Integrand size = 32, antiderivative size = 32 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{c^2 e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {657, 643} \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{c^2 e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Rule 643
Rule 657
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx}{c} \\ & = -\frac {1}{c^2 e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{c^2 e \sqrt {c (d+e x)^2}} \]
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Time = 2.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62
method | result | size |
risch | \(-\frac {1}{c^{2} \sqrt {c \left (e x +d \right )^{2}}\, e}\) | \(20\) |
pseudoelliptic | \(-\frac {1}{c^{2} \sqrt {c \left (e x +d \right )^{2}}\, e}\) | \(20\) |
gosper | \(-\frac {\left (e x +d \right )^{4}}{e \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(35\) |
default | \(-\frac {\left (e x +d \right )^{4}}{e \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(35\) |
trager | \(\frac {x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{c^{3} d \left (e x +d \right )^{2}}\) | \(38\) |
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Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c^{3} e^{3} x^{2} + 2 \, c^{3} d e^{2} x + c^{3} d^{2} e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (32) = 64\).
Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\begin {cases} - \frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{3} d^{2} e + 2 c^{3} d e^{2} x + c^{3} e^{3} x^{2}} & \text {for}\: e \neq 0 \\\frac {d^{3} x}{\left (c d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (30) = 60\).
Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.97 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {c^{2} d^{3} e^{4}}{\left (c e^{2}\right )^{\frac {9}{2}} {\left (x + \frac {d}{e}\right )}^{4}} - \frac {e x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c} + \frac {8 \, c d^{2} e^{3}}{3 \, \left (c e^{2}\right )^{\frac {7}{2}} {\left (x + \frac {d}{e}\right )}^{3}} - \frac {5 \, d^{2}}{3 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c e} - \frac {2 \, d e^{2}}{\left (c e^{2}\right )^{\frac {5}{2}} {\left (x + \frac {d}{e}\right )}^{2}} + \frac {d^{3}}{\left (c e^{2}\right )^{\frac {5}{2}} {\left (x + \frac {d}{e}\right )}^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{{\left (e x + d\right )} c^{\frac {5}{2}} e \mathrm {sgn}\left (e x + d\right )} \]
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Time = 10.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c^3\,e\,{\left (d+e\,x\right )}^2} \]
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