Integrand size = 15, antiderivative size = 151 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=-\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}-\frac {2 d^2 (c+d x)^{11}}{455 (b c-a d)^3 (a+b x)^{13}}+\frac {d^3 (c+d x)^{11}}{1365 (b c-a d)^4 (a+b x)^{12}}-\frac {d^4 (c+d x)^{11}}{15015 (b c-a d)^5 (a+b x)^{11}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=-\frac {d^4 (c+d x)^{11}}{15015 (a+b x)^{11} (b c-a d)^5}+\frac {d^3 (c+d x)^{11}}{1365 (a+b x)^{12} (b c-a d)^4}-\frac {2 d^2 (c+d x)^{11}}{455 (a+b x)^{13} (b c-a d)^3}+\frac {2 d (c+d x)^{11}}{105 (a+b x)^{14} (b c-a d)^2}-\frac {(c+d x)^{11}}{15 (a+b x)^{15} (b c-a d)} \]
[In]
[Out]
Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}-\frac {(4 d) \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx}{15 (b c-a d)} \\ & = -\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}+\frac {\left (2 d^2\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx}{35 (b c-a d)^2} \\ & = -\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}-\frac {2 d^2 (c+d x)^{11}}{455 (b c-a d)^3 (a+b x)^{13}}-\frac {\left (4 d^3\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx}{455 (b c-a d)^3} \\ & = -\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}-\frac {2 d^2 (c+d x)^{11}}{455 (b c-a d)^3 (a+b x)^{13}}+\frac {d^3 (c+d x)^{11}}{1365 (b c-a d)^4 (a+b x)^{12}}+\frac {d^4 \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx}{1365 (b c-a d)^4} \\ & = -\frac {(c+d x)^{11}}{15 (b c-a d) (a+b x)^{15}}+\frac {2 d (c+d x)^{11}}{105 (b c-a d)^2 (a+b x)^{14}}-\frac {2 d^2 (c+d x)^{11}}{455 (b c-a d)^3 (a+b x)^{13}}+\frac {d^3 (c+d x)^{11}}{1365 (b c-a d)^4 (a+b x)^{12}}-\frac {d^4 (c+d x)^{11}}{15015 (b c-a d)^5 (a+b x)^{11}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(690\) vs. \(2(151)=302\).
Time = 0.17 (sec) , antiderivative size = 690, normalized size of antiderivative = 4.57 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=-\frac {a^{10} d^{10}+5 a^9 b d^9 (c+3 d x)+15 a^8 b^2 d^8 \left (c^2+5 c d x+7 d^2 x^2\right )+5 a^7 b^3 d^7 \left (7 c^3+45 c^2 d x+105 c d^2 x^2+91 d^3 x^3\right )+35 a^6 b^4 d^6 \left (2 c^4+15 c^3 d x+45 c^2 d^2 x^2+65 c d^3 x^3+39 d^4 x^4\right )+21 a^5 b^5 d^5 \left (6 c^5+50 c^4 d x+175 c^3 d^2 x^2+325 c^2 d^3 x^3+325 c d^4 x^4+143 d^5 x^5\right )+35 a^4 b^6 d^4 \left (6 c^6+54 c^5 d x+210 c^4 d^2 x^2+455 c^3 d^3 x^3+585 c^2 d^4 x^4+429 c d^5 x^5+143 d^6 x^6\right )+5 a^3 b^7 d^3 \left (66 c^7+630 c^6 d x+2646 c^5 d^2 x^2+6370 c^4 d^3 x^3+9555 c^3 d^4 x^4+9009 c^2 d^5 x^5+5005 c d^6 x^6+1287 d^7 x^7\right )+15 a^2 b^8 d^2 \left (33 c^8+330 c^7 d x+1470 c^6 d^2 x^2+3822 c^5 d^3 x^3+6370 c^4 d^4 x^4+7007 c^3 d^5 x^5+5005 c^2 d^6 x^6+2145 c d^7 x^7+429 d^8 x^8\right )+5 a b^9 d \left (143 c^9+1485 c^8 d x+6930 c^7 d^2 x^2+19110 c^6 d^3 x^3+34398 c^5 d^4 x^4+42042 c^4 d^5 x^5+35035 c^3 d^6 x^6+19305 c^2 d^7 x^7+6435 c d^8 x^8+1001 d^9 x^9\right )+b^{10} \left (1001 c^{10}+10725 c^9 d x+51975 c^8 d^2 x^2+150150 c^7 d^3 x^3+286650 c^6 d^4 x^4+378378 c^5 d^5 x^5+350350 c^4 d^6 x^6+225225 c^3 d^7 x^7+96525 c^2 d^8 x^8+25025 c d^9 x^9+3003 d^{10} x^{10}\right )}{15015 b^{11} (a+b x)^{15}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(141)=282\).
Time = 0.25 (sec) , antiderivative size = 831, normalized size of antiderivative = 5.50
method | result | size |
risch | \(\frac {-\frac {a^{10} d^{10}+5 a^{9} b c \,d^{9}+15 a^{8} b^{2} c^{2} d^{8}+35 a^{7} b^{3} c^{3} d^{7}+70 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}+330 a^{3} b^{7} c^{7} d^{3}+495 a^{2} b^{8} c^{8} d^{2}+715 a \,b^{9} c^{9} d +1001 b^{10} c^{10}}{15015 b^{11}}-\frac {d \left (a^{9} d^{9}+5 a^{8} b c \,d^{8}+15 a^{7} b^{2} c^{2} d^{7}+35 a^{6} b^{3} c^{3} d^{6}+70 a^{5} b^{4} c^{4} d^{5}+126 a^{4} b^{5} c^{5} d^{4}+210 a^{3} b^{6} c^{6} d^{3}+330 a^{2} b^{7} c^{7} d^{2}+495 a \,b^{8} c^{8} d +715 b^{9} c^{9}\right ) x}{1001 b^{10}}-\frac {d^{2} \left (a^{8} d^{8}+5 a^{7} b c \,d^{7}+15 a^{6} b^{2} c^{2} d^{6}+35 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}+126 a^{3} b^{5} c^{5} d^{3}+210 a^{2} b^{6} c^{6} d^{2}+330 a \,b^{7} c^{7} d +495 b^{8} c^{8}\right ) x^{2}}{143 b^{9}}-\frac {d^{3} \left (a^{7} d^{7}+5 a^{6} b c \,d^{6}+15 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+126 a^{2} b^{5} c^{5} d^{2}+210 a \,b^{6} c^{6} d +330 b^{7} c^{7}\right ) x^{3}}{33 b^{8}}-\frac {d^{4} \left (a^{6} d^{6}+5 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}+35 a^{3} b^{3} c^{3} d^{3}+70 a^{2} b^{4} c^{4} d^{2}+126 a \,b^{5} c^{5} d +210 b^{6} c^{6}\right ) x^{4}}{11 b^{7}}-\frac {d^{5} \left (a^{5} d^{5}+5 a^{4} b c \,d^{4}+15 a^{3} b^{2} c^{2} d^{3}+35 a^{2} b^{3} c^{3} d^{2}+70 a \,b^{4} c^{4} d +126 b^{5} c^{5}\right ) x^{5}}{5 b^{6}}-\frac {d^{6} \left (a^{4} d^{4}+5 a^{3} b c \,d^{3}+15 a^{2} b^{2} c^{2} d^{2}+35 a \,b^{3} c^{3} d +70 b^{4} c^{4}\right ) x^{6}}{3 b^{5}}-\frac {3 d^{7} \left (a^{3} d^{3}+5 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d +35 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}-\frac {3 d^{8} \left (a^{2} d^{2}+5 a b c d +15 b^{2} c^{2}\right ) x^{8}}{7 b^{3}}-\frac {d^{9} \left (a d +5 b c \right ) x^{9}}{3 b^{2}}-\frac {d^{10} x^{10}}{5 b}}{\left (b x +a \right )^{15}}\) | \(831\) |
default | \(-\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 )}{11 b^{11} \left (b x +a \right )^{11}}-\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 )}{13 b^{11} \left (b x +a \right )^{13}}-\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 )}{3 b^{11} \left (b x +a \right )^{9}}+\frac {5 d^{9} \left (a d -b c \right )}{3 b^{11} \left (b x +a \right )^{6}}+\frac {15 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 )^{8}}+\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 )}{7 b^{11} \left (b x +a \right )^{14}}+\frac {10 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 )^{12}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{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 )}{5 b^{11} \left (b x +a \right )^{10}}-\frac {d^{10}}{5 b^{11} \left (b x +a \right )^{5}}-\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}}{15 b^{11} \left (b x +a \right )^{15}}\) | \(867\) |
norman | \(\frac {\frac {-a^{10} b^{4} d^{10}-5 a^{9} b^{5} c \,d^{9}-15 a^{8} b^{6} c^{2} d^{8}-35 a^{7} b^{7} c^{3} d^{7}-70 a^{6} b^{8} c^{4} d^{6}-126 a^{5} b^{9} c^{5} d^{5}-210 a^{4} b^{10} c^{6} d^{4}-330 a^{3} c^{7} d^{3} b^{11}-495 a^{2} b^{12} c^{8} d^{2}-715 a \,b^{13} c^{9} d -1001 b^{14} c^{10}}{15015 b^{15}}+\frac {\left (-a^{9} b^{4} d^{10}-5 a^{8} b^{5} c \,d^{9}-15 a^{7} b^{6} c^{2} d^{8}-35 a^{6} b^{7} c^{3} d^{7}-70 a^{5} b^{8} c^{4} d^{6}-126 a^{4} b^{9} c^{5} d^{5}-210 a^{3} b^{10} c^{6} d^{4}-330 a^{2} b^{11} c^{7} d^{3}-495 a \,b^{12} c^{8} d^{2}-715 b^{13} c^{9} d \right ) x}{1001 b^{14}}+\frac {\left (-a^{8} b^{4} d^{10}-5 a^{7} b^{5} c \,d^{9}-15 a^{6} b^{6} c^{2} d^{8}-35 a^{5} b^{7} c^{3} d^{7}-70 a^{4} b^{8} c^{4} d^{6}-126 a^{3} b^{9} c^{5} d^{5}-210 a^{2} b^{10} c^{6} d^{4}-330 a \,c^{7} d^{3} b^{11}-495 b^{12} c^{8} d^{2}\right ) x^{2}}{143 b^{13}}+\frac {\left (-a^{7} b^{4} d^{10}-5 a^{6} b^{5} c \,d^{9}-15 a^{5} b^{6} c^{2} d^{8}-35 a^{4} b^{7} c^{3} d^{7}-70 a^{3} b^{8} c^{4} d^{6}-126 a^{2} b^{9} c^{5} d^{5}-210 a \,b^{10} c^{6} d^{4}-330 b^{11} c^{7} d^{3}\right ) x^{3}}{33 b^{12}}+\frac {\left (-a^{6} b^{4} d^{10}-5 a^{5} b^{5} c \,d^{9}-15 a^{4} b^{6} c^{2} d^{8}-35 a^{3} b^{7} c^{3} d^{7}-70 a^{2} b^{8} c^{4} d^{6}-126 a \,b^{9} c^{5} d^{5}-210 b^{10} c^{6} d^{4}\right ) x^{4}}{11 b^{11}}+\frac {\left (-a^{5} b^{4} d^{10}-5 a^{4} b^{5} c \,d^{9}-15 a^{3} b^{6} c^{2} d^{8}-35 a^{2} b^{7} c^{3} d^{7}-70 a \,b^{8} c^{4} d^{6}-126 b^{9} c^{5} d^{5}\right ) x^{5}}{5 b^{10}}+\frac {\left (-a^{4} b^{4} d^{10}-5 a^{3} b^{5} c \,d^{9}-15 a^{2} b^{6} c^{2} d^{8}-35 a \,b^{7} c^{3} d^{7}-70 b^{8} c^{4} d^{6}\right ) x^{6}}{3 b^{9}}+\frac {3 \left (-a^{3} b^{4} d^{10}-5 a^{2} b^{5} c \,d^{9}-15 a \,b^{6} c^{2} d^{8}-35 b^{7} c^{3} d^{7}\right ) x^{7}}{7 b^{8}}+\frac {3 \left (-a^{2} b^{4} d^{10}-5 a \,b^{5} c \,d^{9}-15 b^{6} c^{2} d^{8}\right ) x^{8}}{7 b^{7}}+\frac {\left (-a \,b^{4} d^{10}-5 b^{5} c \,d^{9}\right ) x^{9}}{3 b^{6}}-\frac {d^{10} x^{10}}{5 b}}{\left (b x +a \right )^{15}}\) | \(909\) |
gosper | \(-\frac {3003 x^{10} d^{10} b^{10}+5005 x^{9} a \,b^{9} d^{10}+25025 x^{9} b^{10} c \,d^{9}+6435 x^{8} a^{2} b^{8} d^{10}+32175 x^{8} a \,b^{9} c \,d^{9}+96525 x^{8} b^{10} c^{2} d^{8}+6435 x^{7} a^{3} b^{7} d^{10}+32175 x^{7} a^{2} b^{8} c \,d^{9}+96525 x^{7} a \,b^{9} c^{2} d^{8}+225225 x^{7} b^{10} c^{3} d^{7}+5005 x^{6} a^{4} b^{6} d^{10}+25025 x^{6} a^{3} b^{7} c \,d^{9}+75075 x^{6} a^{2} b^{8} c^{2} d^{8}+175175 x^{6} a \,b^{9} c^{3} d^{7}+350350 x^{6} b^{10} c^{4} d^{6}+3003 x^{5} a^{5} b^{5} d^{10}+15015 x^{5} a^{4} b^{6} c \,d^{9}+45045 x^{5} a^{3} b^{7} c^{2} d^{8}+105105 x^{5} a^{2} b^{8} c^{3} d^{7}+210210 x^{5} a \,b^{9} c^{4} d^{6}+378378 x^{5} b^{10} c^{5} d^{5}+1365 x^{4} a^{6} b^{4} d^{10}+6825 x^{4} a^{5} b^{5} c \,d^{9}+20475 x^{4} a^{4} b^{6} c^{2} d^{8}+47775 x^{4} a^{3} b^{7} c^{3} d^{7}+95550 x^{4} a^{2} b^{8} c^{4} d^{6}+171990 x^{4} a \,b^{9} c^{5} d^{5}+286650 x^{4} b^{10} c^{6} d^{4}+455 x^{3} a^{7} b^{3} d^{10}+2275 x^{3} a^{6} b^{4} c \,d^{9}+6825 x^{3} a^{5} b^{5} c^{2} d^{8}+15925 x^{3} a^{4} b^{6} c^{3} d^{7}+31850 x^{3} a^{3} b^{7} c^{4} d^{6}+57330 x^{3} a^{2} b^{8} c^{5} d^{5}+95550 x^{3} a \,b^{9} c^{6} d^{4}+150150 x^{3} b^{10} c^{7} d^{3}+105 x^{2} a^{8} b^{2} d^{10}+525 x^{2} a^{7} b^{3} c \,d^{9}+1575 x^{2} a^{6} b^{4} c^{2} d^{8}+3675 x^{2} a^{5} b^{5} c^{3} d^{7}+7350 x^{2} a^{4} b^{6} c^{4} d^{6}+13230 x^{2} a^{3} b^{7} c^{5} d^{5}+22050 x^{2} a^{2} b^{8} c^{6} d^{4}+34650 x^{2} a \,b^{9} c^{7} d^{3}+51975 x^{2} b^{10} c^{8} d^{2}+15 x \,a^{9} b \,d^{10}+75 x \,a^{8} b^{2} c \,d^{9}+225 x \,a^{7} b^{3} c^{2} d^{8}+525 x \,a^{6} b^{4} c^{3} d^{7}+1050 x \,a^{5} b^{5} c^{4} d^{6}+1890 x \,a^{4} b^{6} c^{5} d^{5}+3150 x \,a^{3} b^{7} c^{6} d^{4}+4950 x \,a^{2} b^{8} c^{7} d^{3}+7425 x a \,b^{9} c^{8} d^{2}+10725 x \,b^{10} c^{9} d +a^{10} d^{10}+5 a^{9} b c \,d^{9}+15 a^{8} b^{2} c^{2} d^{8}+35 a^{7} b^{3} c^{3} d^{7}+70 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}+330 a^{3} b^{7} c^{7} d^{3}+495 a^{2} b^{8} c^{8} d^{2}+715 a \,b^{9} c^{9} d +1001 b^{10} c^{10}}{15015 b^{11} \left (b x +a \right )^{15}}\) | \(962\) |
parallelrisch | \(\frac {-3003 d^{10} x^{10} b^{14}-5005 a \,b^{13} d^{10} x^{9}-25025 b^{14} c \,d^{9} x^{9}-6435 a^{2} b^{12} d^{10} x^{8}-32175 a \,b^{13} c \,d^{9} x^{8}-96525 b^{14} c^{2} d^{8} x^{8}-6435 a^{3} b^{11} d^{10} x^{7}-32175 a^{2} b^{12} c \,d^{9} x^{7}-96525 a \,b^{13} c^{2} d^{8} x^{7}-225225 b^{14} c^{3} d^{7} x^{7}-5005 a^{4} b^{10} d^{10} x^{6}-25025 a^{3} b^{11} c \,d^{9} x^{6}-75075 a^{2} b^{12} c^{2} d^{8} x^{6}-175175 a \,b^{13} c^{3} d^{7} x^{6}-350350 b^{14} c^{4} d^{6} x^{6}-3003 a^{5} b^{9} d^{10} x^{5}-15015 a^{4} b^{10} c \,d^{9} x^{5}-45045 a^{3} b^{11} c^{2} d^{8} x^{5}-105105 a^{2} b^{12} c^{3} d^{7} x^{5}-210210 a \,b^{13} c^{4} d^{6} x^{5}-378378 b^{14} c^{5} d^{5} x^{5}-1365 a^{6} b^{8} d^{10} x^{4}-6825 a^{5} b^{9} c \,d^{9} x^{4}-20475 a^{4} b^{10} c^{2} d^{8} x^{4}-47775 a^{3} b^{11} c^{3} d^{7} x^{4}-95550 a^{2} b^{12} c^{4} d^{6} x^{4}-171990 a \,b^{13} c^{5} d^{5} x^{4}-286650 b^{14} c^{6} d^{4} x^{4}-455 a^{7} b^{7} d^{10} x^{3}-2275 a^{6} b^{8} c \,d^{9} x^{3}-6825 a^{5} b^{9} c^{2} d^{8} x^{3}-15925 a^{4} b^{10} c^{3} d^{7} x^{3}-31850 a^{3} b^{11} c^{4} d^{6} x^{3}-57330 a^{2} b^{12} c^{5} d^{5} x^{3}-95550 a \,b^{13} c^{6} d^{4} x^{3}-150150 b^{14} c^{7} d^{3} x^{3}-105 a^{8} b^{6} d^{10} x^{2}-525 a^{7} b^{7} c \,d^{9} x^{2}-1575 a^{6} b^{8} c^{2} d^{8} x^{2}-3675 a^{5} b^{9} c^{3} d^{7} x^{2}-7350 a^{4} b^{10} c^{4} d^{6} x^{2}-13230 a^{3} b^{11} c^{5} d^{5} x^{2}-22050 a^{2} b^{12} c^{6} d^{4} x^{2}-34650 a \,b^{13} c^{7} d^{3} x^{2}-51975 b^{14} c^{8} d^{2} x^{2}-15 a^{9} b^{5} d^{10} x -75 a^{8} b^{6} c \,d^{9} x -225 a^{7} b^{7} c^{2} d^{8} x -525 a^{6} b^{8} c^{3} d^{7} x -1050 a^{5} b^{9} c^{4} d^{6} x -1890 a^{4} b^{10} c^{5} d^{5} x -3150 a^{3} b^{11} c^{6} d^{4} x -4950 a^{2} b^{12} c^{7} d^{3} x -7425 a \,b^{13} c^{8} d^{2} x -10725 b^{14} c^{9} d x -a^{10} b^{4} d^{10}-5 a^{9} b^{5} c \,d^{9}-15 a^{8} b^{6} c^{2} d^{8}-35 a^{7} b^{7} c^{3} d^{7}-70 a^{6} b^{8} c^{4} d^{6}-126 a^{5} b^{9} c^{5} d^{5}-210 a^{4} b^{10} c^{6} d^{4}-330 a^{3} c^{7} d^{3} b^{11}-495 a^{2} b^{12} c^{8} d^{2}-715 a \,b^{13} c^{9} d -1001 b^{14} c^{10}}{15015 b^{15} \left (b x +a \right )^{15}}\) | \(970\) |
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (141) = 282\).
Time = 0.23 (sec) , antiderivative size = 1019, normalized size of antiderivative = 6.75 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=-\frac {3003 \, b^{10} d^{10} x^{10} + 1001 \, b^{10} c^{10} + 715 \, a b^{9} c^{9} d + 495 \, a^{2} b^{8} c^{8} d^{2} + 330 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 70 \, a^{6} b^{4} c^{4} d^{6} + 35 \, a^{7} b^{3} c^{3} d^{7} + 15 \, a^{8} b^{2} c^{2} d^{8} + 5 \, a^{9} b c d^{9} + a^{10} d^{10} + 5005 \, {\left (5 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 6435 \, {\left (15 \, b^{10} c^{2} d^{8} + 5 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 6435 \, {\left (35 \, b^{10} c^{3} d^{7} + 15 \, a b^{9} c^{2} d^{8} + 5 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 5005 \, {\left (70 \, b^{10} c^{4} d^{6} + 35 \, a b^{9} c^{3} d^{7} + 15 \, a^{2} b^{8} c^{2} d^{8} + 5 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 3003 \, {\left (126 \, b^{10} c^{5} d^{5} + 70 \, a b^{9} c^{4} d^{6} + 35 \, a^{2} b^{8} c^{3} d^{7} + 15 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 1365 \, {\left (210 \, b^{10} c^{6} d^{4} + 126 \, a b^{9} c^{5} d^{5} + 70 \, a^{2} b^{8} c^{4} d^{6} + 35 \, a^{3} b^{7} c^{3} d^{7} + 15 \, a^{4} b^{6} c^{2} d^{8} + 5 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 455 \, {\left (330 \, b^{10} c^{7} d^{3} + 210 \, a b^{9} c^{6} d^{4} + 126 \, a^{2} b^{8} c^{5} d^{5} + 70 \, a^{3} b^{7} c^{4} d^{6} + 35 \, a^{4} b^{6} c^{3} d^{7} + 15 \, a^{5} b^{5} c^{2} d^{8} + 5 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 105 \, {\left (495 \, b^{10} c^{8} d^{2} + 330 \, a b^{9} c^{7} d^{3} + 210 \, a^{2} b^{8} c^{6} d^{4} + 126 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} + 35 \, a^{5} b^{5} c^{3} d^{7} + 15 \, a^{6} b^{4} c^{2} d^{8} + 5 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 15 \, {\left (715 \, b^{10} c^{9} d + 495 \, a b^{9} c^{8} d^{2} + 330 \, a^{2} b^{8} c^{7} d^{3} + 210 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 70 \, a^{5} b^{5} c^{4} d^{6} + 35 \, a^{6} b^{4} c^{3} d^{7} + 15 \, a^{7} b^{3} c^{2} d^{8} + 5 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{15015 \, {\left (b^{26} x^{15} + 15 \, a b^{25} x^{14} + 105 \, a^{2} b^{24} x^{13} + 455 \, a^{3} b^{23} x^{12} + 1365 \, a^{4} b^{22} x^{11} + 3003 \, a^{5} b^{21} x^{10} + 5005 \, a^{6} b^{20} x^{9} + 6435 \, a^{7} b^{19} x^{8} + 6435 \, a^{8} b^{18} x^{7} + 5005 \, a^{9} b^{17} x^{6} + 3003 \, a^{10} b^{16} x^{5} + 1365 \, a^{11} b^{15} x^{4} + 455 \, a^{12} b^{14} x^{3} + 105 \, a^{13} b^{13} x^{2} + 15 \, a^{14} b^{12} x + a^{15} b^{11}\right )}} \]
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Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (141) = 282\).
Time = 0.25 (sec) , antiderivative size = 1019, normalized size of antiderivative = 6.75 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=-\frac {3003 \, b^{10} d^{10} x^{10} + 1001 \, b^{10} c^{10} + 715 \, a b^{9} c^{9} d + 495 \, a^{2} b^{8} c^{8} d^{2} + 330 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 70 \, a^{6} b^{4} c^{4} d^{6} + 35 \, a^{7} b^{3} c^{3} d^{7} + 15 \, a^{8} b^{2} c^{2} d^{8} + 5 \, a^{9} b c d^{9} + a^{10} d^{10} + 5005 \, {\left (5 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 6435 \, {\left (15 \, b^{10} c^{2} d^{8} + 5 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 6435 \, {\left (35 \, b^{10} c^{3} d^{7} + 15 \, a b^{9} c^{2} d^{8} + 5 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 5005 \, {\left (70 \, b^{10} c^{4} d^{6} + 35 \, a b^{9} c^{3} d^{7} + 15 \, a^{2} b^{8} c^{2} d^{8} + 5 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 3003 \, {\left (126 \, b^{10} c^{5} d^{5} + 70 \, a b^{9} c^{4} d^{6} + 35 \, a^{2} b^{8} c^{3} d^{7} + 15 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 1365 \, {\left (210 \, b^{10} c^{6} d^{4} + 126 \, a b^{9} c^{5} d^{5} + 70 \, a^{2} b^{8} c^{4} d^{6} + 35 \, a^{3} b^{7} c^{3} d^{7} + 15 \, a^{4} b^{6} c^{2} d^{8} + 5 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 455 \, {\left (330 \, b^{10} c^{7} d^{3} + 210 \, a b^{9} c^{6} d^{4} + 126 \, a^{2} b^{8} c^{5} d^{5} + 70 \, a^{3} b^{7} c^{4} d^{6} + 35 \, a^{4} b^{6} c^{3} d^{7} + 15 \, a^{5} b^{5} c^{2} d^{8} + 5 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 105 \, {\left (495 \, b^{10} c^{8} d^{2} + 330 \, a b^{9} c^{7} d^{3} + 210 \, a^{2} b^{8} c^{6} d^{4} + 126 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} + 35 \, a^{5} b^{5} c^{3} d^{7} + 15 \, a^{6} b^{4} c^{2} d^{8} + 5 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 15 \, {\left (715 \, b^{10} c^{9} d + 495 \, a b^{9} c^{8} d^{2} + 330 \, a^{2} b^{8} c^{7} d^{3} + 210 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 70 \, a^{5} b^{5} c^{4} d^{6} + 35 \, a^{6} b^{4} c^{3} d^{7} + 15 \, a^{7} b^{3} c^{2} d^{8} + 5 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{15015 \, {\left (b^{26} x^{15} + 15 \, a b^{25} x^{14} + 105 \, a^{2} b^{24} x^{13} + 455 \, a^{3} b^{23} x^{12} + 1365 \, a^{4} b^{22} x^{11} + 3003 \, a^{5} b^{21} x^{10} + 5005 \, a^{6} b^{20} x^{9} + 6435 \, a^{7} b^{19} x^{8} + 6435 \, a^{8} b^{18} x^{7} + 5005 \, a^{9} b^{17} x^{6} + 3003 \, a^{10} b^{16} x^{5} + 1365 \, a^{11} b^{15} x^{4} + 455 \, a^{12} b^{14} x^{3} + 105 \, a^{13} b^{13} x^{2} + 15 \, a^{14} b^{12} x + a^{15} b^{11}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (141) = 282\).
Time = 0.32 (sec) , antiderivative size = 961, normalized size of antiderivative = 6.36 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=-\frac {3003 \, b^{10} d^{10} x^{10} + 25025 \, b^{10} c d^{9} x^{9} + 5005 \, a b^{9} d^{10} x^{9} + 96525 \, b^{10} c^{2} d^{8} x^{8} + 32175 \, a b^{9} c d^{9} x^{8} + 6435 \, a^{2} b^{8} d^{10} x^{8} + 225225 \, b^{10} c^{3} d^{7} x^{7} + 96525 \, a b^{9} c^{2} d^{8} x^{7} + 32175 \, a^{2} b^{8} c d^{9} x^{7} + 6435 \, a^{3} b^{7} d^{10} x^{7} + 350350 \, b^{10} c^{4} d^{6} x^{6} + 175175 \, a b^{9} c^{3} d^{7} x^{6} + 75075 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 25025 \, a^{3} b^{7} c d^{9} x^{6} + 5005 \, a^{4} b^{6} d^{10} x^{6} + 378378 \, b^{10} c^{5} d^{5} x^{5} + 210210 \, a b^{9} c^{4} d^{6} x^{5} + 105105 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 45045 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 15015 \, a^{4} b^{6} c d^{9} x^{5} + 3003 \, a^{5} b^{5} d^{10} x^{5} + 286650 \, b^{10} c^{6} d^{4} x^{4} + 171990 \, a b^{9} c^{5} d^{5} x^{4} + 95550 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 47775 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 20475 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 6825 \, a^{5} b^{5} c d^{9} x^{4} + 1365 \, a^{6} b^{4} d^{10} x^{4} + 150150 \, b^{10} c^{7} d^{3} x^{3} + 95550 \, a b^{9} c^{6} d^{4} x^{3} + 57330 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 31850 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 15925 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 6825 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 2275 \, a^{6} b^{4} c d^{9} x^{3} + 455 \, a^{7} b^{3} d^{10} x^{3} + 51975 \, b^{10} c^{8} d^{2} x^{2} + 34650 \, a b^{9} c^{7} d^{3} x^{2} + 22050 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 13230 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 7350 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 3675 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 1575 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 525 \, a^{7} b^{3} c d^{9} x^{2} + 105 \, a^{8} b^{2} d^{10} x^{2} + 10725 \, b^{10} c^{9} d x + 7425 \, a b^{9} c^{8} d^{2} x + 4950 \, a^{2} b^{8} c^{7} d^{3} x + 3150 \, a^{3} b^{7} c^{6} d^{4} x + 1890 \, a^{4} b^{6} c^{5} d^{5} x + 1050 \, a^{5} b^{5} c^{4} d^{6} x + 525 \, a^{6} b^{4} c^{3} d^{7} x + 225 \, a^{7} b^{3} c^{2} d^{8} x + 75 \, a^{8} b^{2} c d^{9} x + 15 \, a^{9} b d^{10} x + 1001 \, b^{10} c^{10} + 715 \, a b^{9} c^{9} d + 495 \, a^{2} b^{8} c^{8} d^{2} + 330 \, a^{3} b^{7} c^{7} d^{3} + 210 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 70 \, a^{6} b^{4} c^{4} d^{6} + 35 \, a^{7} b^{3} c^{3} d^{7} + 15 \, a^{8} b^{2} c^{2} d^{8} + 5 \, a^{9} b c d^{9} + a^{10} d^{10}}{15015 \, {\left (b x + a\right )}^{15} b^{11}} \]
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Time = 3.32 (sec) , antiderivative size = 1120, normalized size of antiderivative = 7.42 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx=-\frac {a^{10}\,d^{10}+5\,a^9\,b\,c\,d^9+15\,a^9\,b\,d^{10}\,x+15\,a^8\,b^2\,c^2\,d^8+75\,a^8\,b^2\,c\,d^9\,x+105\,a^8\,b^2\,d^{10}\,x^2+35\,a^7\,b^3\,c^3\,d^7+225\,a^7\,b^3\,c^2\,d^8\,x+525\,a^7\,b^3\,c\,d^9\,x^2+455\,a^7\,b^3\,d^{10}\,x^3+70\,a^6\,b^4\,c^4\,d^6+525\,a^6\,b^4\,c^3\,d^7\,x+1575\,a^6\,b^4\,c^2\,d^8\,x^2+2275\,a^6\,b^4\,c\,d^9\,x^3+1365\,a^6\,b^4\,d^{10}\,x^4+126\,a^5\,b^5\,c^5\,d^5+1050\,a^5\,b^5\,c^4\,d^6\,x+3675\,a^5\,b^5\,c^3\,d^7\,x^2+6825\,a^5\,b^5\,c^2\,d^8\,x^3+6825\,a^5\,b^5\,c\,d^9\,x^4+3003\,a^5\,b^5\,d^{10}\,x^5+210\,a^4\,b^6\,c^6\,d^4+1890\,a^4\,b^6\,c^5\,d^5\,x+7350\,a^4\,b^6\,c^4\,d^6\,x^2+15925\,a^4\,b^6\,c^3\,d^7\,x^3+20475\,a^4\,b^6\,c^2\,d^8\,x^4+15015\,a^4\,b^6\,c\,d^9\,x^5+5005\,a^4\,b^6\,d^{10}\,x^6+330\,a^3\,b^7\,c^7\,d^3+3150\,a^3\,b^7\,c^6\,d^4\,x+13230\,a^3\,b^7\,c^5\,d^5\,x^2+31850\,a^3\,b^7\,c^4\,d^6\,x^3+47775\,a^3\,b^7\,c^3\,d^7\,x^4+45045\,a^3\,b^7\,c^2\,d^8\,x^5+25025\,a^3\,b^7\,c\,d^9\,x^6+6435\,a^3\,b^7\,d^{10}\,x^7+495\,a^2\,b^8\,c^8\,d^2+4950\,a^2\,b^8\,c^7\,d^3\,x+22050\,a^2\,b^8\,c^6\,d^4\,x^2+57330\,a^2\,b^8\,c^5\,d^5\,x^3+95550\,a^2\,b^8\,c^4\,d^6\,x^4+105105\,a^2\,b^8\,c^3\,d^7\,x^5+75075\,a^2\,b^8\,c^2\,d^8\,x^6+32175\,a^2\,b^8\,c\,d^9\,x^7+6435\,a^2\,b^8\,d^{10}\,x^8+715\,a\,b^9\,c^9\,d+7425\,a\,b^9\,c^8\,d^2\,x+34650\,a\,b^9\,c^7\,d^3\,x^2+95550\,a\,b^9\,c^6\,d^4\,x^3+171990\,a\,b^9\,c^5\,d^5\,x^4+210210\,a\,b^9\,c^4\,d^6\,x^5+175175\,a\,b^9\,c^3\,d^7\,x^6+96525\,a\,b^9\,c^2\,d^8\,x^7+32175\,a\,b^9\,c\,d^9\,x^8+5005\,a\,b^9\,d^{10}\,x^9+1001\,b^{10}\,c^{10}+10725\,b^{10}\,c^9\,d\,x+51975\,b^{10}\,c^8\,d^2\,x^2+150150\,b^{10}\,c^7\,d^3\,x^3+286650\,b^{10}\,c^6\,d^4\,x^4+378378\,b^{10}\,c^5\,d^5\,x^5+350350\,b^{10}\,c^4\,d^6\,x^6+225225\,b^{10}\,c^3\,d^7\,x^7+96525\,b^{10}\,c^2\,d^8\,x^8+25025\,b^{10}\,c\,d^9\,x^9+3003\,b^{10}\,d^{10}\,x^{10}}{15015\,a^{15}\,b^{11}+225225\,a^{14}\,b^{12}\,x+1576575\,a^{13}\,b^{13}\,x^2+6831825\,a^{12}\,b^{14}\,x^3+20495475\,a^{11}\,b^{15}\,x^4+45090045\,a^{10}\,b^{16}\,x^5+75150075\,a^9\,b^{17}\,x^6+96621525\,a^8\,b^{18}\,x^7+96621525\,a^7\,b^{19}\,x^8+75150075\,a^6\,b^{20}\,x^9+45090045\,a^5\,b^{21}\,x^{10}+20495475\,a^4\,b^{22}\,x^{11}+6831825\,a^3\,b^{23}\,x^{12}+1576575\,a^2\,b^{24}\,x^{13}+225225\,a\,b^{25}\,x^{14}+15015\,b^{26}\,x^{15}} \]
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