Integrand size = 35, antiderivative size = 77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^9} \, dx=-\frac {\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}+\frac {2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac {c^2 d^2}{4 e^3 (d+e x)^4} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^9} \, dx=\frac {2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac {\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}-\frac {c^2 d^2}{4 e^3 (d+e x)^4} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a e+c d x)^2}{(d+e x)^7} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^7}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^6}+\frac {c^2 d^2}{e^2 (d+e x)^5}\right ) \, dx \\ & = -\frac {\left (c d^2-a e^2\right )^2}{6 e^3 (d+e x)^6}+\frac {2 c d \left (c d^2-a e^2\right )}{5 e^3 (d+e x)^5}-\frac {c^2 d^2}{4 e^3 (d+e x)^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^9} \, dx=-\frac {10 a^2 e^4+4 a c d e^2 (d+6 e x)+c^2 d^2 \left (d^2+6 d e x+15 e^2 x^2\right )}{60 e^3 (d+e x)^6} \]
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Time = 2.45 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {15 x^{2} c^{2} d^{2} e^{2}+24 x a c d \,e^{3}+6 x \,c^{2} d^{3} e +10 a^{2} e^{4}+4 a c \,d^{2} e^{2}+c^{2} d^{4}}{60 e^{3} \left (e x +d \right )^{6}}\) | \(72\) |
risch | \(\frac {-\frac {d^{2} c^{2} x^{2}}{4 e}-\frac {d c \left (4 e^{2} a +c \,d^{2}\right ) x}{10 e^{2}}-\frac {10 a^{2} e^{4}+4 a c \,d^{2} e^{2}+c^{2} d^{4}}{60 e^{3}}}{\left (e x +d \right )^{6}}\) | \(75\) |
parallelrisch | \(\frac {-15 c^{2} d^{2} x^{2} e^{5}-24 a c d \,e^{6} x -6 c^{2} d^{3} e^{4} x -10 a^{2} e^{7}-4 a \,d^{2} e^{5} c -c^{2} d^{4} e^{3}}{60 e^{6} \left (e x +d \right )^{6}}\) | \(78\) |
default | \(-\frac {2 c d \left (e^{2} a -c \,d^{2}\right )}{5 e^{3} \left (e x +d \right )^{5}}-\frac {c^{2} d^{2}}{4 e^{3} \left (e x +d \right )^{4}}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{6 e^{3} \left (e x +d \right )^{6}}\) | \(83\) |
norman | \(\frac {-\frac {d^{2} \left (10 a^{2} e^{9}+4 a c \,d^{2} e^{7}+c^{2} d^{4} e^{5}\right )}{60 e^{8}}-\frac {\left (5 a^{2} e^{9}+26 a c \,d^{2} e^{7}+14 c^{2} d^{4} e^{5}\right ) x^{2}}{30 e^{6}}-\frac {d \left (2 a \,e^{7} c +3 d^{2} e^{5} c^{2}\right ) x^{3}}{5 e^{5}}-\frac {d \left (5 a^{2} e^{9}+8 a c \,d^{2} e^{7}+2 c^{2} d^{4} e^{5}\right ) x}{15 e^{7}}-\frac {e \,c^{2} d^{2} x^{4}}{4}}{\left (e x +d \right )^{8}}\) | \(162\) |
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Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^9} \, dx=-\frac {15 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4} + 6 \, {\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (68) = 136\).
Time = 13.38 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^9} \, dx=\frac {- 10 a^{2} e^{4} - 4 a c d^{2} e^{2} - c^{2} d^{4} - 15 c^{2} d^{2} e^{2} x^{2} + x \left (- 24 a c d e^{3} - 6 c^{2} d^{3} e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \]
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Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^9} \, dx=-\frac {15 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4} + 6 \, {\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x}{60 \, {\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^9} \, dx=-\frac {15 \, c^{2} d^{2} e^{2} x^{2} + 6 \, c^{2} d^{3} e x + 24 \, a c d e^{3} x + c^{2} d^{4} + 4 \, a c d^{2} e^{2} + 10 \, a^{2} e^{4}}{60 \, {\left (e x + d\right )}^{6} e^{3}} \]
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Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^9} \, dx=-\frac {\frac {10\,a^2\,e^4+4\,a\,c\,d^2\,e^2+c^2\,d^4}{60\,e^3}+\frac {c^2\,d^2\,x^2}{4\,e}+\frac {c\,d\,x\,\left (c\,d^2+4\,a\,e^2\right )}{10\,e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]
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