Integrand size = 33, antiderivative size = 26 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx=\frac {c d x}{e}+\left (a-\frac {c d^2}{e^2}\right ) \log (d+e x) \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {24, 45} \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx=\left (a-\frac {c d^2}{e^2}\right ) \log (d+e x)+\frac {c d x}{e} \]
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Rule 24
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {a e^3+c d e^2 x}{d+e x} \, dx}{e^2} \\ & = \frac {\int \left (c d e+\frac {-c d^2 e+a e^3}{d+e x}\right ) \, dx}{e^2} \\ & = \frac {c d x}{e}+\left (a-\frac {c d^2}{e^2}\right ) \log (d+e x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx=\frac {c d x}{e}+\frac {\left (-c d^2+a e^2\right ) \log (d+e x)}{e^2} \]
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Time = 2.38 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {c d x}{e}+\frac {\left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{2}}\) | \(31\) |
risch | \(\frac {c d x}{e}+\ln \left (e x +d \right ) a -\frac {\ln \left (e x +d \right ) c \,d^{2}}{e^{2}}\) | \(32\) |
parallelrisch | \(\frac {\ln \left (e x +d \right ) a \,e^{2}-\ln \left (e x +d \right ) c \,d^{2}+x c d e}{e^{2}}\) | \(34\) |
norman | \(\frac {c d \,x^{2}-\frac {d^{3} c}{e^{2}}}{e x +d}+\frac {\left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{2}}\) | \(48\) |
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Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx=\frac {c d e x - {\left (c d^{2} - a e^{2}\right )} \log \left (e x + d\right )}{e^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx=\frac {c d x}{e} + \frac {\left (a e^{2} - c d^{2}\right ) \log {\left (d + e x \right )}}{e^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx=\frac {c d x}{e} - \frac {{\left (c d^{2} - a e^{2}\right )} \log \left (e x + d\right )}{e^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.58 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx=c d e {\left (\frac {2 \, d \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{3}} + \frac {e x + d}{e^{3}} - \frac {d^{2}}{{\left (e x + d\right )} e^{3}}\right )} - \frac {a d}{e x + d} - \frac {{\left (c d^{2} + a e^{2}\right )} {\left (\frac {\log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e} - \frac {d}{{\left (e x + d\right )} e}\right )}}{e} \]
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Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (a\,e^2-c\,d^2\right )}{e^2}+\frac {c\,d\,x}{e} \]
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