Integrand size = 24, antiderivative size = 73 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3} \]
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Time = 0.04 (sec) , antiderivative size = 73, 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} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^8 (b+2 c x)^8}+\frac {-b^2+4 a c}{8 c^2 d^8 (b+2 c x)^6}+\frac {1}{16 c^2 d^8 (b+2 c x)^4}\right ) \, dx \\ & = -\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=\frac {-15 \left (b^2-4 a c\right )^2+42 \left (b^2-4 a c\right ) (b+2 c x)^2-35 (b+2 c x)^4}{3360 c^3 d^8 (b+2 c x)^7} \]
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Time = 2.64 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {-\frac {4 a c -b^{2}}{80 c^{3} \left (2 c x +b \right )^{5}}-\frac {1}{96 c^{3} \left (2 c x +b \right )^{3}}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{224 c^{3} \left (2 c x +b \right )^{7}}}{d^{8}}\) | \(74\) |
risch | \(\frac {-\frac {x^{4} c}{6}-\frac {b \,x^{3}}{3}-\frac {\left (a c +b^{2}\right ) x^{2}}{5 c}-\frac {b \left (6 a c +b^{2}\right ) x}{30 c^{2}}-\frac {30 a^{2} c^{2}+6 a \,b^{2} c +b^{4}}{420 c^{3}}}{d^{8} \left (2 c x +b \right )^{7}}\) | \(80\) |
gosper | \(-\frac {70 c^{4} x^{4}+140 b \,c^{3} x^{3}+84 x^{2} a \,c^{3}+84 b^{2} c^{2} x^{2}+84 a b \,c^{2} x +14 b^{3} c x +30 a^{2} c^{2}+6 a \,b^{2} c +b^{4}}{420 \left (2 c x +b \right )^{7} d^{8} c^{3}}\) | \(88\) |
norman | \(\frac {\frac {a^{2} x}{b d}+\frac {\left (6 a^{2} c +a \,b^{2}\right ) x^{2}}{b^{2} d}+\frac {\left (60 a^{2} c^{2}+12 a \,b^{2} c +b^{4}\right ) x^{3}}{3 b^{3} d}+\frac {16 c^{3} \left (30 a^{2} c^{2}+6 a \,b^{2} c +b^{4}\right ) x^{6}}{15 b^{6} d}+\frac {8 c^{2} \left (30 a^{2} c^{2}+6 a \,b^{2} c +b^{4}\right ) x^{5}}{5 b^{5} d}+\frac {c \left (240 a^{2} c^{2}+48 a \,b^{2} c +7 b^{4}\right ) x^{4}}{6 b^{4} d}+\frac {32 c^{4} \left (30 a^{2} c^{2}+6 a \,b^{2} c +b^{4}\right ) x^{7}}{105 b^{7} d}}{d^{7} \left (2 c x +b \right )^{7}}\) | \(209\) |
parallelrisch | \(\frac {1920 x^{7} a^{2} c^{6}+384 x^{7} a \,b^{2} c^{5}+64 x^{7} b^{4} c^{4}+6720 x^{6} a^{2} b \,c^{5}+1344 x^{6} a \,b^{3} c^{4}+224 x^{6} b^{5} c^{3}+10080 x^{5} a^{2} b^{2} c^{4}+2016 x^{5} a \,b^{4} c^{3}+336 x^{5} b^{6} c^{2}+8400 x^{4} a^{2} b^{3} c^{3}+1680 x^{4} a \,b^{5} c^{2}+245 x^{4} b^{7} c +4200 x^{3} a^{2} b^{4} c^{2}+840 x^{3} a \,b^{6} c +70 b^{8} x^{3}+1260 x^{2} a^{2} b^{5} c +210 x^{2} a \,b^{7}+210 a^{2} b^{6} x}{210 b^{7} d^{8} \left (2 c x +b \right )^{7}}\) | \(221\) |
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Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (67) = 134\).
Time = 0.30 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.41 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \, {\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \, {\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \, {\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (70) = 140\).
Time = 1.09 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.59 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=\frac {- 30 a^{2} c^{2} - 6 a b^{2} c - b^{4} - 140 b c^{3} x^{3} - 70 c^{4} x^{4} + x^{2} \left (- 84 a c^{3} - 84 b^{2} c^{2}\right ) + x \left (- 84 a b c^{2} - 14 b^{3} c\right )}{420 b^{7} c^{3} d^{8} + 5880 b^{6} c^{4} d^{8} x + 35280 b^{5} c^{5} d^{8} x^{2} + 117600 b^{4} c^{6} d^{8} x^{3} + 235200 b^{3} c^{7} d^{8} x^{4} + 282240 b^{2} c^{8} d^{8} x^{5} + 188160 b c^{9} d^{8} x^{6} + 53760 c^{10} d^{8} x^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (67) = 134\).
Time = 0.19 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.41 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \, {\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \, {\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \, {\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + 84 \, b^{2} c^{2} x^{2} + 84 \, a c^{3} x^{2} + 14 \, b^{3} c x + 84 \, a b c^{2} x + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2}}{420 \, {\left (2 \, c x + b\right )}^{7} c^{3} d^{8}} \]
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Time = 0.12 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.30 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx=-\frac {\frac {30\,a^2\,c^2+6\,a\,b^2\,c+b^4}{420\,c^3}+\frac {b\,x^3}{3}+\frac {c\,x^4}{6}+\frac {x^2\,\left (b^2+a\,c\right )}{5\,c}+\frac {b\,x\,\left (b^2+6\,a\,c\right )}{30\,c^2}}{b^7\,d^8+14\,b^6\,c\,d^8\,x+84\,b^5\,c^2\,d^8\,x^2+280\,b^4\,c^3\,d^8\,x^3+560\,b^3\,c^4\,d^8\,x^4+672\,b^2\,c^5\,d^8\,x^5+448\,b\,c^6\,d^8\,x^6+128\,c^7\,d^8\,x^7} \]
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