Integrand size = 30, antiderivative size = 34 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{3 c e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {643} \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{3 c e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Rule 643
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3 c e \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c (d+e x)^2}}{3 c^3 e (d+e x)^4} \]
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Time = 2.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-\frac {1}{3 c^{2} \left (e x +d \right )^{2} \sqrt {c \left (e x +d \right )^{2}}\, e}\) | \(27\) |
pseudoelliptic | \(-\frac {1}{3 c^{2} \left (e x +d \right )^{2} \sqrt {c \left (e x +d \right )^{2}}\, e}\) | \(27\) |
gosper | \(-\frac {\left (e x +d \right )^{2}}{3 e \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(35\) |
default | \(-\frac {\left (e x +d \right )^{2}}{3 e \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(35\) |
trager | \(\frac {\left (x^{2} e^{2}+3 d e x +3 d^{2}\right ) x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{3 c^{3} d^{3} \left (e x +d \right )^{4}}\) | \(57\) |
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (30) = 60\).
Time = 0.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.44 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \, {\left (c^{3} e^{5} x^{4} + 4 \, c^{3} d e^{4} x^{3} + 6 \, c^{3} d^{2} e^{3} x^{2} + 4 \, c^{3} d^{3} e^{2} x + c^{3} d^{4} e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (32) = 64\).
Time = 0.44 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.65 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\begin {cases} - \frac {1}{3 c^{2} d^{2} e \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} + 6 c^{2} d e^{2} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} + 3 c^{2} e^{3} x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {d x}{\left (c d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{3 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c e} \]
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{3 \, {\left (e x + d\right )}^{3} c^{\frac {5}{2}} e \mathrm {sgn}\left (e x + d\right )} \]
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Time = 9.73 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{3\,c^3\,e\,{\left (d+e\,x\right )}^4} \]
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