Integrand size = 32, antiderivative size = 29 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {c \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]
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Time = 0.02 (sec) , antiderivative size = 29, 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} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {c \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \]
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Rule 643
Rule 657
Rubi steps \begin{align*} \text {integral}& = c^2 \int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx \\ & = \frac {c \sqrt {c d^2+2 c d e x+c e^2 x^2}}{e} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.62 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {c \sqrt {c (d+e x)^2}}{e} \]
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Time = 2.49 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59
method | result | size |
pseudoelliptic | \(\frac {c \sqrt {c \left (e x +d \right )^{2}}}{e}\) | \(17\) |
risch | \(\frac {c \sqrt {c \left (e x +d \right )^{2}}\, x}{e x +d}\) | \(22\) |
default | \(\frac {x \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{\left (e x +d \right )^{3}}\) | \(32\) |
trager | \(\frac {c x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{e x +d}\) | \(33\) |
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none
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c x}{e x + d} \]
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Time = 1.48 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=c \left (\begin {cases} \frac {x \sqrt {c d^{2}}}{d} & \text {for}\: e = 0 \\\frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text {otherwise} \end {cases}\right ) \]
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Exception generated. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx={\left (c x \mathrm {sgn}\left (e x + d\right ) + \frac {c d \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]
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Timed out. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^3} \,d x \]
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