^3+165 c d^4 x^4+132 d^5 x^5\right )+a^4 b^6 d^4 \left (7 c^6+72 c^5 d x+330 c^4 d^2 x^2+880 c^3 d^3 x^3+1485 c^2 d^4 x^4+1584 c d^5 x^5+924 d^6 x^6\right )+4 a^3 b^7 d^3 \left (2 c^7+21 c^6 d x+99 c^5 d^2 x^2+275 c^4 d^3 x^3+495 c^3 d^4 x^4+594 c^2 d^5 x^5+462 c d^6 x^6+198 d^7 x^7\right )+3 a^2 b^8 d^2 \left (3 c^8+32 c^7 d x+154 c^6 d^2 x^2+440 c^5 d^3 x^3+825 c^4 d^4 x^4+1056 c^3 d^5 x^5+924 c^2 d^6 x^6+528 c d^7 x^7+165 d^8 x^8\right )+2 a b^9 d \left (5 c^9+54 c^8 d x+264 c^7 d^2 x^2+770 c^6 d^3 x^3+1485 c^5 d^4 x^4+1980 c^4 d^5 x^5+1848 c^3 d^6 x^6+1188 c^2 d^7 x^7+495 c d^8 x^8+110 d^9 x^9\right )+b^{10} \left (11 c^{10}+120 c^9 d x+594 c^8 d^2 x^2+1760 c^7 d^3 x^3+3465 c^6 d^4 x^4+4752 c^5 d^5 x^5+4620 c^4 d^6 x^6+3168 c^3 d^7 x^7+1485 c^2 d^8 x^8+440 c d^9 x^9+66 d^{10} x^{10}\right )}{132 b^{11} (a+b x)^{12}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(54)=108\).
Time = 0.23 (sec) , antiderivative size = 831, normalized size of antiderivative = 14.33
method | result | size |
risch | \(\frac {-\frac {d^{10} x^{10}}{2 b}-\frac {5 d^{9} \left (a d +2 b c \right ) x^{9}}{3 b^{2}}-\frac {15 d^{8} \left (a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) x^{8}}{4 b^{3}}-\frac {6 d^{7} \left (a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) x^{7}}{b^{4}}-\frac {7 d^{6} \left (a^{4} d^{4}+2 a^{3} b c \,d^{3}+3 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +5 b^{4} c^{4}\right ) x^{6}}{b^{5}}-\frac {6 d^{5} \left (a^{5} d^{5}+2 a^{4} b c \,d^{4}+3 a^{3} b^{2} c^{2} d^{3}+4 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +6 b^{5} c^{5}\right ) x^{5}}{b^{6}}-\frac {15 d^{4} \left (a^{6} d^{6}+2 a^{5} b c \,d^{5}+3 a^{4} b^{2} c^{2} d^{4}+4 a^{3} b^{3} c^{3} d^{3}+5 a^{2} b^{4} c^{4} d^{2}+6 a \,b^{5} c^{5} d +7 b^{6} c^{6}\right ) x^{4}}{4 b^{7}}-\frac {5 d^{3} \left (a^{7} d^{7}+2 a^{6} b c \,d^{6}+3 a^{5} b^{2} c^{2} d^{5}+4 a^{4} b^{3} c^{3} d^{4}+5 a^{3} b^{4} c^{4} d^{3}+6 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +8 b^{7} c^{7}\right ) x^{3}}{3 b^{8}}-\frac {d^{2} \left (a^{8} d^{8}+2 a^{7} b c \,d^{7}+3 a^{6} b^{2} c^{2} d^{6}+4 a^{5} b^{3} c^{3} d^{5}+5 a^{4} b^{4} c^{4} d^{4}+6 a^{3} b^{5} c^{5} d^{3}+7 a^{2} b^{6} c^{6} d^{2}+8 a \,b^{7} c^{7} d +9 b^{8} c^{8}\right ) x^{2}}{2 b^{9}}-\frac {d \left (a^{9} d^{9}+2 a^{8} b c \,d^{8}+3 a^{7} b^{2} c^{2} d^{7}+4 a^{6} b^{3} c^{3} d^{6}+5 a^{5} b^{4} c^{4} d^{5}+6 a^{4} b^{5} c^{5} d^{4}+7 a^{3} b^{6} c^{6} d^{3}+8 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d +10 b^{9} c^{9}\right ) x}{11 b^{10}}-\frac {a^{10} d^{10}+2 a^{9} b c \,d^{9}+3 a^{8} b^{2} c^{2} d^{8}+4 a^{7} b^{3} c^{3} d^{7}+5 a^{6} b^{4} c^{4} d^{6}+6 a^{5} b^{5} c^{5} d^{5}+7 a^{4} b^{6} c^{6} d^{4}+8 a^{3} b^{7} c^{7} d^{3}+9 a^{2} b^{8} c^{8} d^{2}+10 a \,b^{9} c^{9} d +11 b^{10} c^{10}}{132 b^{11}}}{\left (b x +a \right )^{12}}\) | \(831\) |
default | \(\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 )}{11 b^{11} \left (b x +a \right )^{11}}+\frac {10 d^{9} \left (a d -b c \right )}{3 b^{11} \left (b x +a \right )^{3}}+\frac {40 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 )}{3 b^{11} \left (b x +a \right )^{9}}-\frac {35 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 )^{6}}-\frac {105 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 )}{4 b^{11} \left (b x +a \right )^{8}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 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}}{12 b^{11} \left (b x +a \right )^{12}}+\frac {36 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 )^{7}}-\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 )}{2 b^{11} \left (b x +a \right )^{10}}-\frac {d^{10}}{2 b^{11} \left (b x +a \right )^{2}}+\frac {24 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 )^{5}}\) | \(867\) |
norman | \(\frac {-\frac {d^{10} x^{10}}{2 b}+\frac {5 \left (-a b \,d^{10}-2 b^{2} c \,d^{9}\right ) x^{9}}{3 b^{3}}+\frac {15 \left (-a^{2} b \,d^{10}-2 a \,b^{2} c \,d^{9}-3 b^{3} c^{2} d^{8}\right ) x^{8}}{4 b^{4}}+\frac {6 \left (-a^{3} b \,d^{10}-2 a^{2} b^{2} c \,d^{9}-3 a \,b^{3} c^{2} d^{8}-4 b^{4} c^{3} d^{7}\right ) x^{7}}{b^{5}}+\frac {7 \left (-a^{4} b \,d^{10}-2 a^{3} b^{2} c \,d^{9}-3 a^{2} b^{3} c^{2} d^{8}-4 a \,b^{4} c^{3} d^{7}-5 b^{5} c^{4} d^{6}\right ) x^{6}}{b^{6}}+\frac {6 \left (-a^{5} b \,d^{10}-2 a^{4} b^{2} c \,d^{9}-3 a^{3} b^{3} c^{2} d^{8}-4 a^{2} b^{4} c^{3} d^{7}-5 a \,b^{5} c^{4} d^{6}-6 b^{6} c^{5} d^{5}\right ) x^{5}}{b^{7}}+\frac {15 \left (-a^{6} b \,d^{10}-2 a^{5} b^{2} c \,d^{9}-3 a^{4} b^{3} c^{2} d^{8}-4 a^{3} b^{4} c^{3} d^{7}-5 a^{2} b^{5} c^{4} d^{6}-6 a \,b^{6} c^{5} d^{5}-7 b^{7} c^{6} d^{4}\right ) x^{4}}{4 b^{8}}+\frac {5 \left (-a^{7} b \,d^{10}-2 a^{6} b^{2} c \,d^{9}-3 a^{5} b^{3} c^{2} d^{8}-4 a^{4} b^{4} c^{3} d^{7}-5 a^{3} b^{5} c^{4} d^{6}-6 a^{2} b^{6} c^{5} d^{5}-7 a \,b^{7} c^{6} d^{4}-8 b^{8} c^{7} d^{3}\right ) x^{3}}{3 b^{9}}+\frac {\left (-a^{8} b \,d^{10}-2 a^{7} b^{2} c \,d^{9}-3 a^{6} b^{3} c^{2} d^{8}-4 a^{5} b^{4} c^{3} d^{7}-5 a^{4} b^{5} c^{4} d^{6}-6 a^{3} b^{6} c^{5} d^{5}-7 a^{2} b^{7} c^{6} d^{4}-8 a \,b^{8} c^{7} d^{3}-9 b^{9} c^{8} d^{2}\right ) x^{2}}{2 b^{10}}+\frac {\left (-a^{9} b \,d^{10}-2 a^{8} b^{2} c \,d^{9}-3 a^{7} b^{3} c^{2} d^{8}-4 a^{6} b^{4} c^{3} d^{7}-5 a^{5} b^{5} c^{4} d^{6}-6 a^{4} b^{6} c^{5} d^{5}-7 a^{3} b^{7} c^{6} d^{4}-8 a^{2} b^{8} c^{7} d^{3}-9 a \,b^{9} c^{8} d^{2}-10 b^{10} c^{9} d \right ) x}{11 b^{11}}+\frac {-a^{10} b \,d^{10}-2 a^{9} b^{2} c \,d^{9}-3 a^{8} b^{3} c^{2} d^{8}-4 a^{7} b^{4} c^{3} d^{7}-5 a^{6} b^{5} c^{4} d^{6}-6 a^{5} b^{6} c^{5} d^{5}-7 a^{4} b^{7} c^{6} d^{4}-8 a^{3} b^{8} c^{7} d^{3}-9 a^{2} b^{9} c^{8} d^{2}-10 a \,b^{10} c^{9} d -11 b^{11} c^{10}}{132 b^{12}}}{\left (b x +a \right )^{12}}\) | \(889\) |
gosper | \(-\frac {66 x^{10} d^{10} b^{10}+220 x^{9} a \,b^{9} d^{10}+440 x^{9} b^{10} c \,d^{9}+495 x^{8} a^{2} b^{8} d^{10}+990 x^{8} a \,b^{9} c \,d^{9}+1485 x^{8} b^{10} c^{2} d^{8}+792 x^{7} a^{3} b^{7} d^{10}+1584 x^{7} a^{2} b^{8} c \,d^{9}+2376 x^{7} a \,b^{9} c^{2} d^{8}+3168 x^{7} b^{10} c^{3} d^{7}+924 x^{6} a^{4} b^{6} d^{10}+1848 x^{6} a^{3} b^{7} c \,d^{9}+2772 x^{6} a^{2} b^{8} c^{2} d^{8}+3696 x^{6} a \,b^{9} c^{3} d^{7}+4620 x^{6} b^{10} c^{4} d^{6}+792 x^{5} a^{5} b^{5} d^{10}+1584 x^{5} a^{4} b^{6} c \,d^{9}+2376 x^{5} a^{3} b^{7} c^{2} d^{8}+3168 x^{5} a^{2} b^{8} c^{3} d^{7}+3960 x^{5} a \,b^{9} c^{4} d^{6}+4752 x^{5} b^{10} c^{5} d^{5}+495 x^{4} a^{6} b^{4} d^{10}+990 x^{4} a^{5} b^{5} c \,d^{9}+1485 x^{4} a^{4} b^{6} c^{2} d^{8}+1980 x^{4} a^{3} b^{7} c^{3} d^{7}+2475 x^{4} a^{2} b^{8} c^{4} d^{6}+2970 x^{4} a \,b^{9} c^{5} d^{5}+3465 x^{4} b^{10} c^{6} d^{4}+220 x^{3} a^{7} b^{3} d^{10}+440 x^{3} a^{6} b^{4} c \,d^{9}+660 x^{3} a^{5} b^{5} c^{2} d^{8}+880 x^{3} a^{4} b^{6} c^{3} d^{7}+1100 x^{3} a^{3} b^{7} c^{4} d^{6}+1320 x^{3} a^{2} b^{8} c^{5} d^{5}+1540 x^{3} a \,b^{9} c^{6} d^{4}+1760 x^{3} b^{10} c^{7} d^{3}+66 x^{2} a^{8} b^{2} d^{10}+132 x^{2} a^{7} b^{3} c \,d^{9}+198 x^{2} a^{6} b^{4} c^{2} d^{8}+264 x^{2} a^{5} b^{5} c^{3} d^{7}+330 x^{2} a^{4} b^{6} c^{4} d^{6}+396 x^{2} a^{3} b^{7} c^{5} d^{5}+462 x^{2} a^{2} b^{8} c^{6} d^{4}+528 x^{2} a \,b^{9} c^{7} d^{3}+594 x^{2} b^{10} c^{8} d^{2}+12 x \,a^{9} b \,d^{10}+24 x \,a^{8} b^{2} c \,d^{9}+36 x \,a^{7} b^{3} c^{2} d^{8}+48 x \,a^{6} b^{4} c^{3} d^{7}+60 x \,a^{5} b^{5} c^{4} d^{6}+72 x \,a^{4} b^{6} c^{5} d^{5}+84 x \,a^{3} b^{7} c^{6} d^{4}+96 x \,a^{2} b^{8} c^{7} d^{3}+108 x a \,b^{9} c^{8} d^{2}+120 x \,b^{10} c^{9} d +a^{10} d^{10}+2 a^{9} b c \,d^{9}+3 a^{8} b^{2} c^{2} d^{8}+4 a^{7} b^{3} c^{3} d^{7}+5 a^{6} b^{4} c^{4} d^{6}+6 a^{5} b^{5} c^{5} d^{5}+7 a^{4} b^{6} c^{6} d^{4}+8 a^{3} b^{7} c^{7} d^{3}+9 a^{2} b^{8} c^{8} d^{2}+10 a \,b^{9} c^{9} d +11 b^{10} c^{10}}{132 b^{11} \left (b x +a \right )^{12}}\) | \(962\) |
parallelrisch | \(\frac {-66 d^{10} x^{10} b^{11}-220 a \,b^{10} d^{10} x^{9}-440 b^{11} c \,d^{9} x^{9}-495 a^{2} b^{9} d^{10} x^{8}-990 a \,b^{10} c \,d^{9} x^{8}-1485 b^{11} c^{2} d^{8} x^{8}-792 a^{3} b^{8} d^{10} x^{7}-1584 a^{2} b^{9} c \,d^{9} x^{7}-2376 a \,b^{10} c^{2} d^{8} x^{7}-3168 b^{11} c^{3} d^{7} x^{7}-924 a^{4} b^{7} d^{10} x^{6}-1848 a^{3} b^{8} c \,d^{9} x^{6}-2772 a^{2} b^{9} c^{2} d^{8} x^{6}-3696 a \,b^{10} c^{3} d^{7} x^{6}-4620 b^{11} c^{4} d^{6} x^{6}-792 a^{5} b^{6} d^{10} x^{5}-1584 a^{4} b^{7} c \,d^{9} x^{5}-2376 a^{3} b^{8} c^{2} d^{8} x^{5}-3168 a^{2} b^{9} c^{3} d^{7} x^{5}-3960 a \,b^{10} c^{4} d^{6} x^{5}-4752 b^{11} c^{5} d^{5} x^{5}-495 a^{6} b^{5} d^{10} x^{4}-990 a^{5} b^{6} c \,d^{9} x^{4}-1485 a^{4} b^{7} c^{2} d^{8} x^{4}-1980 a^{3} b^{8} c^{3} d^{7} x^{4}-2475 a^{2} b^{9} c^{4} d^{6} x^{4}-2970 a \,b^{10} c^{5} d^{5} x^{4}-3465 b^{11} c^{6} d^{4} x^{4}-220 a^{7} b^{4} d^{10} x^{3}-440 a^{6} b^{5} c \,d^{9} x^{3}-660 a^{5} b^{6} c^{2} d^{8} x^{3}-880 a^{4} b^{7} c^{3} d^{7} x^{3}-1100 a^{3} b^{8} c^{4} d^{6} x^{3}-1320 a^{2} b^{9} c^{5} d^{5} x^{3}-1540 a \,b^{10} c^{6} d^{4} x^{3}-1760 b^{11} c^{7} d^{3} x^{3}-66 a^{8} b^{3} d^{10} x^{2}-132 a^{7} b^{4} c \,d^{9} x^{2}-198 a^{6} b^{5} c^{2} d^{8} x^{2}-264 a^{5} b^{6} c^{3} d^{7} x^{2}-330 a^{4} b^{7} c^{4} d^{6} x^{2}-396 a^{3} b^{8} c^{5} d^{5} x^{2}-462 a^{2} b^{9} c^{6} d^{4} x^{2}-528 a \,b^{10} c^{7} d^{3} x^{2}-594 b^{11} c^{8} d^{2} x^{2}-12 a^{9} b^{2} d^{10} x -24 a^{8} b^{3} c \,d^{9} x -36 a^{7} b^{4} c^{2} d^{8} x -48 a^{6} b^{5} c^{3} d^{7} x -60 a^{5} b^{6} c^{4} d^{6} x -72 a^{4} b^{7} c^{5} d^{5} x -84 a^{3} b^{8} c^{6} d^{4} x -96 a^{2} b^{9} c^{7} d^{3} x -108 a \,b^{10} c^{8} d^{2} x -120 b^{11} c^{9} d x -a^{10} b \,d^{10}-2 a^{9} b^{2} c \,d^{9}-3 a^{8} b^{3} c^{2} d^{8}-4 a^{7} b^{4} c^{3} d^{7}-5 a^{6} b^{5} c^{4} d^{6}-6 a^{5} b^{6} c^{5} d^{5}-7 a^{4} b^{7} c^{6} d^{4}-8 a^{3} b^{8} c^{7} d^{3}-9 a^{2} b^{9} c^{8} d^{2}-10 a \,b^{10} c^{9} d -11 b^{11} c^{10}}{132 b^{12} \left (b x +a \right )^{12}}\) | \(968\) |
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Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (54) = 108\).
Time = 0.24 (sec) , antiderivative size = 986, normalized size of antiderivative = 17.00 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=-\frac {66 \, b^{10} d^{10} x^{10} + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10} + 220 \, {\left (2 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 495 \, {\left (3 \, b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 792 \, {\left (4 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 2 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 924 \, {\left (5 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 3 \, a^{2} b^{8} c^{2} d^{8} + 2 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 792 \, {\left (6 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 4 \, a^{2} b^{8} c^{3} d^{7} + 3 \, a^{3} b^{7} c^{2} d^{8} + 2 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 495 \, {\left (7 \, b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 4 \, a^{3} b^{7} c^{3} d^{7} + 3 \, a^{4} b^{6} c^{2} d^{8} + 2 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 220 \, {\left (8 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 6 \, a^{2} b^{8} c^{5} d^{5} + 5 \, a^{3} b^{7} c^{4} d^{6} + 4 \, a^{4} b^{6} c^{3} d^{7} + 3 \, a^{5} b^{5} c^{2} d^{8} + 2 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 66 \, {\left (9 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 7 \, a^{2} b^{8} c^{6} d^{4} + 6 \, a^{3} b^{7} c^{5} d^{5} + 5 \, a^{4} b^{6} c^{4} d^{6} + 4 \, a^{5} b^{5} c^{3} d^{7} + 3 \, a^{6} b^{4} c^{2} d^{8} + 2 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 12 \, {\left (10 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 8 \, a^{2} b^{8} c^{7} d^{3} + 7 \, a^{3} b^{7} c^{6} d^{4} + 6 \, a^{4} b^{6} c^{5} d^{5} + 5 \, a^{5} b^{5} c^{4} d^{6} + 4 \, a^{6} b^{4} c^{3} d^{7} + 3 \, a^{7} b^{3} c^{2} d^{8} + 2 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{132 \, {\left (b^{23} x^{12} + 12 \, a b^{22} x^{11} + 66 \, a^{2} b^{21} x^{10} + 220 \, a^{3} b^{20} x^{9} + 495 \, a^{4} b^{19} x^{8} + 792 \, a^{5} b^{18} x^{7} + 924 \, a^{6} b^{17} x^{6} + 792 \, a^{7} b^{16} x^{5} + 495 \, a^{8} b^{15} x^{4} + 220 \, a^{9} b^{14} x^{3} + 66 \, a^{10} b^{13} x^{2} + 12 \, a^{11} b^{12} x + a^{12} b^{11}\right )}} \]
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Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (54) = 108\).
Time = 0.26 (sec) , antiderivative size = 986, normalized size of antiderivative = 17.00 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=-\frac {66 \, b^{10} d^{10} x^{10} + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10} + 220 \, {\left (2 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 495 \, {\left (3 \, b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 792 \, {\left (4 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 2 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 924 \, {\left (5 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 3 \, a^{2} b^{8} c^{2} d^{8} + 2 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 792 \, {\left (6 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 4 \, a^{2} b^{8} c^{3} d^{7} + 3 \, a^{3} b^{7} c^{2} d^{8} + 2 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 495 \, {\left (7 \, b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 4 \, a^{3} b^{7} c^{3} d^{7} + 3 \, a^{4} b^{6} c^{2} d^{8} + 2 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 220 \, {\left (8 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 6 \, a^{2} b^{8} c^{5} d^{5} + 5 \, a^{3} b^{7} c^{4} d^{6} + 4 \, a^{4} b^{6} c^{3} d^{7} + 3 \, a^{5} b^{5} c^{2} d^{8} + 2 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 66 \, {\left (9 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 7 \, a^{2} b^{8} c^{6} d^{4} + 6 \, a^{3} b^{7} c^{5} d^{5} + 5 \, a^{4} b^{6} c^{4} d^{6} + 4 \, a^{5} b^{5} c^{3} d^{7} + 3 \, a^{6} b^{4} c^{2} d^{8} + 2 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 12 \, {\left (10 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 8 \, a^{2} b^{8} c^{7} d^{3} + 7 \, a^{3} b^{7} c^{6} d^{4} + 6 \, a^{4} b^{6} c^{5} d^{5} + 5 \, a^{5} b^{5} c^{4} d^{6} + 4 \, a^{6} b^{4} c^{3} d^{7} + 3 \, a^{7} b^{3} c^{2} d^{8} + 2 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{132 \, {\left (b^{23} x^{12} + 12 \, a b^{22} x^{11} + 66 \, a^{2} b^{21} x^{10} + 220 \, a^{3} b^{20} x^{9} + 495 \, a^{4} b^{19} x^{8} + 792 \, a^{5} b^{18} x^{7} + 924 \, a^{6} b^{17} x^{6} + 792 \, a^{7} b^{16} x^{5} + 495 \, a^{8} b^{15} x^{4} + 220 \, a^{9} b^{14} x^{3} + 66 \, a^{10} b^{13} x^{2} + 12 \, a^{11} b^{12} x + a^{12} b^{11}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (54) = 108\).
Time = 0.34 (sec) , antiderivative size = 961, normalized size of antiderivative = 16.57 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=-\frac {66 \, b^{10} d^{10} x^{10} + 440 \, b^{10} c d^{9} x^{9} + 220 \, a b^{9} d^{10} x^{9} + 1485 \, b^{10} c^{2} d^{8} x^{8} + 990 \, a b^{9} c d^{9} x^{8} + 495 \, a^{2} b^{8} d^{10} x^{8} + 3168 \, b^{10} c^{3} d^{7} x^{7} + 2376 \, a b^{9} c^{2} d^{8} x^{7} + 1584 \, a^{2} b^{8} c d^{9} x^{7} + 792 \, a^{3} b^{7} d^{10} x^{7} + 4620 \, b^{10} c^{4} d^{6} x^{6} + 3696 \, a b^{9} c^{3} d^{7} x^{6} + 2772 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 1848 \, a^{3} b^{7} c d^{9} x^{6} + 924 \, a^{4} b^{6} d^{10} x^{6} + 4752 \, b^{10} c^{5} d^{5} x^{5} + 3960 \, a b^{9} c^{4} d^{6} x^{5} + 3168 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 2376 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 1584 \, a^{4} b^{6} c d^{9} x^{5} + 792 \, a^{5} b^{5} d^{10} x^{5} + 3465 \, b^{10} c^{6} d^{4} x^{4} + 2970 \, a b^{9} c^{5} d^{5} x^{4} + 2475 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 1980 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 1485 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 990 \, a^{5} b^{5} c d^{9} x^{4} + 495 \, a^{6} b^{4} d^{10} x^{4} + 1760 \, b^{10} c^{7} d^{3} x^{3} + 1540 \, a b^{9} c^{6} d^{4} x^{3} + 1320 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 1100 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 880 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 660 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 440 \, a^{6} b^{4} c d^{9} x^{3} + 220 \, a^{7} b^{3} d^{10} x^{3} + 594 \, b^{10} c^{8} d^{2} x^{2} + 528 \, a b^{9} c^{7} d^{3} x^{2} + 462 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 396 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 330 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 264 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 198 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 132 \, a^{7} b^{3} c d^{9} x^{2} + 66 \, a^{8} b^{2} d^{10} x^{2} + 120 \, b^{10} c^{9} d x + 108 \, a b^{9} c^{8} d^{2} x + 96 \, a^{2} b^{8} c^{7} d^{3} x + 84 \, a^{3} b^{7} c^{6} d^{4} x + 72 \, a^{4} b^{6} c^{5} d^{5} x + 60 \, a^{5} b^{5} c^{4} d^{6} x + 48 \, a^{6} b^{4} c^{3} d^{7} x + 36 \, a^{7} b^{3} c^{2} d^{8} x + 24 \, a^{8} b^{2} c d^{9} x + 12 \, a^{9} b d^{10} x + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10}}{132 \, {\left (b x + a\right )}^{12} b^{11}} \]
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Time = 0.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=\frac {{\left (c+d\,x\right )}^{11}\,\left (12\,a\,d-11\,b\,c+b\,d\,x\right )}{132\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^{12}} \]
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