Integrand size = 30, antiderivative size = 24 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=-\frac {(d+e x)^{-3+m}}{c^2 e (3-m)} \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=-\frac {(d+e x)^{m-3}}{c^2 e (3-m)} \]
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Rule 12
Rule 27
Rule 32
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^{-4+m}}{c^2} \, dx \\ & = \frac {\int (d+e x)^{-4+m} \, dx}{c^2} \\ & = -\frac {(d+e x)^{-3+m}}{c^2 e (3-m)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {(d+e x)^{-3+m}}{c^2 e (-3+m)} \]
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Time = 3.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {\left (e x +d \right )^{m}}{c^{2} e \left (-3+m \right ) \left (e x +d \right )^{3}}\) | \(27\) |
parallelrisch | \(\frac {\left (e x +d \right )^{m}}{c^{2} e \left (-3+m \right ) \left (e x +d \right )^{3}}\) | \(27\) |
norman | \(\frac {{\mathrm e}^{m \ln \left (e x +d \right )}}{c^{2} e \left (-3+m \right ) \left (e x +d \right )^{3}}\) | \(29\) |
gosper | \(\frac {\left (e x +d \right )^{-1+m}}{c^{2} e \left (-3+m \right ) \left (x^{2} e^{2}+2 d e x +d^{2}\right )}\) | \(40\) |
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (21) = 42\).
Time = 0.39 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.17 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {{\left (e x + d\right )}^{m}}{c^{2} d^{3} e m - 3 \, c^{2} d^{3} e + {\left (c^{2} e^{4} m - 3 \, c^{2} e^{4}\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} m - 3 \, c^{2} d e^{3}\right )} x^{2} + 3 \, {\left (c^{2} d^{2} e^{2} m - 3 \, c^{2} d^{2} e^{2}\right )} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (17) = 34\).
Time = 0.80 (sec) , antiderivative size = 136, normalized size of antiderivative = 5.67 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\begin {cases} \frac {x}{c^{2} d} & \text {for}\: e = 0 \wedge m = 3 \\\frac {d^{m} x}{c^{2} d^{4}} & \text {for}\: e = 0 \\\frac {\log {\left (\frac {d}{e} + x \right )}}{c^{2} e} & \text {for}\: m = 3 \\\frac {\left (d + e x\right )^{m}}{c^{2} d^{3} e m - 3 c^{2} d^{3} e + 3 c^{2} d^{2} e^{2} m x - 9 c^{2} d^{2} e^{2} x + 3 c^{2} d e^{3} m x^{2} - 9 c^{2} d e^{3} x^{2} + c^{2} e^{4} m x^{3} - 3 c^{2} e^{4} x^{3}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (21) = 42\).
Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {{\left (e x + d\right )}^{m}}{c^{2} e^{4} {\left (m - 3\right )} x^{3} + 3 \, c^{2} d e^{3} {\left (m - 3\right )} x^{2} + 3 \, c^{2} d^{2} e^{2} {\left (m - 3\right )} x + c^{2} d^{3} e {\left (m - 3\right )}} \]
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\[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{2}} \,d x } \]
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Time = 9.71 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {{\left (d+e\,x\right )}^m}{c^2\,e^4\,\left (m-3\right )\,\left (x^3+\frac {d^3}{e^3}+\frac {3\,d\,x^2}{e}+\frac {3\,d^2\,x}{e^2}\right )} \]
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