Integrand size = 32, antiderivative size = 42 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {(d+e x) \log (d+e x)}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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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} \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {(d+e x) \log (d+e x)}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]
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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} \\ & = \frac {\left (c d e+c e^2 x\right ) \int \frac {1}{c d e+c e^2 x} \, dx}{c \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ & = \frac {(d+e x) \log (d+e x)}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {(d+e x) \log (d+e x)}{c e \sqrt {c (d+e x)^2}} \]
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Time = 2.59 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {\left (e x +d \right ) \ln \left (e x +d \right )}{c \sqrt {c \left (e x +d \right )^{2}}\, e}\) | \(30\) |
default | \(\frac {\left (e x +d \right )^{3} \ln \left (e x +d \right )}{\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}} e}\) | \(40\) |
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Time = 0.37 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{c^{2} e^{2} x + c^{2} d e} \]
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Time = 1.83 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.64 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\left (\frac {d}{e} + x\right ) \log {\left (\frac {d}{e} + x \right )}}{c \sqrt {c e^{2} \left (\frac {d}{e} + x\right )^{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (40) = 80\).
Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.93 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {3 \, c^{2} d^{2} e^{4}}{2 \, \left (c e^{2}\right )^{\frac {7}{2}} {\left (x + \frac {d}{e}\right )}^{2}} + \frac {2 \, c d e^{3} x}{\left (c e^{2}\right )^{\frac {5}{2}} {\left (x + \frac {d}{e}\right )}^{2}} + \frac {e^{2} \log \left (x + \frac {d}{e}\right )}{\left (c e^{2}\right )^{\frac {3}{2}}} - \frac {2 \, d}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} + \frac {d^{2}}{2 \, \left (c e^{2}\right )^{\frac {3}{2}} {\left (x + \frac {d}{e}\right )}^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.52 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\log \left ({\left | e x + d \right |}\right )}{c^{\frac {3}{2}} e \mathrm {sgn}\left (e x + d\right )} \]
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Timed out. \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}} \,d x \]
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