Integrand size = 32, antiderivative size = 42 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {(d+e x) \log (d+e x)}{c^2 e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {656, 622, 31} \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {(d+e x) \log (d+e x)}{c^2 e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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Rule 31
Rule 622
Rule 656
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx}{c^2} \\ & = \frac {\left (c d e+c e^2 x\right ) \int \frac {1}{c d e+c e^2 x} \, dx}{c^2 \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ & = \frac {(d+e x) \log (d+e x)}{c^2 e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {(d+e x) \log (d+e x)}{c^2 e \sqrt {c (d+e x)^2}} \]
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Time = 2.70 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {\left (e x +d \right ) \ln \left (e x +d \right )}{c^{2} \sqrt {c \left (e x +d \right )^{2}}\, e}\) | \(30\) |
default | \(\frac {\left (e x +d \right )^{5} \ln \left (e x +d \right )}{\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}} e}\) | \(40\) |
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Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{c^{3} e^{2} x + c^{3} d e} \]
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Time = 3.54 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.69 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {\left (\frac {d}{e} + x\right ) \log {\left (\frac {d}{e} + x \right )}}{c^{2} \sqrt {c e^{2} \left (\frac {d}{e} + x\right )^{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (40) = 80\).
Time = 0.24 (sec) , antiderivative size = 449, normalized size of antiderivative = 10.69 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {1}{12} \, e^{4} {\left (\frac {48 \, \sqrt {c} d e^{3} x^{3} + 108 \, \sqrt {c} d^{2} e^{2} x^{2} + 88 \, \sqrt {c} d^{3} e x + 25 \, \sqrt {c} d^{4}}{c^{3} e^{9} x^{4} + 4 \, c^{3} d e^{8} x^{3} + 6 \, c^{3} d^{2} e^{7} x^{2} + 4 \, c^{3} d^{3} e^{6} x + c^{3} d^{4} e^{5}} + \frac {12 \, \log \left (e x + d\right )}{c^{\frac {5}{2}} e^{5}}\right )} - \frac {1}{3} \, d e^{3} {\left (\frac {3 \, c^{2} d^{3} e}{\left (c e^{2}\right )^{\frac {9}{2}} {\left (x + \frac {d}{e}\right )}^{4}} + \frac {12 \, x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c e^{2}} - \frac {8 \, c d^{2}}{\left (c e^{2}\right )^{\frac {7}{2}} {\left (x + \frac {d}{e}\right )}^{3}} + \frac {8 \, d^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c e^{4}} + \frac {6 \, d}{\left (c e^{2}\right )^{\frac {5}{2}} e {\left (x + \frac {d}{e}\right )}^{2}} - \frac {6 \, d^{3}}{\left (c e^{2}\right )^{\frac {5}{2}} e^{3} {\left (x + \frac {d}{e}\right )}^{4}}\right )} - \frac {1}{2} \, d^{2} e^{2} {\left (\frac {3 \, c^{2} d^{2} e^{2}}{\left (c e^{2}\right )^{\frac {9}{2}} {\left (x + \frac {d}{e}\right )}^{4}} - \frac {8 \, c d e}{\left (c e^{2}\right )^{\frac {7}{2}} {\left (x + \frac {d}{e}\right )}^{3}} + \frac {6}{\left (c e^{2}\right )^{\frac {5}{2}} {\left (x + \frac {d}{e}\right )}^{2}}\right )} - \frac {1}{3} \, d^{3} e {\left (\frac {4}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c e^{2}} - \frac {3 \, d}{\left (c e^{2}\right )^{\frac {5}{2}} e {\left (x + \frac {d}{e}\right )}^{4}}\right )} - \frac {d^{4}}{4 \, \left (c e^{2}\right )^{\frac {5}{2}} {\left (x + \frac {d}{e}\right )}^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.52 \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\frac {\log \left ({\left | e x + d \right |}\right )}{c^{\frac {5}{2}} e \mathrm {sgn}\left (e x + d\right )} \]
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Timed out. \[ \int \frac {(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}} \,d x \]
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