Integrand size = 32, antiderivative size = 37 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {c (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
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Time = 0.01 (sec) , antiderivative size = 37, 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 = {656, 623} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {c (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
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Rule 623
Rule 656
Rubi steps \begin{align*} \text {integral}& = c \int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx \\ & = \frac {c (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {c^2 x (d+e x) (2 d+e x)}{2 \sqrt {c (d+e x)^2}} \]
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Time = 2.69 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08
method | result | size |
gosper | \(\frac {x \left (e x +2 d \right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{2 \left (e x +d \right )^{3}}\) | \(40\) |
default | \(\frac {x \left (e x +2 d \right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{2 \left (e x +d \right )^{3}}\) | \(40\) |
trager | \(\frac {c x \left (e x +2 d \right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{2 e x +2 d}\) | \(41\) |
risch | \(\frac {c \sqrt {c \left (e x +d \right )^{2}}\, e \,x^{2}}{2 e x +2 d}+\frac {c \sqrt {c \left (e x +d \right )^{2}}\, d x}{e x +d}\) | \(49\) |
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Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (c e x^{2} + 2 \, c d x\right )}}{2 \, {\left (e x + d\right )}} \]
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Time = 1.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.97 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=c \left (\begin {cases} \left (\frac {d}{2 e} + \frac {x}{2}\right ) \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} & \text {for}\: c e^{2} \neq 0 \\\frac {\left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3 c d e} & \text {for}\: c d e \neq 0 \\x \sqrt {c d^{2}} & \text {otherwise} \end {cases}\right ) \]
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Exception generated. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {1}{2} \, {\left (c e x^{2} \mathrm {sgn}\left (e x + d\right ) + 2 \, c d x \mathrm {sgn}\left (e x + d\right ) + \frac {c d^{2} \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]
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Timed out. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,d x \]
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