Integrand size = 19, antiderivative size = 64 \[ \int \frac {(d+e x)^3}{b x+c x^2} \, dx=\frac {e^2 (3 c d-b e) x}{c^2}+\frac {e^3 x^2}{2 c}+\frac {d^3 \log (x)}{b}-\frac {(c d-b e)^3 \log (b+c x)}{b c^3} \]
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Time = 0.04 (sec) , antiderivative size = 64, 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} \[ \int \frac {(d+e x)^3}{b x+c x^2} \, dx=-\frac {(c d-b e)^3 \log (b+c x)}{b c^3}+\frac {e^2 x (3 c d-b e)}{c^2}+\frac {d^3 \log (x)}{b}+\frac {e^3 x^2}{2 c} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^2 (3 c d-b e)}{c^2}+\frac {d^3}{b x}+\frac {e^3 x}{c}+\frac {(-c d+b e)^3}{b c^2 (b+c x)}\right ) \, dx \\ & = \frac {e^2 (3 c d-b e) x}{c^2}+\frac {e^3 x^2}{2 c}+\frac {d^3 \log (x)}{b}-\frac {(c d-b e)^3 \log (b+c x)}{b c^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^3}{b x+c x^2} \, dx=\frac {b c e^2 x (6 c d-2 b e+c e x)+2 c^3 d^3 \log (x)-2 (c d-b e)^3 \log (b+c x)}{2 b c^3} \]
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Time = 1.88 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.33
method | result | size |
default | \(-\frac {e^{2} \left (-\frac {1}{2} c e \,x^{2}+b e x -3 c d x \right )}{c^{2}}+\frac {d^{3} \ln \left (x \right )}{b}+\frac {\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \ln \left (c x +b \right )}{c^{3} b}\) | \(85\) |
norman | \(\frac {e^{3} x^{2}}{2 c}-\frac {e^{2} \left (b e -3 c d \right ) x}{c^{2}}+\frac {d^{3} \ln \left (x \right )}{b}+\frac {\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \ln \left (c x +b \right )}{c^{3} b}\) | \(88\) |
parallelrisch | \(\frac {e^{3} x^{2} c^{2} b +2 d^{3} \ln \left (x \right ) c^{3}+2 \ln \left (c x +b \right ) b^{3} e^{3}-6 \ln \left (c x +b \right ) b^{2} c d \,e^{2}+6 \ln \left (c x +b \right ) b \,c^{2} d^{2} e -2 \ln \left (c x +b \right ) c^{3} d^{3}-2 x \,b^{2} c \,e^{3}+6 x b \,c^{2} d \,e^{2}}{2 c^{3} b}\) | \(112\) |
risch | \(\frac {e^{3} x^{2}}{2 c}-\frac {e^{3} b x}{c^{2}}+\frac {3 d \,e^{2} x}{c}+\frac {d^{3} \ln \left (x \right )}{b}+\frac {b^{2} \ln \left (-c x -b \right ) e^{3}}{c^{3}}-\frac {3 b \ln \left (-c x -b \right ) d \,e^{2}}{c^{2}}+\frac {3 \ln \left (-c x -b \right ) d^{2} e}{c}-\frac {\ln \left (-c x -b \right ) d^{3}}{b}\) | \(115\) |
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Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.52 \[ \int \frac {(d+e x)^3}{b x+c x^2} \, dx=\frac {b c^{2} e^{3} x^{2} + 2 \, c^{3} d^{3} \log \left (x\right ) + 2 \, {\left (3 \, b c^{2} d e^{2} - b^{2} c e^{3}\right )} x - 2 \, {\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (c x + b\right )}{2 \, b c^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (54) = 108\).
Time = 0.70 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.75 \[ \int \frac {(d+e x)^3}{b x+c x^2} \, dx=x \left (- \frac {b e^{3}}{c^{2}} + \frac {3 d e^{2}}{c}\right ) + \frac {e^{3} x^{2}}{2 c} + \frac {d^{3} \log {\left (x \right )}}{b} + \frac {\left (b e - c d\right )^{3} \log {\left (x + \frac {- b c^{2} d^{3} + \frac {b \left (b e - c d\right )^{3}}{c}}{b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - 2 c^{3} d^{3}} \right )}}{b c^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x)^3}{b x+c x^2} \, dx=\frac {d^{3} \log \left (x\right )}{b} + \frac {c e^{3} x^{2} + 2 \, {\left (3 \, c d e^{2} - b e^{3}\right )} x}{2 \, c^{2}} - \frac {{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (c x + b\right )}{b c^{3}} \]
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Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x)^3}{b x+c x^2} \, dx=\frac {d^{3} \log \left ({\left | x \right |}\right )}{b} + \frac {c e^{3} x^{2} + 6 \, c d e^{2} x - 2 \, b e^{3} x}{2 \, c^{2}} - \frac {{\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{3}} \]
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Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^3}{b x+c x^2} \, dx=\frac {e^3\,x^2}{2\,c}-x\,\left (\frac {b\,e^3}{c^2}-\frac {3\,d\,e^2}{c}\right )+\frac {d^3\,\ln \left (x\right )}{b}+\frac {\ln \left (b+c\,x\right )\,{\left (b\,e-c\,d\right )}^3}{b\,c^3} \]
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