Integrand size = 32, antiderivative size = 31 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {c^2 \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]
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Time = 0.02 (sec) , antiderivative size = 31, 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} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {c^2 \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]
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Rule 643
Rule 657
Rubi steps \begin{align*} \text {integral}& = c^3 \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx \\ & = \frac {c^2 \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {c^2 \sqrt {c (d+e x)^2}}{e} \]
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Time = 2.59 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(\frac {\sqrt {c \left (e x +d \right )^{2}}\, c^{2}}{e}\) | \(19\) |
risch | \(\frac {c^{2} \sqrt {c \left (e x +d \right )^{2}}\, x}{e x +d}\) | \(24\) |
default | \(\frac {x \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{\left (e x +d \right )^{5}}\) | \(32\) |
trager | \(\frac {c^{2} x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{e x +d}\) | \(35\) |
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none
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c^{2} x}{e x + d} \]
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Time = 2.77 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=c^{2} \left (\begin {cases} \frac {x \sqrt {c d^{2}}}{d} & \text {for}\: e = 0 \\\frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text {otherwise} \end {cases}\right ) \]
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Exception generated. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx={\left (c^{2} x \mathrm {sgn}\left (e x + d\right ) + \frac {c^{2} d \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]
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Timed out. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^5} \,d x \]
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