Integrand size = 17, antiderivative size = 65 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^2} \, dx=-\frac {d}{b^2 x}-\frac {c d-b e}{b^2 (b+c x)}-\frac {(2 c d-b e) \log (x)}{b^3}+\frac {(2 c d-b e) \log (b+c x)}{b^3} \]
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Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {645} \[ \int \frac {d+e x}{\left (b x+c x^2\right )^2} \, dx=-\frac {\log (x) (2 c d-b e)}{b^3}+\frac {(2 c d-b e) \log (b+c x)}{b^3}-\frac {c d-b e}{b^2 (b+c x)}-\frac {d}{b^2 x} \]
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Rule 645
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d}{b^2 x^2}+\frac {-2 c d+b e}{b^3 x}-\frac {c (-c d+b e)}{b^2 (b+c x)^2}-\frac {c (-2 c d+b e)}{b^3 (b+c x)}\right ) \, dx \\ & = -\frac {d}{b^2 x}-\frac {c d-b e}{b^2 (b+c x)}-\frac {(2 c d-b e) \log (x)}{b^3}+\frac {(2 c d-b e) \log (b+c x)}{b^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b d}{x}+\frac {b (-c d+b e)}{b+c x}+(-2 c d+b e) \log (x)+(2 c d-b e) \log (b+c x)}{b^3} \]
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Time = 2.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {d}{b^{2} x}+\frac {\left (b e -2 c d \right ) \ln \left (x \right )}{b^{3}}-\frac {\left (b e -2 c d \right ) \ln \left (c x +b \right )}{b^{3}}+\frac {b e -c d}{b^{2} \left (c x +b \right )}\) | \(63\) |
norman | \(\frac {\frac {c \left (-b e +2 c d \right ) x^{2}}{b^{3}}-\frac {d}{b}}{x \left (c x +b \right )}+\frac {\left (b e -2 c d \right ) \ln \left (x \right )}{b^{3}}-\frac {\left (b e -2 c d \right ) \ln \left (c x +b \right )}{b^{3}}\) | \(70\) |
risch | \(\frac {\frac {\left (b e -2 c d \right ) x}{b^{2}}-\frac {d}{b}}{x \left (c x +b \right )}-\frac {\ln \left (c x +b \right ) e}{b^{2}}+\frac {2 \ln \left (c x +b \right ) c d}{b^{3}}+\frac {\ln \left (-x \right ) e}{b^{2}}-\frac {2 \ln \left (-x \right ) c d}{b^{3}}\) | \(78\) |
parallelrisch | \(\frac {\ln \left (x \right ) x^{2} b \,c^{2} e -2 \ln \left (x \right ) x^{2} c^{3} d -\ln \left (c x +b \right ) x^{2} b \,c^{2} e +2 \ln \left (c x +b \right ) x^{2} c^{3} d +\ln \left (x \right ) x \,b^{2} c e -2 \ln \left (x \right ) x b \,c^{2} d -\ln \left (c x +b \right ) x \,b^{2} c e +2 \ln \left (c x +b \right ) x b \,c^{2} d +b^{2} c e x -2 b \,c^{2} d x -b^{2} c d}{b^{3} c x \left (c x +b \right )}\) | \(141\) |
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Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.71 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^2} \, dx=-\frac {b^{2} d + {\left (2 \, b c d - b^{2} e\right )} x - {\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} + {\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (c x + b\right ) + {\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} + {\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (x\right )}{b^{3} c x^{2} + b^{4} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (54) = 108\).
Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.97 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^2} \, dx=\frac {- b d + x \left (b e - 2 c d\right )}{b^{3} x + b^{2} c x^{2}} + \frac {\left (b e - 2 c d\right ) \log {\left (x + \frac {b^{2} e - 2 b c d - b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} - \frac {\left (b e - 2 c d\right ) \log {\left (x + \frac {b^{2} e - 2 b c d + b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.06 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^2} \, dx=-\frac {b d + {\left (2 \, c d - b e\right )} x}{b^{2} c x^{2} + b^{3} x} + \frac {{\left (2 \, c d - b e\right )} \log \left (c x + b\right )}{b^{3}} - \frac {{\left (2 \, c d - b e\right )} \log \left (x\right )}{b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.14 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^2} \, dx=-\frac {{\left (2 \, c d - b e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac {2 \, c d x - b e x + b d}{{\left (c x^{2} + b x\right )} b^{2}} + \frac {{\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} \]
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Time = 9.56 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^2} \, dx=-\frac {\frac {d}{b}-\frac {x\,\left (b\,e-2\,c\,d\right )}{b^2}}{c\,x^2+b\,x}-\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )\,\left (b\,e-2\,c\,d\right )}{b^3} \]
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