Integrand size = 24, antiderivative size = 101 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=\frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5} \]
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Time = 0.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=\frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 d^{12} (b+2 c x)^{12}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^{12} (b+2 c x)^{10}}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^{12} (b+2 c x)^8}+\frac {1}{64 c^3 d^{12} (b+2 c x)^6}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=\frac {105 \left (b^2-4 a c\right )^3-385 \left (b^2-4 a c\right )^2 (b+2 c x)^2+495 \left (b^2-4 a c\right ) (b+2 c x)^4-231 (b+2 c x)^6}{147840 c^4 d^{12} (b+2 c x)^{11}} \]
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Time = 2.52 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {-\frac {1}{640 c^{4} \left (2 c x +b \right )^{5}}-\frac {12 a c -3 b^{2}}{896 c^{4} \left (2 c x +b \right )^{7}}-\frac {64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{1408 c^{4} \left (2 c x +b \right )^{11}}-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{1152 c^{4} \left (2 c x +b \right )^{9}}}{d^{12}}\) | \(121\) |
risch | \(\frac {-\frac {c^{2} x^{6}}{10}-\frac {3 b c \,x^{5}}{10}+\left (-\frac {3 a c}{14}-\frac {9 b^{2}}{28}\right ) x^{4}-\frac {b \left (3 a c +b^{2}\right ) x^{3}}{7 c}-\frac {\left (7 a^{2} c^{2}+10 a \,b^{2} c +b^{4}\right ) x^{2}}{42 c^{2}}-\frac {b \left (70 a^{2} c^{2}+10 a \,b^{2} c +b^{4}\right ) x}{420 c^{3}}-\frac {420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}}{9240 c^{4}}}{d^{12} \left (2 c x +b \right )^{11}}\) | \(148\) |
gosper | \(-\frac {924 c^{6} x^{6}+2772 b \,c^{5} x^{5}+1980 a \,c^{5} x^{4}+2970 b^{2} c^{4} x^{4}+3960 a b \,c^{4} x^{3}+1320 x^{3} b^{3} c^{3}+1540 a^{2} c^{4} x^{2}+2200 a \,b^{2} c^{3} x^{2}+220 x^{2} b^{4} c^{2}+1540 a^{2} b \,c^{3} x +220 x a \,b^{3} c^{2}+22 x \,b^{5} c +420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}}{9240 \left (2 c x +b \right )^{11} d^{12} c^{4}}\) | \(166\) |
norman | \(\frac {\frac {a^{3} x}{b d}+\frac {\left (60 a^{3} c^{2}+10 a^{2} b^{2} c +a \,b^{4}\right ) x^{3}}{b^{3} d}+\frac {\left (20 c \,a^{3}+3 a^{2} b^{2}\right ) x^{2}}{2 b^{2} d}+\frac {\left (960 c^{3} a^{3}+160 a^{2} b^{2} c^{2}+22 a \,b^{4} c +b^{6}\right ) x^{4}}{4 b^{4} d}+\frac {128 c^{6} \left (420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{10}}{105 b^{10} d}+\frac {64 c^{5} \left (420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{9}}{21 b^{9} d}+\frac {32 c^{4} \left (420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{8}}{7 b^{8} d}+\frac {32 c^{3} \left (420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{7}}{7 b^{7} d}+\frac {c \left (6720 c^{3} a^{3}+1120 a^{2} b^{2} c^{2}+160 a \,b^{4} c +13 b^{6}\right ) x^{5}}{10 b^{5} d}+\frac {c^{2} \left (13440 c^{3} a^{3}+2240 a^{2} b^{2} c^{2}+320 a \,b^{4} c +31 b^{6}\right ) x^{6}}{10 b^{6} d}+\frac {256 c^{7} \left (420 c^{3} a^{3}+70 a^{2} b^{2} c^{2}+10 a \,b^{4} c +b^{6}\right ) x^{11}}{1155 b^{11} d}}{d^{11} \left (2 c x +b \right )^{11}}\) | \(435\) |
parallelrisch | \(\frac {430080 x^{11} a^{3} c^{10}+71680 x^{11} a^{2} b^{2} c^{9}+10240 x^{11} a \,b^{4} c^{8}+1024 x^{11} b^{6} c^{7}+2365440 x^{10} a^{3} b \,c^{9}+394240 x^{10} a^{2} b^{3} c^{8}+56320 x^{10} a \,b^{5} c^{7}+5632 x^{10} b^{7} c^{6}+5913600 x^{9} a^{3} b^{2} c^{8}+985600 x^{9} a^{2} b^{4} c^{7}+140800 x^{9} a \,b^{6} c^{6}+14080 x^{9} b^{8} c^{5}+8870400 x^{8} a^{3} b^{3} c^{7}+1478400 x^{8} a^{2} b^{5} c^{6}+211200 x^{8} a \,b^{7} c^{5}+21120 x^{8} b^{9} c^{4}+8870400 x^{7} a^{3} b^{4} c^{6}+1478400 x^{7} a^{2} b^{6} c^{5}+211200 x^{7} a \,b^{8} c^{4}+21120 x^{7} b^{10} c^{3}+6209280 x^{6} a^{3} b^{5} c^{5}+1034880 x^{6} a^{2} b^{7} c^{4}+147840 x^{6} a \,b^{9} c^{3}+14322 x^{6} b^{11} c^{2}+3104640 x^{5} a^{3} b^{6} c^{4}+517440 x^{5} a^{2} b^{8} c^{3}+73920 x^{5} a \,b^{10} c^{2}+6006 x^{5} b^{12} c +1108800 x^{4} a^{3} b^{7} c^{3}+184800 x^{4} a^{2} b^{9} c^{2}+25410 x^{4} a \,b^{11} c +1155 x^{4} b^{13}+277200 x^{3} a^{3} b^{8} c^{2}+46200 x^{3} a^{2} b^{10} c +4620 x^{3} a \,b^{12}+46200 x^{2} a^{3} b^{9} c +6930 x^{2} a^{2} b^{11}+4620 a^{3} b^{10} x}{4620 b^{11} d^{12} \left (2 c x +b \right )^{11}}\) | \(481\) |
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Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (93) = 186\).
Time = 0.33 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.03 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=-\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3} + 990 \, {\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 1320 \, {\left (b^{3} c^{3} + 3 \, a b c^{4}\right )} x^{3} + 220 \, {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} + 22 \, {\left (b^{5} c + 10 \, a b^{3} c^{2} + 70 \, a^{2} b c^{3}\right )} x}{9240 \, {\left (2048 \, c^{15} d^{12} x^{11} + 11264 \, b c^{14} d^{12} x^{10} + 28160 \, b^{2} c^{13} d^{12} x^{9} + 42240 \, b^{3} c^{12} d^{12} x^{8} + 42240 \, b^{4} c^{11} d^{12} x^{7} + 29568 \, b^{5} c^{10} d^{12} x^{6} + 14784 \, b^{6} c^{9} d^{12} x^{5} + 5280 \, b^{7} c^{8} d^{12} x^{4} + 1320 \, b^{8} c^{7} d^{12} x^{3} + 220 \, b^{9} c^{6} d^{12} x^{2} + 22 \, b^{10} c^{5} d^{12} x + b^{11} c^{4} d^{12}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (100) = 200\).
Time = 10.40 (sec) , antiderivative size = 328, normalized size of antiderivative = 3.25 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=\frac {- 420 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} - 10 a b^{4} c - b^{6} - 2772 b c^{5} x^{5} - 924 c^{6} x^{6} + x^{4} \left (- 1980 a c^{5} - 2970 b^{2} c^{4}\right ) + x^{3} \left (- 3960 a b c^{4} - 1320 b^{3} c^{3}\right ) + x^{2} \left (- 1540 a^{2} c^{4} - 2200 a b^{2} c^{3} - 220 b^{4} c^{2}\right ) + x \left (- 1540 a^{2} b c^{3} - 220 a b^{3} c^{2} - 22 b^{5} c\right )}{9240 b^{11} c^{4} d^{12} + 203280 b^{10} c^{5} d^{12} x + 2032800 b^{9} c^{6} d^{12} x^{2} + 12196800 b^{8} c^{7} d^{12} x^{3} + 48787200 b^{7} c^{8} d^{12} x^{4} + 136604160 b^{6} c^{9} d^{12} x^{5} + 273208320 b^{5} c^{10} d^{12} x^{6} + 390297600 b^{4} c^{11} d^{12} x^{7} + 390297600 b^{3} c^{12} d^{12} x^{8} + 260198400 b^{2} c^{13} d^{12} x^{9} + 104079360 b c^{14} d^{12} x^{10} + 18923520 c^{15} d^{12} x^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (93) = 186\).
Time = 0.22 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.03 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=-\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3} + 990 \, {\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 1320 \, {\left (b^{3} c^{3} + 3 \, a b c^{4}\right )} x^{3} + 220 \, {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} + 22 \, {\left (b^{5} c + 10 \, a b^{3} c^{2} + 70 \, a^{2} b c^{3}\right )} x}{9240 \, {\left (2048 \, c^{15} d^{12} x^{11} + 11264 \, b c^{14} d^{12} x^{10} + 28160 \, b^{2} c^{13} d^{12} x^{9} + 42240 \, b^{3} c^{12} d^{12} x^{8} + 42240 \, b^{4} c^{11} d^{12} x^{7} + 29568 \, b^{5} c^{10} d^{12} x^{6} + 14784 \, b^{6} c^{9} d^{12} x^{5} + 5280 \, b^{7} c^{8} d^{12} x^{4} + 1320 \, b^{8} c^{7} d^{12} x^{3} + 220 \, b^{9} c^{6} d^{12} x^{2} + 22 \, b^{10} c^{5} d^{12} x + b^{11} c^{4} d^{12}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=-\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + 2970 \, b^{2} c^{4} x^{4} + 1980 \, a c^{5} x^{4} + 1320 \, b^{3} c^{3} x^{3} + 3960 \, a b c^{4} x^{3} + 220 \, b^{4} c^{2} x^{2} + 2200 \, a b^{2} c^{3} x^{2} + 1540 \, a^{2} c^{4} x^{2} + 22 \, b^{5} c x + 220 \, a b^{3} c^{2} x + 1540 \, a^{2} b c^{3} x + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3}}{9240 \, {\left (2 \, c x + b\right )}^{11} c^{4} d^{12}} \]
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Time = 0.39 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.89 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx=-\frac {\frac {420\,a^3\,c^3+70\,a^2\,b^2\,c^2+10\,a\,b^4\,c+b^6}{9240\,c^4}+x^4\,\left (\frac {9\,b^2}{28}+\frac {3\,a\,c}{14}\right )+\frac {c^2\,x^6}{10}+\frac {x^3\,\left (b^3+3\,a\,c\,b\right )}{7\,c}+\frac {x^2\,\left (7\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{42\,c^2}+\frac {3\,b\,c\,x^5}{10}+\frac {b\,x\,\left (70\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{420\,c^3}}{b^{11}\,d^{12}+22\,b^{10}\,c\,d^{12}\,x+220\,b^9\,c^2\,d^{12}\,x^2+1320\,b^8\,c^3\,d^{12}\,x^3+5280\,b^7\,c^4\,d^{12}\,x^4+14784\,b^6\,c^5\,d^{12}\,x^5+29568\,b^5\,c^6\,d^{12}\,x^6+42240\,b^4\,c^7\,d^{12}\,x^7+42240\,b^3\,c^8\,d^{12}\,x^8+28160\,b^2\,c^9\,d^{12}\,x^9+11264\,b\,c^{10}\,d^{12}\,x^{10}+2048\,c^{11}\,d^{12}\,x^{11}} \]
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