Integrand size = 19, antiderivative size = 87 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^2} \, dx=-\frac {d^3}{b^2 x}-\frac {(c d-b e)^3}{b^2 c^2 (b+c x)}-\frac {d^2 (2 c d-3 b e) \log (x)}{b^3}+\frac {(c d-b e)^2 (2 c d+b e) \log (b+c x)}{b^3 c^2} \]
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Time = 0.06 (sec) , antiderivative size = 87, 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} \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^2} \, dx=\frac {(c d-b e)^2 (b e+2 c d) \log (b+c x)}{b^3 c^2}-\frac {d^2 \log (x) (2 c d-3 b e)}{b^3}-\frac {(c d-b e)^3}{b^2 c^2 (b+c x)}-\frac {d^3}{b^2 x} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^3}{b^2 x^2}+\frac {d^2 (-2 c d+3 b e)}{b^3 x}-\frac {(-c d+b e)^3}{b^2 c (b+c x)^2}+\frac {(-c d+b e)^2 (2 c d+b e)}{b^3 c (b+c x)}\right ) \, dx \\ & = -\frac {d^3}{b^2 x}-\frac {(c d-b e)^3}{b^2 c^2 (b+c x)}-\frac {d^2 (2 c d-3 b e) \log (x)}{b^3}+\frac {(c d-b e)^2 (2 c d+b e) \log (b+c x)}{b^3 c^2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b d^3}{x}+\frac {b (-c d+b e)^3}{c^2 (b+c x)}+d^2 (-2 c d+3 b e) \log (x)+\frac {(c d-b e)^2 (2 c d+b e) \log (b+c x)}{c^2}}{b^3} \]
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Time = 1.90 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.39
method | result | size |
default | \(-\frac {d^{3}}{b^{2} x}+\frac {d^{2} \left (3 b e -2 c d \right ) \ln \left (x \right )}{b^{3}}+\frac {\left (b^{3} e^{3}-3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) \ln \left (c x +b \right )}{b^{3} c^{2}}-\frac {-b^{3} e^{3}+3 b^{2} d \,e^{2} c -3 b \,c^{2} d^{2} e +c^{3} d^{3}}{b^{2} c^{2} \left (c x +b \right )}\) | \(121\) |
norman | \(\frac {-\frac {d^{3}}{b}-\frac {\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right ) x^{2}}{b^{3} c}}{x \left (c x +b \right )}+\frac {d^{2} \left (3 b e -2 c d \right ) \ln \left (x \right )}{b^{3}}+\frac {\left (b^{3} e^{3}-3 b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) \ln \left (c x +b \right )}{b^{3} c^{2}}\) | \(126\) |
risch | \(\frac {\frac {\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right ) x}{b^{2} c^{2}}-\frac {d^{3}}{b}}{x \left (c x +b \right )}+\frac {\ln \left (-c x -b \right ) e^{3}}{c^{2}}-\frac {3 \ln \left (-c x -b \right ) d^{2} e}{b^{2}}+\frac {2 c \ln \left (-c x -b \right ) d^{3}}{b^{3}}+\frac {3 d^{2} \ln \left (x \right ) e}{b^{2}}-\frac {2 d^{3} \ln \left (x \right ) c}{b^{3}}\) | \(140\) |
parallelrisch | \(\frac {3 \ln \left (x \right ) x^{2} b \,c^{3} d^{2} e -2 \ln \left (x \right ) x^{2} c^{4} d^{3}+\ln \left (c x +b \right ) x^{2} b^{3} c \,e^{3}-3 \ln \left (c x +b \right ) x^{2} b \,c^{3} d^{2} e +2 \ln \left (c x +b \right ) x^{2} c^{4} d^{3}+3 \ln \left (x \right ) x \,b^{2} c^{2} d^{2} e -2 \ln \left (x \right ) x b \,c^{3} d^{3}+\ln \left (c x +b \right ) x \,b^{4} e^{3}-3 \ln \left (c x +b \right ) x \,b^{2} c^{2} d^{2} e +2 \ln \left (c x +b \right ) x b \,c^{3} d^{3}+x \,b^{4} e^{3}-3 x \,b^{3} c d \,e^{2}+3 x \,b^{2} c^{2} d^{2} e -2 x b \,c^{3} d^{3}-b^{2} c^{2} d^{3}}{b^{3} c^{2} x \left (c x +b \right )}\) | \(229\) |
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Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (87) = 174\).
Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.28 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^2} \, dx=-\frac {b^{2} c^{2} d^{3} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}\right )} x - {\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{3} c e^{3}\right )} x^{2} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{4} e^{3}\right )} x\right )} \log \left (c x + b\right ) + {\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e\right )} x^{2} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e\right )} x\right )} \log \left (x\right )}{b^{3} c^{3} x^{2} + b^{4} c^{2} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (78) = 156\).
Time = 0.76 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.87 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^2} \, dx=\frac {- b c^{2} d^{3} + x \left (b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - 2 c^{3} d^{3}\right )}{b^{3} c^{2} x + b^{2} c^{3} x^{2}} + \frac {d^{2} \cdot \left (3 b e - 2 c d\right ) \log {\left (x + \frac {- 3 b^{2} c d^{2} e + 2 b c^{2} d^{3} + b c d^{2} \cdot \left (3 b e - 2 c d\right )}{b^{3} e^{3} - 6 b c^{2} d^{2} e + 4 c^{3} d^{3}} \right )}}{b^{3}} + \frac {\left (b e - c d\right )^{2} \left (b e + 2 c d\right ) \log {\left (x + \frac {- 3 b^{2} c d^{2} e + 2 b c^{2} d^{3} + \frac {b \left (b e - c d\right )^{2} \left (b e + 2 c d\right )}{c}}{b^{3} e^{3} - 6 b c^{2} d^{2} e + 4 c^{3} d^{3}} \right )}}{b^{3} c^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.52 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^2} \, dx=-\frac {b c^{2} d^{3} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} x}{b^{2} c^{3} x^{2} + b^{3} c^{2} x} - \frac {{\left (2 \, c d^{3} - 3 \, b d^{2} e\right )} \log \left (x\right )}{b^{3}} + \frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{3} e^{3}\right )} \log \left (c x + b\right )}{b^{3} c^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.48 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^2} \, dx=-\frac {{\left (2 \, c d^{3} - 3 \, b d^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{3} e^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{2}} - \frac {b c^{2} d^{3} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} x}{{\left (c x + b\right )} b^{2} c^{2} x} \]
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Time = 9.66 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.36 \[ \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^2} \, dx=\ln \left (b+c\,x\right )\,\left (\frac {e^3}{c^2}+\frac {2\,c\,d^3}{b^3}-\frac {3\,d^2\,e}{b^2}\right )-\frac {\frac {d^3}{b}-\frac {x\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-2\,c^3\,d^3\right )}{b^2\,c^2}}{c\,x^2+b\,x}+\frac {d^2\,\ln \left (x\right )\,\left (3\,b\,e-2\,c\,d\right )}{b^3} \]
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