Integrand size = 30, antiderivative size = 34 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c e} \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {643} \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c e} \]
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Rule 643
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c e} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {\left (c (d+e x)^2\right )^{7/2}}{7 c e} \]
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Time = 2.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {c^{2} \left (e x +d \right )^{6} \sqrt {c \left (e x +d \right )^{2}}}{7 e}\) | \(27\) |
pseudoelliptic | \(\frac {c^{2} \left (e x +d \right )^{6} \sqrt {c \left (e x +d \right )^{2}}}{7 e}\) | \(27\) |
default | \(\frac {\left (e x +d \right )^{2} \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{7 e}\) | \(35\) |
gosper | \(\frac {x \left (e^{6} x^{6}+7 d \,e^{5} x^{5}+21 d^{2} e^{4} x^{4}+35 x^{3} d^{3} e^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{7 \left (e x +d \right )^{5}}\) | \(95\) |
trager | \(\frac {c^{2} x \left (e^{6} x^{6}+7 d \,e^{5} x^{5}+21 d^{2} e^{4} x^{4}+35 x^{3} d^{3} e^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{7 e x +7 d}\) | \(98\) |
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Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (30) = 60\).
Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.44 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {{\left (c^{2} e^{6} x^{7} + 7 \, c^{2} d e^{5} x^{6} + 21 \, c^{2} d^{2} e^{4} x^{5} + 35 \, c^{2} d^{3} e^{3} x^{4} + 35 \, c^{2} d^{4} e^{2} x^{3} + 21 \, c^{2} d^{5} e x^{2} + 7 \, c^{2} d^{6} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{7 \, {\left (e x + d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (29) = 58\).
Time = 0.36 (sec) , antiderivative size = 287, normalized size of antiderivative = 8.44 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {c^{2} d^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7 e} + \frac {6 c^{2} d^{5} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {15 c^{2} d^{4} e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {20 c^{2} d^{3} e^{2} x^{3} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {15 c^{2} d^{2} e^{3} x^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {6 c^{2} d e^{4} x^{5} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {c^{2} e^{5} x^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\d x \left (c d^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {7}{2}}}{7 \, c e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 4.38 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {1}{7} \, {\left (c^{2} e^{6} x^{7} \mathrm {sgn}\left (e x + d\right ) + 7 \, c^{2} d e^{5} x^{6} \mathrm {sgn}\left (e x + d\right ) + 21 \, c^{2} d^{2} e^{4} x^{5} \mathrm {sgn}\left (e x + d\right ) + 35 \, c^{2} d^{3} e^{3} x^{4} \mathrm {sgn}\left (e x + d\right ) + 35 \, c^{2} d^{4} e^{2} x^{3} \mathrm {sgn}\left (e x + d\right ) + 21 \, c^{2} d^{5} e x^{2} \mathrm {sgn}\left (e x + d\right ) + 7 \, c^{2} d^{6} x \mathrm {sgn}\left (e x + d\right ) + \frac {c^{2} d^{7} \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]
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Time = 9.97 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.56 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {{\left (c\,{\left (d+e\,x\right )}^2\right )}^{7/2}}{7\,c\,e} \]
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