Integrand size = 19, antiderivative size = 193 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=-\frac {1}{2 b^3 d x^2}+\frac {3 c d+b e}{b^4 d^2 x}+\frac {c^3}{2 b^3 (c d-b e) (b+c x)^2}+\frac {c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}-\frac {c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]
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Time = 0.17 (sec) , antiderivative size = 193, 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} \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\frac {c^3 (3 c d-4 b e)}{b^4 (b+c x) (c d-b e)^2}+\frac {b e+3 c d}{b^4 d^2 x}+\frac {c^3}{2 b^3 (b+c x)^2 (c d-b e)}-\frac {1}{2 b^3 d x^2}+\frac {\log (x) \left (b^2 e^2+3 b c d e+6 c^2 d^2\right )}{b^5 d^3}-\frac {c^3 \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{b^3 d x^3}+\frac {-3 c d-b e}{b^4 d^2 x^2}+\frac {6 c^2 d^2+3 b c d e+b^2 e^2}{b^5 d^3 x}+\frac {c^4}{b^3 (-c d+b e) (b+c x)^3}+\frac {c^4 (-3 c d+4 b e)}{b^4 (-c d+b e)^2 (b+c x)^2}+\frac {c^4 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right )}{b^5 (-c d+b e)^3 (b+c x)}+\frac {e^6}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx \\ & = -\frac {1}{2 b^3 d x^2}+\frac {3 c d+b e}{b^4 d^2 x}+\frac {c^3}{2 b^3 (c d-b e) (b+c x)^2}+\frac {c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}-\frac {c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=-\frac {1}{2 b^3 d x^2}+\frac {3 c d+b e}{b^4 d^2 x}-\frac {c^3}{2 b^3 (-c d+b e) (b+c x)^2}+\frac {c^3 (3 c d-4 b e)}{b^4 (c d-b e)^2 (b+c x)}+\frac {\left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^3}+\frac {c^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (b+c x)}{b^5 (-c d+b e)^3}+\frac {e^5 \log (d+e x)}{d^3 (c d-b e)^3} \]
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Time = 1.95 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {1}{2 b^{3} d \,x^{2}}-\frac {-b e -3 c d}{b^{4} d^{2} x}+\frac {\left (b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5} d^{3}}-\frac {c^{3}}{2 \left (b e -c d \right ) b^{3} \left (c x +b \right )^{2}}-\frac {c^{3} \left (4 b e -3 c d \right )}{\left (b e -c d \right )^{2} b^{4} \left (c x +b \right )}+\frac {c^{3} \left (10 b^{2} e^{2}-15 b c d e +6 c^{2} d^{2}\right ) \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{5}}-\frac {e^{5} \ln \left (e x +d \right )}{d^{3} \left (b e -c d \right )^{3}}\) | \(193\) |
norman | \(\frac {\frac {\left (b e +2 c d \right ) x}{b^{2} d^{2}}+\frac {\left (-3 b^{3} c \,e^{3}-2 b^{2} c^{2} d \,e^{2}+18 b \,c^{3} d^{2} e -12 c^{4} d^{3}\right ) c \,x^{3}}{d^{2} b^{4} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {1}{2 b d}+\frac {c^{2} \left (-4 b^{3} c \,e^{3}-3 b^{2} c^{2} d \,e^{2}+27 b \,c^{3} d^{2} e -18 c^{4} d^{3}\right ) x^{4}}{2 d^{2} b^{5} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}}{x^{2} \left (c x +b \right )^{2}}+\frac {\left (b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5} d^{3}}+\frac {c^{3} \left (10 b^{2} e^{2}-15 b c d e +6 c^{2} d^{2}\right ) \ln \left (c x +b \right )}{b^{5} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {e^{5} \ln \left (e x +d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\) | \(346\) |
risch | \(\frac {\frac {c^{2} \left (b^{3} e^{3}+b^{2} d \,e^{2} c -9 b \,c^{2} d^{2} e +6 c^{3} d^{3}\right ) x^{3}}{b^{4} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {c \left (4 b^{3} e^{3}+3 b^{2} d \,e^{2} c -27 b \,c^{2} d^{2} e +18 c^{3} d^{3}\right ) x^{2}}{2 b^{3} d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (b e +2 c d \right ) x}{b^{2} d^{2}}-\frac {1}{2 b d}}{x^{2} \left (c x +b \right )^{2}}-\frac {e^{5} \ln \left (-e x -d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {\ln \left (-x \right ) e^{2}}{b^{3} d^{3}}+\frac {3 \ln \left (-x \right ) c e}{b^{4} d^{2}}+\frac {6 \ln \left (-x \right ) c^{2}}{b^{5} d}+\frac {10 c^{3} \ln \left (c x +b \right ) e^{2}}{b^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {15 c^{4} \ln \left (c x +b \right ) d e}{b^{4} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {6 c^{5} \ln \left (c x +b \right ) d^{2}}{b^{5} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\) | \(443\) |
parallelrisch | \(\frac {-45 x^{4} b \,c^{6} d^{4} e +2 x^{3} b^{4} c^{3} d^{2} e^{3}-2 x \,b^{6} c \,d^{2} e^{3}-6 x \,b^{5} c^{2} d^{3} e^{2}+2 \ln \left (x \right ) x^{4} b^{5} c^{2} e^{5}-2 \ln \left (e x +d \right ) x^{4} b^{5} c^{2} e^{5}+4 \ln \left (x \right ) x^{3} b^{6} c \,e^{5}-4 \ln \left (e x +d \right ) x^{3} b^{6} c \,e^{5}-20 \ln \left (x \right ) x^{4} b^{2} c^{5} d^{3} e^{2}+30 \ln \left (x \right ) x^{4} b \,c^{6} d^{4} e +20 \ln \left (c x +b \right ) x^{4} b^{2} c^{5} d^{3} e^{2}-30 \ln \left (c x +b \right ) x^{4} b \,c^{6} d^{4} e -40 \ln \left (x \right ) x^{3} b^{3} c^{4} d^{3} e^{2}+60 \ln \left (x \right ) x^{3} b^{2} c^{5} d^{4} e +40 \ln \left (c x +b \right ) x^{3} b^{3} c^{4} d^{3} e^{2}+b^{4} c^{3} d^{5}-b^{7} d^{2} e^{3}+3 b^{6} c \,d^{3} e^{2}-3 b^{5} c^{2} d^{4} e -60 \ln \left (c x +b \right ) x^{3} b^{2} c^{5} d^{4} e -20 \ln \left (x \right ) x^{2} b^{4} c^{3} d^{3} e^{2}+30 \ln \left (x \right ) x^{2} b^{3} c^{4} d^{4} e +20 \ln \left (c x +b \right ) x^{2} b^{4} c^{3} d^{3} e^{2}-30 \ln \left (c x +b \right ) x^{2} b^{3} c^{4} d^{4} e -6 x^{3} b^{5} c^{2} d \,e^{4}+40 x^{3} b^{3} c^{4} d^{3} e^{2}-60 x^{3} b^{2} c^{5} d^{4} e -12 \ln \left (x \right ) x^{4} c^{7} d^{5}+12 \ln \left (c x +b \right ) x^{4} c^{7} d^{5}+24 x^{3} b \,c^{6} d^{5}-4 x \,b^{3} c^{4} d^{5}+18 x^{4} c^{7} d^{5}+2 x \,b^{7} d \,e^{4}+2 \ln \left (x \right ) x^{2} b^{7} e^{5}-2 \ln \left (e x +d \right ) x^{2} b^{7} e^{5}+10 x \,b^{4} c^{3} d^{4} e -12 \ln \left (x \right ) x^{2} b^{2} c^{5} d^{5}+12 \ln \left (c x +b \right ) x^{2} b^{2} c^{5} d^{5}-24 \ln \left (x \right ) x^{3} b \,c^{6} d^{5}+24 \ln \left (c x +b \right ) x^{3} b \,c^{6} d^{5}-4 x^{4} b^{4} c^{3} d \,e^{4}+x^{4} b^{3} c^{4} d^{2} e^{3}+30 x^{4} b^{2} c^{5} d^{3} e^{2}}{2 \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \left (c x +b \right )^{2} x^{2} b^{5} d^{3}}\) | \(746\) |
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Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (189) = 378\).
Time = 18.68 (sec) , antiderivative size = 716, normalized size of antiderivative = 3.71 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=-\frac {b^{4} c^{3} d^{5} - 3 \, b^{5} c^{2} d^{4} e + 3 \, b^{6} c d^{3} e^{2} - b^{7} d^{2} e^{3} - 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{5} c^{2} d e^{4}\right )} x^{3} - {\left (18 \, b^{2} c^{5} d^{5} - 45 \, b^{3} c^{4} d^{4} e + 30 \, b^{4} c^{3} d^{3} e^{2} + b^{5} c^{2} d^{2} e^{3} - 4 \, b^{6} c d e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{3} c^{4} d^{5} - 5 \, b^{4} c^{3} d^{4} e + 3 \, b^{5} c^{2} d^{3} e^{2} + b^{6} c d^{2} e^{3} - b^{7} d e^{4}\right )} x + 2 \, {\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2}\right )} x^{4} + 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2}\right )} x^{3} + {\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left (b^{5} c^{2} e^{5} x^{4} + 2 \, b^{6} c e^{5} x^{3} + b^{7} e^{5} x^{2}\right )} \log \left (e x + d\right ) - 2 \, {\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2} - b^{5} c^{2} e^{5}\right )} x^{4} + 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{6} c e^{5}\right )} x^{3} + {\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2} - b^{7} e^{5}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (b^{5} c^{5} d^{6} - 3 \, b^{6} c^{4} d^{5} e + 3 \, b^{7} c^{3} d^{4} e^{2} - b^{8} c^{2} d^{3} e^{3}\right )} x^{4} + 2 \, {\left (b^{6} c^{4} d^{6} - 3 \, b^{7} c^{3} d^{5} e + 3 \, b^{8} c^{2} d^{4} e^{2} - b^{9} c d^{3} e^{3}\right )} x^{3} + {\left (b^{7} c^{3} d^{6} - 3 \, b^{8} c^{2} d^{5} e + 3 \, b^{9} c d^{4} e^{2} - b^{10} d^{3} e^{3}\right )} x^{2}\right )}} \]
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Timed out. \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (189) = 378\).
Time = 0.22 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.27 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\frac {e^{5} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {{\left (6 \, c^{5} d^{2} - 15 \, b c^{4} d e + 10 \, b^{2} c^{3} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}} - \frac {b^{3} c^{2} d^{3} - 2 \, b^{4} c d^{2} e + b^{5} d e^{2} - 2 \, {\left (6 \, c^{5} d^{3} - 9 \, b c^{4} d^{2} e + b^{2} c^{3} d e^{2} + b^{3} c^{2} e^{3}\right )} x^{3} - {\left (18 \, b c^{4} d^{3} - 27 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} + 4 \, b^{4} c e^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{5} e^{3}\right )} x}{2 \, {\left ({\left (b^{4} c^{4} d^{4} - 2 \, b^{5} c^{3} d^{3} e + b^{6} c^{2} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b^{5} c^{3} d^{4} - 2 \, b^{6} c^{2} d^{3} e + b^{7} c d^{2} e^{2}\right )} x^{3} + {\left (b^{6} c^{2} d^{4} - 2 \, b^{7} c d^{3} e + b^{8} d^{2} e^{2}\right )} x^{2}\right )}} + \frac {{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left (x\right )}{b^{5} d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (189) = 378\).
Time = 0.27 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.19 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\frac {e^{6} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} - \frac {{\left (6 \, c^{6} d^{2} - 15 \, b c^{5} d e + 10 \, b^{2} c^{4} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4} d^{3} - 3 \, b^{6} c^{3} d^{2} e + 3 \, b^{7} c^{2} d e^{2} - b^{8} c e^{3}} + \frac {{\left (6 \, c^{2} d^{2} + 3 \, b c d e + b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5} d^{3}} - \frac {b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - 2 \, {\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - b^{4} c^{2} d e^{4}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} + b^{4} c^{2} d^{2} e^{3} - 4 \, b^{5} c d e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + b^{5} c d^{2} e^{3} - b^{6} d e^{4}\right )} x}{2 \, {\left (c d - b e\right )}^{3} {\left (c x + b\right )}^{2} b^{4} d^{3} x^{2}} \]
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Time = 10.16 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(d+e x) \left (b x+c x^2\right )^3} \, dx=\frac {\frac {x\,\left (b\,e+2\,c\,d\right )}{b^2\,d^2}-\frac {1}{2\,b\,d}+\frac {x^2\,\left (4\,b^3\,c\,e^3+3\,b^2\,c^2\,d\,e^2-27\,b\,c^3\,d^2\,e+18\,c^4\,d^3\right )}{2\,b^3\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x^3\,\left (b^3\,c^2\,e^3+b^2\,c^3\,d\,e^2-9\,b\,c^4\,d^2\,e+6\,c^5\,d^3\right )}{b^4\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}+\frac {\ln \left (b+c\,x\right )\,\left (10\,b^2\,c^3\,e^2-15\,b\,c^4\,d\,e+6\,c^5\,d^2\right )}{b^8\,e^3-3\,b^7\,c\,d\,e^2+3\,b^6\,c^2\,d^2\,e-b^5\,c^3\,d^3}-\frac {e^5\,\ln \left (d+e\,x\right )}{d^3\,{\left (b\,e-c\,d\right )}^3}+\frac {\ln \left (x\right )\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5\,d^3} \]
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