Integrand size = 32, antiderivative size = 39 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e} \]
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Time = 0.02 (sec) , antiderivative size = 39, 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} \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e} \]
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Rule 623
Rule 656
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx}{c^3} \\ & = \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 c^3 e} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {x (d+e x) (2 d+e x)}{2 c^2 \sqrt {c (d+e x)^2}} \]
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Time = 3.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03
method | result | size |
gosper | \(\frac {x \left (e x +2 d \right ) \left (e x +d \right )^{5}}{2 \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(40\) |
default | \(\frac {x \left (e x +2 d \right ) \left (e x +d \right )^{5}}{2 \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(40\) |
trager | \(\frac {x \left (e x +2 d \right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{2 c^{3} \left (e x +d \right )}\) | \(43\) |
risch | \(\frac {\left (e x +d \right ) e \,x^{2}}{2 c^{2} \sqrt {c \left (e x +d \right )^{2}}}+\frac {\left (e x +d \right ) d x}{c^{2} \sqrt {c \left (e x +d \right )^{2}}}\) | \(49\) |
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Time = 0.36 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x^{2} + 2 \, d x\right )}}{2 \, {\left (c^{3} e x + c^{3} d\right )}} \]
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Time = 4.71 (sec) , antiderivative size = 224, normalized size of antiderivative = 5.74 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {\begin {cases} \left (\frac {d}{2 c e} + \frac {x}{2 c}\right ) \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} & \text {for}\: c e^{2} \neq 0 \\\frac {2 d^{2} \sqrt {c d^{2} + 2 c d e x} + \frac {2 \left (- c d^{2} \sqrt {c d^{2} + 2 c d e x} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3}\right )}{c} + \frac {c^{2} d^{4} \sqrt {c d^{2} + 2 c d e x} - \frac {2 c d^{2} \left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {5}{2}}}{5}}{2 c^{2} d^{2}}}{2 c d e} & \text {for}\: c d e \neq 0 \\\frac {d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}}{\sqrt {c d^{2}}} & \text {otherwise} \end {cases}}{c^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (35) = 70\).
Time = 0.21 (sec) , antiderivative size = 232, normalized size of antiderivative = 5.95 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {e^{4} x^{5}}{2 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c} + \frac {5 \, d e^{3} x^{4}}{2 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c} - \frac {25 \, c^{2} d^{6} e^{4}}{4 \, \left (c e^{2}\right )^{\frac {9}{2}} {\left (x + \frac {d}{e}\right )}^{4}} - \frac {10 \, d^{3} e x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c} + \frac {50 \, c d^{5} e^{3}}{3 \, \left (c e^{2}\right )^{\frac {7}{2}} {\left (x + \frac {d}{e}\right )}^{3}} - \frac {26 \, d^{5}}{3 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c e} - \frac {25 \, d^{4} e^{2}}{2 \, \left (c e^{2}\right )^{\frac {5}{2}} {\left (x + \frac {d}{e}\right )}^{2}} + \frac {25 \, d^{6}}{4 \, \left (c e^{2}\right )^{\frac {5}{2}} {\left (x + \frac {d}{e}\right )}^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.59 \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {e x^{2} + 2 \, d x}{2 \, c^{\frac {5}{2}} \mathrm {sgn}\left (e x + d\right )} \]
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Timed out. \[ \int \frac {(d+e x)^6}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^6}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}} \,d x \]
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