Integrand size = 24, antiderivative size = 36 \[ \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e} \]
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Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {623} \[ \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e} \]
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Rule 623
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 e} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69 \[ \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {(d+e x) \left (c (d+e x)^2\right )^{5/2}}{6 e} \]
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Time = 2.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {c^{2} \left (e x +d \right )^{5} \sqrt {c \left (e x +d \right )^{2}}}{6 e}\) | \(27\) |
default | \(\frac {\left (e x +d \right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{6 e}\) | \(33\) |
gosper | \(\frac {x \left (e^{5} x^{5}+6 x^{4} d \,e^{4}+15 d^{2} e^{3} x^{3}+20 d^{3} e^{2} x^{2}+15 d^{4} e x +6 d^{5}\right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{6 \left (e x +d \right )^{5}}\) | \(84\) |
trager | \(\frac {c^{2} x \left (e^{5} x^{5}+6 x^{4} d \,e^{4}+15 d^{2} e^{3} x^{3}+20 d^{3} e^{2} x^{2}+15 d^{4} e x +6 d^{5}\right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{6 e x +6 d}\) | \(87\) |
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (32) = 64\).
Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.86 \[ \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {{\left (c^{2} e^{5} x^{6} + 6 \, c^{2} d e^{4} x^{5} + 15 \, c^{2} d^{2} e^{3} x^{4} + 20 \, c^{2} d^{3} e^{2} x^{3} + 15 \, c^{2} d^{4} e x^{2} + 6 \, c^{2} d^{5} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{6 \, {\left (e x + d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (34) = 68\).
Time = 1.48 (sec) , antiderivative size = 828, normalized size of antiderivative = 23.00 \[ \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=c^{2} d^{4} \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 ) + 4 c^{2} d^{3} 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 ) + 6 c^{2} d^{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 ) + 4 c^{2} d e^{3} \left (\begin {cases} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} \left (- \frac {d^{4}}{20 e^{4}} + \frac {d^{3} x}{20 e^{3}} - \frac {d^{2} x^{2}}{20 e^{2}} + \frac {d x^{3}}{20 e} + \frac {x^{4}}{5}\right ) & \text {for}\: c e^{2} \neq 0 \\\frac {- \frac {c^{3} d^{6} \left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} d^{4} \left (c d^{2} + 2 c d e x\right )^{\frac {5}{2}}}{5} - \frac {3 c d^{2} \left (c d^{2} + 2 c d e x\right )^{\frac {7}{2}}}{7} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {9}{2}}}{9}}{8 c^{4} d^{4} e^{4}} & \text {for}\: c d e \neq 0 \\\frac {x^{4} \sqrt {c d^{2}}}{4} & \text {otherwise} \end {cases}\right ) + c^{2} e^{4} \left (\begin {cases} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} \left (\frac {d^{5}}{30 e^{5}} - \frac {d^{4} x}{30 e^{4}} + \frac {d^{3} x^{2}}{30 e^{3}} - \frac {d^{2} x^{3}}{30 e^{2}} + \frac {d x^{4}}{30 e} + \frac {x^{5}}{6}\right ) & \text {for}\: c e^{2} \neq 0 \\\frac {\frac {c^{4} d^{8} \left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} d^{6} \left (c d^{2} + 2 c d e x\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} d^{4} \left (c d^{2} + 2 c d e x\right )^{\frac {7}{2}}}{7} - \frac {4 c d^{2} \left (c d^{2} + 2 c d e x\right )^{\frac {9}{2}}}{9} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {11}{2}}}{11}}{16 c^{5} d^{5} e^{5}} & \text {for}\: c d e \neq 0 \\\frac {x^{5} \sqrt {c d^{2}}}{5} & \text {otherwise} \end {cases}\right ) \]
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Exception generated. \[ \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (32) = 64\).
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.72 \[ \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (3 \, {\left (e x^{2} + 2 \, d x\right )} c^{2} d^{4} \mathrm {sgn}\left (e x + d\right ) + \frac {c^{2} d^{6} \mathrm {sgn}\left (e x + d\right )}{e} + 3 \, {\left (e x^{2} + 2 \, d x\right )}^{2} c^{2} d^{2} e \mathrm {sgn}\left (e x + d\right ) + {\left (e x^{2} + 2 \, d x\right )}^{3} c^{2} e^{2} \mathrm {sgn}\left (e x + d\right )\right )} \sqrt {c} \]
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Time = 9.69 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {\left (x\,e^2+d\,e\right )\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{6\,e^2} \]
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