Integrand size = 15, antiderivative size = 198 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {d^7}{6 b^8 (a+b x)^6} \]
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Time = 0.10 (sec) , antiderivative size = 198, 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} \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {d^7}{6 b^8 (a+b x)^6} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^7}{b^7 (a+b x)^{14}}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^{13}}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^{12}}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^{11}}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^{10}}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^9}+\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^8}+\frac {d^7}{b^7 (a+b x)^7}\right ) \, dx \\ & = -\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {d^7}{6 b^8 (a+b x)^6} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.86 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {a^7 d^7+a^6 b d^6 (6 c+13 d x)+3 a^5 b^2 d^5 \left (7 c^2+26 c d x+26 d^2 x^2\right )+a^4 b^3 d^4 \left (56 c^3+273 c^2 d x+468 c d^2 x^2+286 d^3 x^3\right )+a^3 b^4 d^3 \left (126 c^4+728 c^3 d x+1638 c^2 d^2 x^2+1716 c d^3 x^3+715 d^4 x^4\right )+3 a^2 b^5 d^2 \left (84 c^5+546 c^4 d x+1456 c^3 d^2 x^2+2002 c^2 d^3 x^3+1430 c d^4 x^4+429 d^5 x^5\right )+a b^6 d \left (462 c^6+3276 c^5 d x+9828 c^4 d^2 x^2+16016 c^3 d^3 x^3+15015 c^2 d^4 x^4+7722 c d^5 x^5+1716 d^6 x^6\right )+b^7 \left (792 c^7+6006 c^6 d x+19656 c^5 d^2 x^2+36036 c^4 d^3 x^3+40040 c^3 d^4 x^4+27027 c^2 d^5 x^5+10296 c d^6 x^6+1716 d^7 x^7\right )}{10296 b^8 (a+b x)^{13}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(184)=368\).
Time = 0.23 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.21
method | result | size |
risch | \(\frac {-\frac {d^{7} x^{7}}{6 b}-\frac {d^{6} \left (a d +6 b c \right ) x^{6}}{6 b^{2}}-\frac {d^{5} \left (a^{2} d^{2}+6 a b c d +21 b^{2} c^{2}\right ) x^{5}}{8 b^{3}}-\frac {5 d^{4} \left (a^{3} d^{3}+6 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d +56 b^{3} c^{3}\right ) x^{4}}{72 b^{4}}-\frac {d^{3} \left (a^{4} d^{4}+6 a^{3} b c \,d^{3}+21 a^{2} b^{2} c^{2} d^{2}+56 a \,b^{3} c^{3} d +126 b^{4} c^{4}\right ) x^{3}}{36 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+6 a^{4} b c \,d^{4}+21 a^{3} b^{2} c^{2} d^{3}+56 a^{2} b^{3} c^{3} d^{2}+126 a \,b^{4} c^{4} d +252 b^{5} c^{5}\right ) x^{2}}{132 b^{6}}-\frac {d \left (a^{6} d^{6}+6 a^{5} b c \,d^{5}+21 a^{4} b^{2} c^{2} d^{4}+56 a^{3} b^{3} c^{3} d^{3}+126 a^{2} b^{4} c^{4} d^{2}+252 a \,b^{5} c^{5} d +462 b^{6} c^{6}\right ) x}{792 b^{7}}-\frac {a^{7} d^{7}+6 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+56 a^{4} b^{3} c^{3} d^{4}+126 a^{3} b^{4} c^{4} d^{3}+252 a^{2} b^{5} c^{5} d^{2}+462 a \,b^{6} c^{6} d +792 b^{7} c^{7}}{10296 b^{8}}}{\left (b x +a \right )^{13}}\) | \(438\) |
default | \(\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{11 b^{8} \left (b x +a \right )^{11}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{13 b^{8} \left (b x +a \right )^{13}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{9 b^{8} \left (b x +a \right )^{9}}-\frac {d^{7}}{6 b^{8} \left (b x +a \right )^{6}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{8 b^{8} \left (b x +a \right )^{8}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{12 b^{8} \left (b x +a \right )^{12}}+\frac {d^{6} \left (a d -b c \right )}{b^{8} \left (b x +a \right )^{7}}-\frac {7 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{2 b^{8} \left (b x +a \right )^{10}}\) | \(463\) |
norman | \(\frac {-\frac {d^{7} x^{7}}{6 b}+\frac {\left (-a \,b^{5} d^{7}-6 b^{6} c \,d^{6}\right ) x^{6}}{6 b^{7}}+\frac {\left (-a^{2} b^{5} d^{7}-6 a \,b^{6} c \,d^{6}-21 b^{7} c^{2} d^{5}\right ) x^{5}}{8 b^{8}}+\frac {5 \left (-a^{3} b^{5} d^{7}-6 a^{2} b^{6} c \,d^{6}-21 a \,b^{7} c^{2} d^{5}-56 b^{8} c^{3} d^{4}\right ) x^{4}}{72 b^{9}}+\frac {\left (-a^{4} b^{5} d^{7}-6 a^{3} b^{6} c \,d^{6}-21 a^{2} b^{7} c^{2} d^{5}-56 a \,b^{8} c^{3} d^{4}-126 b^{9} c^{4} d^{3}\right ) x^{3}}{36 b^{10}}+\frac {\left (-a^{5} b^{5} d^{7}-6 a^{4} b^{6} c \,d^{6}-21 a^{3} b^{7} c^{2} d^{5}-56 a^{2} b^{8} c^{3} d^{4}-126 a \,b^{9} c^{4} d^{3}-252 b^{10} c^{5} d^{2}\right ) x^{2}}{132 b^{11}}+\frac {\left (-a^{6} b^{5} d^{7}-6 a^{5} b^{6} c \,d^{6}-21 a^{4} b^{7} c^{2} d^{5}-56 a^{3} b^{8} c^{3} d^{4}-126 a^{2} b^{9} c^{4} d^{3}-252 a \,b^{10} c^{5} d^{2}-462 b^{11} c^{6} d \right ) x}{792 b^{12}}+\frac {-a^{7} b^{5} d^{7}-6 a^{6} b^{6} c \,d^{6}-21 a^{5} b^{7} c^{2} d^{5}-56 a^{4} b^{8} c^{3} d^{4}-126 a^{3} b^{9} c^{4} d^{3}-252 a^{2} b^{10} c^{5} d^{2}-462 a \,c^{6} d \,b^{11}-792 b^{12} c^{7}}{10296 b^{13}}}{\left (b x +a \right )^{13}}\) | \(492\) |
gosper | \(-\frac {1716 x^{7} d^{7} b^{7}+1716 x^{6} a \,b^{6} d^{7}+10296 x^{6} b^{7} c \,d^{6}+1287 x^{5} a^{2} b^{5} d^{7}+7722 x^{5} a \,b^{6} c \,d^{6}+27027 x^{5} b^{7} c^{2} d^{5}+715 x^{4} a^{3} b^{4} d^{7}+4290 x^{4} a^{2} b^{5} c \,d^{6}+15015 x^{4} a \,b^{6} c^{2} d^{5}+40040 x^{4} b^{7} c^{3} d^{4}+286 x^{3} a^{4} b^{3} d^{7}+1716 x^{3} a^{3} b^{4} c \,d^{6}+6006 x^{3} a^{2} b^{5} c^{2} d^{5}+16016 x^{3} a \,b^{6} c^{3} d^{4}+36036 x^{3} b^{7} c^{4} d^{3}+78 x^{2} a^{5} b^{2} d^{7}+468 x^{2} a^{4} b^{3} c \,d^{6}+1638 x^{2} a^{3} b^{4} c^{2} d^{5}+4368 x^{2} a^{2} b^{5} c^{3} d^{4}+9828 x^{2} a \,b^{6} c^{4} d^{3}+19656 x^{2} b^{7} c^{5} d^{2}+13 x \,a^{6} b \,d^{7}+78 x \,a^{5} b^{2} c \,d^{6}+273 x \,a^{4} b^{3} c^{2} d^{5}+728 x \,a^{3} b^{4} c^{3} d^{4}+1638 x \,a^{2} b^{5} c^{4} d^{3}+3276 x a \,b^{6} c^{5} d^{2}+6006 x \,b^{7} c^{6} d +a^{7} d^{7}+6 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+56 a^{4} b^{3} c^{3} d^{4}+126 a^{3} b^{4} c^{4} d^{3}+252 a^{2} b^{5} c^{5} d^{2}+462 a \,b^{6} c^{6} d +792 b^{7} c^{7}}{10296 b^{8} \left (b x +a \right )^{13}}\) | \(497\) |
parallelrisch | \(\frac {-1716 d^{7} x^{7} b^{12}-1716 a \,b^{11} d^{7} x^{6}-10296 b^{12} c \,d^{6} x^{6}-1287 a^{2} b^{10} d^{7} x^{5}-7722 a \,b^{11} c \,d^{6} x^{5}-27027 b^{12} c^{2} d^{5} x^{5}-715 a^{3} b^{9} d^{7} x^{4}-4290 a^{2} b^{10} c \,d^{6} x^{4}-15015 a \,b^{11} c^{2} d^{5} x^{4}-40040 b^{12} c^{3} d^{4} x^{4}-286 a^{4} b^{8} d^{7} x^{3}-1716 a^{3} b^{9} c \,d^{6} x^{3}-6006 a^{2} b^{10} c^{2} d^{5} x^{3}-16016 a \,b^{11} c^{3} d^{4} x^{3}-36036 b^{12} c^{4} d^{3} x^{3}-78 a^{5} b^{7} d^{7} x^{2}-468 a^{4} b^{8} c \,d^{6} x^{2}-1638 a^{3} b^{9} c^{2} d^{5} x^{2}-4368 a^{2} b^{10} c^{3} d^{4} x^{2}-9828 a \,b^{11} c^{4} d^{3} x^{2}-19656 b^{12} c^{5} d^{2} x^{2}-13 a^{6} b^{6} d^{7} x -78 a^{5} b^{7} c \,d^{6} x -273 a^{4} b^{8} c^{2} d^{5} x -728 a^{3} b^{9} c^{3} d^{4} x -1638 a^{2} b^{10} c^{4} d^{3} x -3276 a \,b^{11} c^{5} d^{2} x -6006 b^{12} c^{6} d x -a^{7} b^{5} d^{7}-6 a^{6} b^{6} c \,d^{6}-21 a^{5} b^{7} c^{2} d^{5}-56 a^{4} b^{8} c^{3} d^{4}-126 a^{3} b^{9} c^{4} d^{3}-252 a^{2} b^{10} c^{5} d^{2}-462 a \,c^{6} d \,b^{11}-792 b^{12} c^{7}}{10296 b^{13} \left (b x +a \right )^{13}}\) | \(505\) |
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Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (184) = 368\).
Time = 0.23 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.99 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \, {\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \, {\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \, {\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \, {\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \, {\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \, {\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \, {\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \]
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Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (184) = 368\).
Time = 0.24 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.99 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \, {\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \, {\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \, {\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \, {\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \, {\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \, {\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \, {\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (184) = 368\).
Time = 0.30 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.51 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {1716 \, b^{7} d^{7} x^{7} + 10296 \, b^{7} c d^{6} x^{6} + 1716 \, a b^{6} d^{7} x^{6} + 27027 \, b^{7} c^{2} d^{5} x^{5} + 7722 \, a b^{6} c d^{6} x^{5} + 1287 \, a^{2} b^{5} d^{7} x^{5} + 40040 \, b^{7} c^{3} d^{4} x^{4} + 15015 \, a b^{6} c^{2} d^{5} x^{4} + 4290 \, a^{2} b^{5} c d^{6} x^{4} + 715 \, a^{3} b^{4} d^{7} x^{4} + 36036 \, b^{7} c^{4} d^{3} x^{3} + 16016 \, a b^{6} c^{3} d^{4} x^{3} + 6006 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 1716 \, a^{3} b^{4} c d^{6} x^{3} + 286 \, a^{4} b^{3} d^{7} x^{3} + 19656 \, b^{7} c^{5} d^{2} x^{2} + 9828 \, a b^{6} c^{4} d^{3} x^{2} + 4368 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 1638 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 468 \, a^{4} b^{3} c d^{6} x^{2} + 78 \, a^{5} b^{2} d^{7} x^{2} + 6006 \, b^{7} c^{6} d x + 3276 \, a b^{6} c^{5} d^{2} x + 1638 \, a^{2} b^{5} c^{4} d^{3} x + 728 \, a^{3} b^{4} c^{3} d^{4} x + 273 \, a^{4} b^{3} c^{2} d^{5} x + 78 \, a^{5} b^{2} c d^{6} x + 13 \, a^{6} b d^{7} x + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7}}{10296 \, {\left (b x + a\right )}^{13} b^{8}} \]
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Time = 0.42 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.88 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {\frac {a^7\,d^7+6\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5+56\,a^4\,b^3\,c^3\,d^4+126\,a^3\,b^4\,c^4\,d^3+252\,a^2\,b^5\,c^5\,d^2+462\,a\,b^6\,c^6\,d+792\,b^7\,c^7}{10296\,b^8}+\frac {d^7\,x^7}{6\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+6\,a^4\,b\,c\,d^4+21\,a^3\,b^2\,c^2\,d^3+56\,a^2\,b^3\,c^3\,d^2+126\,a\,b^4\,c^4\,d+252\,b^5\,c^5\right )}{132\,b^6}+\frac {5\,d^4\,x^4\,\left (a^3\,d^3+6\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d+56\,b^3\,c^3\right )}{72\,b^4}+\frac {d^6\,x^6\,\left (a\,d+6\,b\,c\right )}{6\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+6\,a^3\,b\,c\,d^3+21\,a^2\,b^2\,c^2\,d^2+56\,a\,b^3\,c^3\,d+126\,b^4\,c^4\right )}{36\,b^5}+\frac {d\,x\,\left (a^6\,d^6+6\,a^5\,b\,c\,d^5+21\,a^4\,b^2\,c^2\,d^4+56\,a^3\,b^3\,c^3\,d^3+126\,a^2\,b^4\,c^4\,d^2+252\,a\,b^5\,c^5\,d+462\,b^6\,c^6\right )}{792\,b^7}+\frac {d^5\,x^5\,\left (a^2\,d^2+6\,a\,b\,c\,d+21\,b^2\,c^2\right )}{8\,b^3}}{a^{13}+13\,a^{12}\,b\,x+78\,a^{11}\,b^2\,x^2+286\,a^{10}\,b^3\,x^3+715\,a^9\,b^4\,x^4+1287\,a^8\,b^5\,x^5+1716\,a^7\,b^6\,x^6+1716\,a^6\,b^7\,x^7+1287\,a^5\,b^8\,x^8+715\,a^4\,b^9\,x^9+286\,a^3\,b^{10}\,x^{10}+78\,a^2\,b^{11}\,x^{11}+13\,a\,b^{12}\,x^{12}+b^{13}\,x^{13}} \]
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