Integrand size = 28, antiderivative size = 11 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx=\frac {c \log (d+e x)}{e} \]
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Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {24, 21, 31} \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx=\frac {c \log (d+e x)}{e} \]
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Rule 21
Rule 24
Rule 31
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {c d e^2+c e^3 x}{(d+e x)^2} \, dx}{e^2} \\ & = c \int \frac {1}{d+e x} \, dx \\ & = \frac {c \log (d+e x)}{e} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx=\frac {c \log (d+e x)}{e} \]
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Time = 2.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {c \ln \left (e x +d \right )}{e}\) | \(12\) |
norman | \(\frac {c \ln \left (e x +d \right )}{e}\) | \(12\) |
risch | \(\frac {c \ln \left (e x +d \right )}{e}\) | \(12\) |
parallelrisch | \(\frac {c \ln \left (e x +d \right )}{e}\) | \(12\) |
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none
Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx=\frac {c \log \left (e x + d\right )}{e} \]
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Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx=\frac {c \log {\left (d + e x \right )}}{e} \]
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none
Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx=\frac {c \log \left (e x + d\right )}{e} \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx=\frac {c \log \left ({\left | e x + d \right |}\right )}{e} \]
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Time = 9.51 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^3} \, dx=\frac {c\,\ln \left (d+e\,x\right )}{e} \]
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