Integrand size = 26, antiderivative size = 173 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {(b d-a e)^6}{12 e^7 (d+e x)^{12}}+\frac {6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac {3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac {20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac {15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac {6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac {b^6}{6 e^7 (d+e x)^6} \]
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Time = 0.08 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=\frac {6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac {15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac {20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac {3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac {6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac {(b d-a e)^6}{12 e^7 (d+e x)^{12}}-\frac {b^6}{6 e^7 (d+e x)^6} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^6}{(d+e x)^{13}} \, dx \\ & = \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{13}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{12}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{11}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{10}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^9}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^8}+\frac {b^6}{e^6 (d+e x)^7}\right ) \, dx \\ & = -\frac {(b d-a e)^6}{12 e^7 (d+e x)^{12}}+\frac {6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac {3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac {20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac {15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac {6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac {b^6}{6 e^7 (d+e x)^6} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {462 a^6 e^6+252 a^5 b e^5 (d+12 e x)+126 a^4 b^2 e^4 \left (d^2+12 d e x+66 e^2 x^2\right )+56 a^3 b^3 e^3 \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+21 a^2 b^4 e^2 \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+6 a b^5 e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+b^6 \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )}{5544 e^7 (d+e x)^{12}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(334\) vs. \(2(159)=318\).
Time = 2.24 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.94
method | result | size |
risch | \(\frac {-\frac {b^{6} x^{6}}{6 e}-\frac {b^{5} \left (6 a e +b d \right ) x^{5}}{7 e^{2}}-\frac {5 b^{4} \left (21 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) x^{4}}{56 e^{3}}-\frac {5 b^{3} \left (56 a^{3} e^{3}+21 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{126 e^{4}}-\frac {b^{2} \left (126 e^{4} a^{4}+56 b \,e^{3} d \,a^{3}+21 b^{2} e^{2} d^{2} a^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{84 e^{5}}-\frac {b \left (252 a^{5} e^{5}+126 a^{4} b d \,e^{4}+56 a^{3} b^{2} d^{2} e^{3}+21 a^{2} b^{3} d^{3} e^{2}+6 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{462 e^{6}}-\frac {462 a^{6} e^{6}+252 a^{5} b d \,e^{5}+126 a^{4} b^{2} d^{2} e^{4}+56 a^{3} b^{3} d^{3} e^{3}+21 a^{2} b^{4} d^{4} e^{2}+6 a \,b^{5} d^{5} e +b^{6} d^{6}}{5544 e^{7}}}{\left (e x +d \right )^{12}}\) | \(335\) |
default | \(-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{12 e^{7} \left (e x +d \right )^{12}}-\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{11 e^{7} \left (e x +d \right )^{11}}-\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{8 e^{7} \left (e x +d \right )^{8}}-\frac {3 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{2 e^{7} \left (e x +d \right )^{10}}-\frac {6 b^{5} \left (a e -b d \right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {b^{6}}{6 e^{7} \left (e x +d \right )^{6}}-\frac {20 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{9 e^{7} \left (e x +d \right )^{9}}\) | \(357\) |
norman | \(\frac {-\frac {b^{6} x^{6}}{6 e}-\frac {\left (6 a \,b^{5} e^{6}+b^{6} d \,e^{5}\right ) x^{5}}{7 e^{7}}-\frac {5 \left (21 a^{2} b^{4} e^{7}+6 a \,b^{5} d \,e^{6}+b^{6} d^{2} e^{5}\right ) x^{4}}{56 e^{8}}-\frac {5 \left (56 a^{3} b^{3} e^{8}+21 a^{2} b^{4} d \,e^{7}+6 a \,b^{5} d^{2} e^{6}+b^{6} d^{3} e^{5}\right ) x^{3}}{126 e^{9}}-\frac {\left (126 a^{4} b^{2} e^{9}+56 a^{3} b^{3} d \,e^{8}+21 a^{2} b^{4} d^{2} e^{7}+6 a \,b^{5} d^{3} e^{6}+b^{6} d^{4} e^{5}\right ) x^{2}}{84 e^{10}}-\frac {\left (252 a^{5} b \,e^{10}+126 a^{4} b^{2} d \,e^{9}+56 a^{3} b^{3} d^{2} e^{8}+21 a^{2} b^{4} d^{3} e^{7}+6 a \,b^{5} d^{4} e^{6}+b^{6} d^{5} e^{5}\right ) x}{462 e^{11}}-\frac {462 a^{6} e^{11}+252 a^{5} b d \,e^{10}+126 a^{4} b^{2} d^{2} e^{9}+56 a^{3} b^{3} d^{3} e^{8}+21 a^{2} b^{4} d^{4} e^{7}+6 a \,b^{5} d^{5} e^{6}+b^{6} d^{6} e^{5}}{5544 e^{12}}}{\left (e x +d \right )^{12}}\) | \(375\) |
gosper | \(-\frac {924 x^{6} b^{6} e^{6}+4752 x^{5} a \,b^{5} e^{6}+792 x^{5} b^{6} d \,e^{5}+10395 x^{4} a^{2} b^{4} e^{6}+2970 x^{4} a \,b^{5} d \,e^{5}+495 x^{4} b^{6} d^{2} e^{4}+12320 x^{3} a^{3} b^{3} e^{6}+4620 x^{3} a^{2} b^{4} d \,e^{5}+1320 x^{3} a \,b^{5} d^{2} e^{4}+220 x^{3} b^{6} d^{3} e^{3}+8316 x^{2} a^{4} b^{2} e^{6}+3696 x^{2} a^{3} b^{3} d \,e^{5}+1386 x^{2} a^{2} b^{4} d^{2} e^{4}+396 x^{2} a \,b^{5} d^{3} e^{3}+66 x^{2} b^{6} d^{4} e^{2}+3024 x \,a^{5} b \,e^{6}+1512 x \,a^{4} b^{2} d \,e^{5}+672 x \,a^{3} b^{3} d^{2} e^{4}+252 x \,a^{2} b^{4} d^{3} e^{3}+72 x a \,b^{5} d^{4} e^{2}+12 x \,b^{6} d^{5} e +462 a^{6} e^{6}+252 a^{5} b d \,e^{5}+126 a^{4} b^{2} d^{2} e^{4}+56 a^{3} b^{3} d^{3} e^{3}+21 a^{2} b^{4} d^{4} e^{2}+6 a \,b^{5} d^{5} e +b^{6} d^{6}}{5544 e^{7} \left (e x +d \right )^{12}}\) | \(376\) |
parallelrisch | \(\frac {-924 b^{6} x^{6} e^{11}-4752 a \,b^{5} e^{11} x^{5}-792 b^{6} d \,e^{10} x^{5}-10395 a^{2} b^{4} e^{11} x^{4}-2970 a \,b^{5} d \,e^{10} x^{4}-495 b^{6} d^{2} e^{9} x^{4}-12320 a^{3} b^{3} e^{11} x^{3}-4620 a^{2} b^{4} d \,e^{10} x^{3}-1320 a \,b^{5} d^{2} e^{9} x^{3}-220 b^{6} d^{3} e^{8} x^{3}-8316 a^{4} b^{2} e^{11} x^{2}-3696 a^{3} b^{3} d \,e^{10} x^{2}-1386 a^{2} b^{4} d^{2} e^{9} x^{2}-396 a \,b^{5} d^{3} e^{8} x^{2}-66 b^{6} d^{4} e^{7} x^{2}-3024 a^{5} b \,e^{11} x -1512 a^{4} b^{2} d \,e^{10} x -672 a^{3} b^{3} d^{2} e^{9} x -252 a^{2} b^{4} d^{3} e^{8} x -72 a \,b^{5} d^{4} e^{7} x -12 b^{6} d^{5} e^{6} x -462 a^{6} e^{11}-252 a^{5} b d \,e^{10}-126 a^{4} b^{2} d^{2} e^{9}-56 a^{3} b^{3} d^{3} e^{8}-21 a^{2} b^{4} d^{4} e^{7}-6 a \,b^{5} d^{5} e^{6}-b^{6} d^{6} e^{5}}{5544 e^{12} \left (e x +d \right )^{12}}\) | \(384\) |
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Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (159) = 318\).
Time = 0.31 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.74 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \, {\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \, {\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \, {\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \, {\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \, {\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \, {\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (159) = 318\).
Time = 0.22 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.74 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \, {\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \, {\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \, {\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \, {\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \, {\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \, {\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (159) = 318\).
Time = 0.25 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.17 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {924 \, b^{6} e^{6} x^{6} + 792 \, b^{6} d e^{5} x^{5} + 4752 \, a b^{5} e^{6} x^{5} + 495 \, b^{6} d^{2} e^{4} x^{4} + 2970 \, a b^{5} d e^{5} x^{4} + 10395 \, a^{2} b^{4} e^{6} x^{4} + 220 \, b^{6} d^{3} e^{3} x^{3} + 1320 \, a b^{5} d^{2} e^{4} x^{3} + 4620 \, a^{2} b^{4} d e^{5} x^{3} + 12320 \, a^{3} b^{3} e^{6} x^{3} + 66 \, b^{6} d^{4} e^{2} x^{2} + 396 \, a b^{5} d^{3} e^{3} x^{2} + 1386 \, a^{2} b^{4} d^{2} e^{4} x^{2} + 3696 \, a^{3} b^{3} d e^{5} x^{2} + 8316 \, a^{4} b^{2} e^{6} x^{2} + 12 \, b^{6} d^{5} e x + 72 \, a b^{5} d^{4} e^{2} x + 252 \, a^{2} b^{4} d^{3} e^{3} x + 672 \, a^{3} b^{3} d^{2} e^{4} x + 1512 \, a^{4} b^{2} d e^{5} x + 3024 \, a^{5} b e^{6} x + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6}}{5544 \, {\left (e x + d\right )}^{12} e^{7}} \]
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Time = 10.10 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{13}} \, dx=-\frac {\frac {462\,a^6\,e^6+252\,a^5\,b\,d\,e^5+126\,a^4\,b^2\,d^2\,e^4+56\,a^3\,b^3\,d^3\,e^3+21\,a^2\,b^4\,d^4\,e^2+6\,a\,b^5\,d^5\,e+b^6\,d^6}{5544\,e^7}+\frac {b^6\,x^6}{6\,e}+\frac {5\,b^3\,x^3\,\left (56\,a^3\,e^3+21\,a^2\,b\,d\,e^2+6\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{126\,e^4}+\frac {b\,x\,\left (252\,a^5\,e^5+126\,a^4\,b\,d\,e^4+56\,a^3\,b^2\,d^2\,e^3+21\,a^2\,b^3\,d^3\,e^2+6\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{462\,e^6}+\frac {b^5\,x^5\,\left (6\,a\,e+b\,d\right )}{7\,e^2}+\frac {b^2\,x^2\,\left (126\,a^4\,e^4+56\,a^3\,b\,d\,e^3+21\,a^2\,b^2\,d^2\,e^2+6\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{84\,e^5}+\frac {5\,b^4\,x^4\,\left (21\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )}{56\,e^3}}{d^{12}+12\,d^{11}\,e\,x+66\,d^{10}\,e^2\,x^2+220\,d^9\,e^3\,x^3+495\,d^8\,e^4\,x^4+792\,d^7\,e^5\,x^5+924\,d^6\,e^6\,x^6+792\,d^5\,e^7\,x^7+495\,d^4\,e^8\,x^8+220\,d^3\,e^9\,x^9+66\,d^2\,e^{10}\,x^{10}+12\,d\,e^{11}\,x^{11}+e^{12}\,x^{12}} \]
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