Integrand size = 35, antiderivative size = 39 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{\sqrt {d+e x}}+\frac {2 c d \sqrt {d+e x}}{e^2} \]
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Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {24, 45} \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {2 c d \sqrt {d+e x}}{e^2}-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{\sqrt {d+e x}} \]
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Rule 24
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {a e^3+c d e^2 x}{(d+e x)^{3/2}} \, dx}{e^2} \\ & = \frac {\int \left (\frac {-c d^2 e+a e^3}{(d+e x)^{3/2}}+\frac {c d e}{\sqrt {d+e x}}\right ) \, dx}{e^2} \\ & = -\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{\sqrt {d+e x}}+\frac {2 c d \sqrt {d+e x}}{e^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {-2 a e^2+2 c d (2 d+e x)}{e^2 \sqrt {d+e x}} \]
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Time = 2.56 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79
method | result | size |
gosper | \(-\frac {2 \left (-x c d e +e^{2} a -2 c \,d^{2}\right )}{\sqrt {e x +d}\, e^{2}}\) | \(31\) |
trager | \(-\frac {2 \left (-x c d e +e^{2} a -2 c \,d^{2}\right )}{\sqrt {e x +d}\, e^{2}}\) | \(31\) |
pseudoelliptic | \(-\frac {2 \left (-x c d e +e^{2} a -2 c \,d^{2}\right )}{\sqrt {e x +d}\, e^{2}}\) | \(31\) |
derivativedivides | \(\frac {2 c d \sqrt {e x +d}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{2}}\) | \(38\) |
default | \(\frac {2 c d \sqrt {e x +d}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{2}}\) | \(38\) |
risch | \(\frac {2 c d \sqrt {e x +d}}{e^{2}}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{e^{2} \sqrt {e x +d}}\) | \(40\) |
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Time = 0.38 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} \sqrt {e x + d}}{e^{3} x + d e^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.49 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\begin {cases} - \frac {2 a}{\sqrt {d + e x}} + \frac {4 c d^{2}}{e^{2} \sqrt {d + e x}} + \frac {2 c d x}{e \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c x^{2}}{2 \sqrt {d}} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.08 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {\sqrt {e x + d} c d}{e} + \frac {c d^{2} - a e^{2}}{\sqrt {e x + d} e}\right )}}{e} \]
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, \sqrt {e x + d} c d}{e^{2}} + \frac {2 \, {\left (c d^{2} - a e^{2}\right )}}{\sqrt {e x + d} e^{2}} \]
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Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx=\frac {4\,c\,d^2+2\,c\,x\,d\,e-2\,a\,e^2}{e^2\,\sqrt {d+e\,x}} \]
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