Integrand size = 19, antiderivative size = 73 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^2} \, dx=-\frac {d^2}{b^2 x}-\frac {(c d-b e)^2}{b^2 c (b+c x)}-\frac {2 d (c d-b e) \log (x)}{b^3}+\frac {2 d (c d-b e) \log (b+c x)}{b^3} \]
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Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^2} \, dx=-\frac {2 d \log (x) (c d-b e)}{b^3}+\frac {2 d (c d-b e) \log (b+c x)}{b^3}-\frac {(c d-b e)^2}{b^2 c (b+c x)}-\frac {d^2}{b^2 x} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2}{b^2 x^2}+\frac {2 d (-c d+b e)}{b^3 x}+\frac {(-c d+b e)^2}{b^2 (b+c x)^2}-\frac {2 c d (-c d+b e)}{b^3 (b+c x)}\right ) \, dx \\ & = -\frac {d^2}{b^2 x}-\frac {(c d-b e)^2}{b^2 c (b+c x)}-\frac {2 d (c d-b e) \log (x)}{b^3}+\frac {2 d (c d-b e) \log (b+c x)}{b^3} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b d^2}{x}-\frac {b (c d-b e)^2}{c (b+c x)}+2 d (-c d+b e) \log (x)+2 d (c d-b e) \log (b+c x)}{b^3} \]
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Time = 1.89 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.16
method | result | size |
default | \(-\frac {d^{2}}{b^{2} x}+\frac {2 d \left (b e -c d \right ) \ln \left (x \right )}{b^{3}}-\frac {b^{2} e^{2}-2 b c d e +c^{2} d^{2}}{b^{2} c \left (c x +b \right )}-\frac {2 d \left (b e -c d \right ) \ln \left (c x +b \right )}{b^{3}}\) | \(85\) |
norman | \(\frac {\frac {\left (b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}\right ) x^{2}}{b^{3}}-\frac {d^{2}}{b}}{x \left (c x +b \right )}+\frac {2 d \left (b e -c d \right ) \ln \left (x \right )}{b^{3}}-\frac {2 d \left (b e -c d \right ) \ln \left (c x +b \right )}{b^{3}}\) | \(87\) |
risch | \(\frac {-\frac {\left (b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}\right ) x}{b^{2} c}-\frac {d^{2}}{b}}{x \left (c x +b \right )}+\frac {2 d \ln \left (-x \right ) e}{b^{2}}-\frac {2 d^{2} \ln \left (-x \right ) c}{b^{3}}-\frac {2 d \ln \left (c x +b \right ) e}{b^{2}}+\frac {2 d^{2} \ln \left (c x +b \right ) c}{b^{3}}\) | \(105\) |
parallelrisch | \(\frac {2 \ln \left (x \right ) x^{2} b \,c^{2} d e -2 \ln \left (x \right ) x^{2} c^{3} d^{2}-2 \ln \left (c x +b \right ) x^{2} b \,c^{2} d e +2 \ln \left (c x +b \right ) x^{2} c^{3} d^{2}+2 \ln \left (x \right ) x \,b^{2} c d e -2 \ln \left (x \right ) x b \,c^{2} d^{2}-2 \ln \left (c x +b \right ) x \,b^{2} c d e +2 \ln \left (c x +b \right ) x b \,c^{2} d^{2}-x \,b^{3} e^{2}+2 x \,b^{2} c d e -2 b \,c^{2} d^{2} x -b^{2} c \,d^{2}}{b^{3} c x \left (c x +b \right )}\) | \(170\) |
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Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (73) = 146\).
Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.04 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^2} \, dx=-\frac {b^{2} c d^{2} + {\left (2 \, b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} x - 2 \, {\left ({\left (c^{3} d^{2} - b c^{2} d e\right )} x^{2} + {\left (b c^{2} d^{2} - b^{2} c d e\right )} x\right )} \log \left (c x + b\right ) + 2 \, {\left ({\left (c^{3} d^{2} - b c^{2} d e\right )} x^{2} + {\left (b c^{2} d^{2} - b^{2} c d e\right )} x\right )} \log \left (x\right )}{b^{3} c^{2} x^{2} + b^{4} c x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (63) = 126\).
Time = 0.39 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.37 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^2} \, dx=\frac {- b c d^{2} + x \left (- b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{b^{3} c x + b^{2} c^{2} x^{2}} + \frac {2 d \left (b e - c d\right ) \log {\left (x + \frac {2 b^{2} d e - 2 b c d^{2} - 2 b d \left (b e - c d\right )}{4 b c d e - 4 c^{2} d^{2}} \right )}}{b^{3}} - \frac {2 d \left (b e - c d\right ) \log {\left (x + \frac {2 b^{2} d e - 2 b c d^{2} + 2 b d \left (b e - c d\right )}{4 b c d e - 4 c^{2} d^{2}} \right )}}{b^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^2} \, dx=-\frac {b c d^{2} + {\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} x}{b^{2} c^{2} x^{2} + b^{3} c x} + \frac {2 \, {\left (c d^{2} - b d e\right )} \log \left (c x + b\right )}{b^{3}} - \frac {2 \, {\left (c d^{2} - b d e\right )} \log \left (x\right )}{b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.36 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^2} \, dx=-\frac {2 \, {\left (c d^{2} - b d e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {2 \, {\left (c^{2} d^{2} - b c d e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} - \frac {2 \, c^{2} d^{2} x - 2 \, b c d e x + b^{2} e^{2} x + b c d^{2}}{{\left (c x^{2} + b x\right )} b^{2} c} \]
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Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^2}{\left (b x+c x^2\right )^2} \, dx=\frac {4\,d\,\mathrm {atanh}\left (\frac {2\,d\,\left (b\,e-c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (2\,c\,d^2-2\,b\,d\,e\right )}\right )\,\left (b\,e-c\,d\right )}{b^3}-\frac {\frac {d^2}{b}+\frac {x\,\left (b^2\,e^2-2\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^2\,c}}{c\,x^2+b\,x} \]
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