Integrand size = 30, antiderivative size = 17 \[ \int \frac {(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {(d+e x)^2}{2 c^3 e} \]
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Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 9} \[ \int \frac {(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {(d+e x)^2}{2 c^3 e} \]
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Rule 9
Rule 27
Rubi steps \begin{align*} \text {integral}& = \int \frac {d+e x}{c^3} \, dx \\ & = \frac {(d+e x)^2}{2 c^3 e} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {d x+\frac {e x^2}{2}}{c^3} \]
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Time = 2.91 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(\frac {x \left (e x +2 d \right )}{2 c^{3}}\) | \(14\) |
default | \(\frac {\frac {1}{2} e \,x^{2}+d x}{c^{3}}\) | \(15\) |
parallelrisch | \(\frac {e \,x^{2}+2 d x}{2 c^{3}}\) | \(16\) |
risch | \(\frac {e \,x^{2}}{2 c^{3}}+\frac {d x}{c^{3}}\) | \(17\) |
norman | \(\frac {-\frac {10 d^{7}}{e c}-\frac {49 d^{6} x}{c}+\frac {e^{6} x^{7}}{2 c}+\frac {7 d \,e^{5} x^{6}}{2 c}-\frac {35 e^{3} d^{3} x^{4}}{c}-\frac {175 d^{4} e^{2} x^{3}}{2 c}-\frac {189 d^{5} e \,x^{2}}{2 c}}{c^{2} \left (e x +d \right )^{5}}\) | \(96\) |
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none
Time = 0.36 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {e x^{2} + 2 \, d x}{2 \, c^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {d x}{c^{3}} + \frac {e x^{2}}{2 c^{3}} \]
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none
Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {e x^{2} + 2 \, d x}{2 \, c^{3}} \]
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none
Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {e x^{2} + 2 \, d x}{2 \, c^{3}} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx=\frac {x\,\left (2\,d+e\,x\right )}{2\,c^3} \]
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