Integrand size = 30, antiderivative size = 32 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 32, 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} \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Rule 643
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{c 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 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{c e \sqrt {c (d+e x)^2}} \]
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Time = 2.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62
method | result | size |
risch | \(-\frac {1}{c \sqrt {c \left (e x +d \right )^{2}}\, e}\) | \(20\) |
pseudoelliptic | \(-\frac {1}{c \sqrt {c \left (e x +d \right )^{2}}\, e}\) | \(20\) |
gosper | \(-\frac {\left (e x +d \right )^{2}}{e \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(35\) |
default | \(-\frac {\left (e x +d \right )^{2}}{e \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(35\) |
trager | \(\frac {x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{c^{2} d \left (e x +d \right )^{2}}\) | \(38\) |
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Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c^{2} e^{3} x^{2} + 2 \, c^{2} d e^{2} x + c^{2} d^{2} e} \]
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Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\begin {cases} - \frac {1}{c e \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}} & \text {for}\: e \neq 0 \\\frac {d x}{\left (c d^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{{\left (e x + d\right )} c^{\frac {3}{2}} e \mathrm {sgn}\left (e x + d\right )} \]
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Time = 10.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c^2\,e\,{\left (d+e\,x\right )}^2} \]
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