Integrand size = 33, antiderivative size = 33 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=-\frac {a-\frac {c d^2}{e^2}}{d+e x}+\frac {c d \log (d+e x)}{e^2} \]
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Time = 0.02 (sec) , antiderivative size = 33, 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} \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d \log (d+e x)}{e^2}-\frac {a-\frac {c d^2}{e^2}}{d+e x} \]
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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)^2} \, dx}{e^2} \\ & = \frac {\int \left (\frac {-c d^2 e+a e^3}{(d+e x)^2}+\frac {c d e}{d+e x}\right ) \, dx}{e^2} \\ & = -\frac {a-\frac {c d^2}{e^2}}{d+e x}+\frac {c d \log (d+e x)}{e^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d^2-a e^2}{e^2 (d+e x)}+\frac {c d \log (d+e x)}{e^2} \]
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Time = 2.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15
method | result | size |
default | \(-\frac {e^{2} a -c \,d^{2}}{e^{2} \left (e x +d \right )}+\frac {c d \ln \left (e x +d \right )}{e^{2}}\) | \(38\) |
risch | \(-\frac {a}{e x +d}+\frac {c \,d^{2}}{e^{2} \left (e x +d \right )}+\frac {c d \ln \left (e x +d \right )}{e^{2}}\) | \(39\) |
parallelrisch | \(\frac {\ln \left (e x +d \right ) x c d e +\ln \left (e x +d \right ) c \,d^{2}-e^{2} a +c \,d^{2}}{e^{2} \left (e x +d \right )}\) | \(46\) |
norman | \(\frac {\frac {\left (a d \,e^{2}-d^{3} c \right ) x^{2}}{d^{2}}+\frac {\left (a d \,e^{2}-d^{3} c \right ) x}{d e}}{\left (e x +d \right )^{2}}+\frac {c d \ln \left (e x +d \right )}{e^{2}}\) | \(64\) |
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Time = 0.35 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d^{2} - a e^{2} + {\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{e^{3} x + d e^{2}} \]
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Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d \log {\left (d + e x \right )}}{e^{2}} + \frac {- a e^{2} + c d^{2}}{d e^{2} + e^{3} x} \]
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Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d \log \left (e x + d\right )}{e^{2}} + \frac {c d^{2} - a e^{2}}{e^{3} x + d e^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c d \log \left ({\left | e x + d \right |}\right )}{e^{2}} + \frac {c d^{2} - a e^{2}}{{\left (e x + d\right )} e^{2}} \]
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Time = 9.81 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx=\frac {c\,d\,\ln \left (d+e\,x\right )}{e^2}-\frac {a\,e^2-c\,d^2}{e^2\,\left (d+e\,x\right )} \]
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