Integrand size = 32, antiderivative size = 37 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e} \]
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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} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e} \]
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Rule 623
Rule 656
Rubi steps \begin{align*} \text {integral}& = c \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx \\ & = \frac {c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {c (d+e x) \left (c (d+e x)^2\right )^{3/2}}{4 e} \]
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Time = 2.90 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {c^{2} \left (e x +d \right )^{3} \sqrt {c \left (e x +d \right )^{2}}}{4 e}\) | \(27\) |
default | \(\frac {\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{4 \left (e x +d \right ) e}\) | \(35\) |
gosper | \(\frac {x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{4 \left (e x +d \right )^{5}}\) | \(62\) |
trager | \(\frac {c^{2} x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{4 e x +4 d}\) | \(65\) |
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (33) = 66\).
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.03 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {{\left (c^{2} e^{3} x^{4} + 4 \, c^{2} d e^{2} x^{3} + 6 \, c^{2} d^{2} e x^{2} + 4 \, c^{2} d^{3} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \, {\left (e x + d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (34) = 68\).
Time = 1.68 (sec) , antiderivative size = 371, normalized size of antiderivative = 10.03 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=c^{2} d^{2} \left (\begin {cases} \left (\frac {d}{2 e} + \frac {x}{2}\right ) \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} & \text {for}\: c e^{2} \neq 0 \\\frac {\left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3 c d e} & \text {for}\: c d e \neq 0 \\x \sqrt {c d^{2}} & \text {otherwise} \end {cases}\right ) + 2 c^{2} d e \left (\begin {cases} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} \left (- \frac {d^{2}}{6 e^{2}} + \frac {d x}{6 e} + \frac {x^{2}}{3}\right ) & \text {for}\: c e^{2} \neq 0 \\\frac {- \frac {c d^{2} \left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {5}{2}}}{5}}{2 c^{2} d^{2} e^{2}} & \text {for}\: c d e \neq 0 \\\frac {x^{2} \sqrt {c d^{2}}}{2} & \text {otherwise} \end {cases}\right ) + c^{2} e^{2} \left (\begin {cases} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} \left (\frac {d^{3}}{12 e^{3}} - \frac {d^{2} x}{12 e^{2}} + \frac {d x^{2}}{12 e} + \frac {x^{3}}{4}\right ) & \text {for}\: c e^{2} \neq 0 \\\frac {\frac {c^{2} d^{4} \left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3} - \frac {2 c d^{2} \left (c d^{2} + 2 c d e x\right )^{\frac {5}{2}}}{5} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {7}{2}}}{7}}{4 c^{3} d^{3} e^{3}} & \text {for}\: c d e \neq 0 \\\frac {x^{3} \sqrt {c d^{2}}}{3} & \text {otherwise} \end {cases}\right ) \]
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none
Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {5}{2}}}{4 \, {\left (e^{2} x + d e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (33) = 66\).
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.41 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {1}{4} \, {\left (c^{2} e^{3} x^{4} \mathrm {sgn}\left (e x + d\right ) + 4 \, c^{2} d e^{2} x^{3} \mathrm {sgn}\left (e x + d\right ) + 6 \, c^{2} d^{2} e x^{2} \mathrm {sgn}\left (e x + d\right ) + 4 \, c^{2} d^{3} x \mathrm {sgn}\left (e x + d\right ) + \frac {c^{2} d^{4} \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]
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Timed out. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \]
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