Integrand size = 35, antiderivative size = 77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^5}+\frac {c d \left (c d^2-a e^2\right )}{2 e^3 (d+e x)^4}-\frac {c^2 d^2}{3 e^3 (d+e x)^3} \]
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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} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=\frac {c d \left (c d^2-a e^2\right )}{2 e^3 (d+e x)^4}-\frac {\left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^5}-\frac {c^2 d^2}{3 e^3 (d+e x)^3} \]
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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)^6} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^6}-\frac {2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)^5}+\frac {c^2 d^2}{e^2 (d+e x)^4}\right ) \, dx \\ & = -\frac {\left (c d^2-a e^2\right )^2}{5 e^3 (d+e x)^5}+\frac {c d \left (c d^2-a e^2\right )}{2 e^3 (d+e x)^4}-\frac {c^2 d^2}{3 e^3 (d+e x)^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {6 a^2 e^4+3 a c d e^2 (d+5 e x)+c^2 d^2 \left (d^2+5 d e x+10 e^2 x^2\right )}{30 e^3 (d+e x)^5} \]
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Time = 2.97 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {10 x^{2} c^{2} d^{2} e^{2}+15 x a c d \,e^{3}+5 x \,c^{2} d^{3} e +6 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}}{30 e^{3} \left (e x +d \right )^{5}}\) | \(72\) |
risch | \(\frac {-\frac {d^{2} c^{2} x^{2}}{3 e}-\frac {d c \left (3 e^{2} a +c \,d^{2}\right ) x}{6 e^{2}}-\frac {6 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}}{30 e^{3}}}{\left (e x +d \right )^{5}}\) | \(75\) |
parallelrisch | \(\frac {-10 c^{2} d^{2} x^{2} e^{4}-15 a c d \,e^{5} x -5 c^{2} d^{3} e^{3} x -6 a^{2} e^{6}-3 a c \,d^{2} e^{4}-c^{2} d^{4} e^{2}}{30 e^{5} \left (e x +d \right )^{5}}\) | \(78\) |
default | \(-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{5 e^{3} \left (e x +d \right )^{5}}-\frac {c^{2} d^{2}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {c d \left (e^{2} a -c \,d^{2}\right )}{2 e^{3} \left (e x +d \right )^{4}}\) | \(83\) |
norman | \(\frac {-\frac {d^{2} \left (6 a^{2} e^{8}+3 a c \,d^{2} e^{6}+c^{2} d^{4} e^{4}\right )}{30 e^{7}}-\frac {\left (2 a^{2} e^{8}+11 a c \,d^{2} e^{6}+7 c^{2} d^{4} e^{4}\right ) x^{2}}{10 e^{5}}-\frac {d \left (3 a c \,e^{6}+5 c^{2} d^{2} e^{4}\right ) x^{3}}{6 e^{4}}-\frac {d \left (12 a^{2} e^{8}+21 a c \,d^{2} e^{6}+7 c^{2} d^{4} e^{4}\right ) x}{30 e^{6}}-\frac {e \,c^{2} d^{2} x^{4}}{3}}{\left (e x +d \right )^{7}}\) | \(162\) |
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Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {10 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 3 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4} + 5 \, {\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
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Time = 3.59 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=\frac {- 6 a^{2} e^{4} - 3 a c d^{2} e^{2} - c^{2} d^{4} - 10 c^{2} d^{2} e^{2} x^{2} + x \left (- 15 a c d e^{3} - 5 c^{2} d^{3} e\right )}{30 d^{5} e^{3} + 150 d^{4} e^{4} x + 300 d^{3} e^{5} x^{2} + 300 d^{2} e^{6} x^{3} + 150 d e^{7} x^{4} + 30 e^{8} x^{5}} \]
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Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {10 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 3 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4} + 5 \, {\left (c^{2} d^{3} e + 3 \, a c d e^{3}\right )} x}{30 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {10 \, c^{2} d^{2} e^{2} x^{2} + 5 \, c^{2} d^{3} e x + 15 \, a c d e^{3} x + c^{2} d^{4} + 3 \, a c d^{2} e^{2} + 6 \, a^{2} e^{4}}{30 \, {\left (e x + d\right )}^{5} e^{3}} \]
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Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\frac {6\,a^2\,e^4+3\,a\,c\,d^2\,e^2+c^2\,d^4}{30\,e^3}+\frac {c^2\,d^2\,x^2}{3\,e}+\frac {c\,d\,x\,\left (c\,d^2+3\,a\,e^2\right )}{6\,e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]
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