Integrand size = 22, antiderivative size = 45 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^7} \, dx=\frac {b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac {1}{32 c^2 d^7 (b+2 c x)^4} \]
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Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {697} \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^7} \, dx=\frac {b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac {1}{32 c^2 d^7 (b+2 c x)^4} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-b^2+4 a c}{4 c d^7 (b+2 c x)^7}+\frac {1}{4 c d^7 (b+2 c x)^5}\right ) \, dx \\ & = \frac {b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac {1}{32 c^2 d^7 (b+2 c x)^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^7} \, dx=\frac {\frac {b^2-4 a c}{48 c^2 (b+2 c x)^6}-\frac {1}{32 c^2 (b+2 c x)^4}}{d^7} \]
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Time = 2.58 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(-\frac {12 c^{2} x^{2}+12 b c x +8 a c +b^{2}}{96 \left (2 c x +b \right )^{6} d^{7} c^{2}}\) | \(38\) |
risch | \(\frac {-\frac {x^{2}}{8}-\frac {b x}{8 c}-\frac {8 a c +b^{2}}{96 c^{2}}}{d^{7} \left (2 c x +b \right )^{6}}\) | \(39\) |
default | \(\frac {-\frac {1}{32 c^{2} \left (2 c x +b \right )^{4}}-\frac {4 a c -b^{2}}{48 c^{2} \left (2 c x +b \right )^{6}}}{d^{7}}\) | \(42\) |
norman | \(\frac {\frac {a x}{b d}+\frac {\left (10 a c +b^{2}\right ) x^{2}}{2 b^{2} d}+\frac {2 c^{3} \left (8 a c +b^{2}\right ) x^{5}}{b^{5} d}+\frac {5 c^{2} \left (8 a c +b^{2}\right ) x^{4}}{2 b^{4} d}+\frac {5 c \left (8 a c +b^{2}\right ) x^{3}}{3 b^{3} d}+\frac {2 c^{4} \left (8 a c +b^{2}\right ) x^{6}}{3 b^{6} d}}{d^{6} \left (2 c x +b \right )^{6}}\) | \(128\) |
parallelrisch | \(\frac {32 x^{6} a \,c^{5}+4 x^{6} b^{2} c^{4}+96 x^{5} a b \,c^{4}+12 x^{5} b^{3} c^{3}+120 a \,b^{2} c^{3} x^{4}+15 c^{2} x^{4} b^{4}+80 x^{3} a \,b^{3} c^{2}+10 x^{3} b^{5} c +30 x^{2} a \,b^{4} c +3 x^{2} b^{6}+6 a \,b^{5} x}{6 b^{6} d^{7} \left (2 c x +b \right )^{6}}\) | \(128\) |
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Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (41) = 82\).
Time = 0.37 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.51 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^7} \, dx=-\frac {12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \, {\left (64 \, c^{8} d^{7} x^{6} + 192 \, b c^{7} d^{7} x^{5} + 240 \, b^{2} c^{6} d^{7} x^{4} + 160 \, b^{3} c^{5} d^{7} x^{3} + 60 \, b^{4} c^{4} d^{7} x^{2} + 12 \, b^{5} c^{3} d^{7} x + b^{6} c^{2} d^{7}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (42) = 84\).
Time = 0.49 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.69 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^7} \, dx=\frac {- 8 a c - b^{2} - 12 b c x - 12 c^{2} x^{2}}{96 b^{6} c^{2} d^{7} + 1152 b^{5} c^{3} d^{7} x + 5760 b^{4} c^{4} d^{7} x^{2} + 15360 b^{3} c^{5} d^{7} x^{3} + 23040 b^{2} c^{6} d^{7} x^{4} + 18432 b c^{7} d^{7} x^{5} + 6144 c^{8} d^{7} x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (41) = 82\).
Time = 0.21 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.51 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^7} \, dx=-\frac {12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \, {\left (64 \, c^{8} d^{7} x^{6} + 192 \, b c^{7} d^{7} x^{5} + 240 \, b^{2} c^{6} d^{7} x^{4} + 160 \, b^{3} c^{5} d^{7} x^{3} + 60 \, b^{4} c^{4} d^{7} x^{2} + 12 \, b^{5} c^{3} d^{7} x + b^{6} c^{2} d^{7}\right )}} \]
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none
Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^7} \, dx=-\frac {12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \, {\left (2 \, c x + b\right )}^{6} c^{2} d^{7}} \]
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Time = 10.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.51 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^7} \, dx=-\frac {\frac {b^2+8\,a\,c}{96\,c^2}+\frac {x^2}{8}+\frac {b\,x}{8\,c}}{b^6\,d^7+12\,b^5\,c\,d^7\,x+60\,b^4\,c^2\,d^7\,x^2+160\,b^3\,c^3\,d^7\,x^3+240\,b^2\,c^4\,d^7\,x^4+192\,b\,c^5\,d^7\,x^5+64\,c^6\,d^7\,x^6} \]
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