Integrand size = 35, antiderivative size = 48 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c d^2-a e^2}{c^2 d^2 (a e+c d x)}+\frac {e \log (a e+c d x)}{c^2 d^2} \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e \log (a e+c d x)}{c^2 d^2}-\frac {c d^2-a e^2}{c^2 d^2 (a e+c d x)} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {d+e x}{(a e+c d x)^2} \, dx \\ & = \int \left (\frac {c d^2-a e^2}{c d (a e+c d x)^2}+\frac {e}{c d (a e+c d x)}\right ) \, dx \\ & = -\frac {c d^2-a e^2}{c^2 d^2 (a e+c d x)}+\frac {e \log (a e+c d x)}{c^2 d^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {-c d^2+a e^2}{c^2 d^2 (a e+c d x)}+\frac {e \log (a e+c d x)}{c^2 d^2} \]
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Time = 2.41 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {e \ln \left (c d x +a e \right )}{c^{2} d^{2}}-\frac {-e^{2} a +c \,d^{2}}{c^{2} d^{2} \left (c d x +a e \right )}\) | \(49\) |
risch | \(\frac {e^{2} a}{c^{2} d^{2} \left (c d x +a e \right )}-\frac {1}{c \left (c d x +a e \right )}+\frac {e \ln \left (c d x +a e \right )}{c^{2} d^{2}}\) | \(55\) |
parallelrisch | \(\frac {\ln \left (c d x +a e \right ) x c d e +\ln \left (c d x +a e \right ) a \,e^{2}+e^{2} a -c \,d^{2}}{c^{2} d^{2} \left (c d x +a e \right )}\) | \(58\) |
norman | \(\frac {\frac {e^{2} a -c \,d^{2}}{c^{2} d}+\frac {\left (e^{4} a -d^{2} e^{2} c \right ) x}{c^{2} d^{2} e}}{\left (c d x +a e \right ) \left (e x +d \right )}+\frac {e \ln \left (c d x +a e \right )}{c^{2} d^{2}}\) | \(83\) |
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none
Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c d^{2} - a e^{2} - {\left (c d e x + a e^{2}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3} x + a c^{2} d^{2} e} \]
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Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {a e^{2} - c d^{2}}{a c^{2} d^{2} e + c^{3} d^{3} x} + \frac {e \log {\left (a e + c d x \right )}}{c^{2} d^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {c d^{2} - a e^{2}}{c^{3} d^{3} x + a c^{2} d^{2} e} + \frac {e \log \left (c d x + a e\right )}{c^{2} d^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e \log \left ({\left | c d x + a e \right |}\right )}{c^{2} d^{2}} - \frac {c d^{2} - a e^{2}}{{\left (c d x + a e\right )} c^{2} d^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {a\,e^2-c\,d^2}{c^2\,d^2\,\left (a\,e+c\,d\,x\right )}+\frac {e\,\ln \left (a\,e+c\,d\,x\right )}{c^2\,d^2} \]
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