Integrand size = 17, antiderivative size = 109 \[ \int \frac {A+B x}{\left (b x+c x^2\right )^3} \, dx=-\frac {A}{2 b^3 x^2}-\frac {b B-3 A c}{b^4 x}-\frac {c (b B-A c)}{2 b^3 (b+c x)^2}-\frac {c (2 b B-3 A c)}{b^4 (b+c x)}-\frac {3 c (b B-2 A c) \log (x)}{b^5}+\frac {3 c (b B-2 A c) \log (b+c x)}{b^5} \]
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Time = 0.08 (sec) , antiderivative size = 109, 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} \[ \int \frac {A+B x}{\left (b x+c x^2\right )^3} \, dx=-\frac {3 c \log (x) (b B-2 A c)}{b^5}+\frac {3 c (b B-2 A c) \log (b+c x)}{b^5}-\frac {b B-3 A c}{b^4 x}-\frac {c (2 b B-3 A c)}{b^4 (b+c x)}-\frac {c (b B-A c)}{2 b^3 (b+c x)^2}-\frac {A}{2 b^3 x^2} \]
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Rule 645
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A}{b^3 x^3}+\frac {b B-3 A c}{b^4 x^2}-\frac {3 c (b B-2 A c)}{b^5 x}+\frac {c^2 (b B-A c)}{b^3 (b+c x)^3}+\frac {c^2 (2 b B-3 A c)}{b^4 (b+c x)^2}+\frac {3 c^2 (b B-2 A c)}{b^5 (b+c x)}\right ) \, dx \\ & = -\frac {A}{2 b^3 x^2}-\frac {b B-3 A c}{b^4 x}-\frac {c (b B-A c)}{2 b^3 (b+c x)^2}-\frac {c (2 b B-3 A c)}{b^4 (b+c x)}-\frac {3 c (b B-2 A c) \log (x)}{b^5}+\frac {3 c (b B-2 A c) \log (b+c x)}{b^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{\left (b x+c x^2\right )^3} \, dx=\frac {-\frac {b \left (b B x \left (2 b^2+9 b c x+6 c^2 x^2\right )+A \left (b^3-4 b^2 c x-18 b c^2 x^2-12 c^3 x^3\right )\right )}{x^2 (b+c x)^2}+6 c (-b B+2 A c) \log (x)+6 c (b B-2 A c) \log (b+c x)}{2 b^5} \]
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Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {A}{2 b^{3} x^{2}}-\frac {-3 A c +B b}{b^{4} x}+\frac {3 c \left (2 A c -B b \right ) \ln \left (x \right )}{b^{5}}-\frac {3 c \left (2 A c -B b \right ) \ln \left (c x +b \right )}{b^{5}}+\frac {c \left (3 A c -2 B b \right )}{b^{4} \left (c x +b \right )}+\frac {\left (A c -B b \right ) c}{2 b^{3} \left (c x +b \right )^{2}}\) | \(107\) |
norman | \(\frac {\frac {\left (2 A c -B b \right ) x}{b^{2}}-\frac {A}{2 b}-\frac {2 c \left (6 A \,c^{2}-3 B b c \right ) x^{3}}{b^{4}}-\frac {c^{2} \left (18 A \,c^{2}-9 B b c \right ) x^{4}}{2 b^{5}}}{x^{2} \left (c x +b \right )^{2}}+\frac {3 c \left (2 A c -B b \right ) \ln \left (x \right )}{b^{5}}-\frac {3 c \left (2 A c -B b \right ) \ln \left (c x +b \right )}{b^{5}}\) | \(116\) |
risch | \(\frac {\frac {3 c^{2} \left (2 A c -B b \right ) x^{3}}{b^{4}}+\frac {9 c \left (2 A c -B b \right ) x^{2}}{2 b^{3}}+\frac {\left (2 A c -B b \right ) x}{b^{2}}-\frac {A}{2 b}}{x^{2} \left (c x +b \right )^{2}}-\frac {6 c^{2} \ln \left (c x +b \right ) A}{b^{5}}+\frac {3 c \ln \left (c x +b \right ) B}{b^{4}}+\frac {6 c^{2} \ln \left (-x \right ) A}{b^{5}}-\frac {3 c \ln \left (-x \right ) B}{b^{4}}\) | \(124\) |
parallelrisch | \(\frac {12 A \ln \left (x \right ) x^{4} c^{4}-12 A \ln \left (c x +b \right ) x^{4} c^{4}-6 B \ln \left (x \right ) x^{4} b \,c^{3}+6 B \ln \left (c x +b \right ) x^{4} b \,c^{3}+24 A \ln \left (x \right ) x^{3} b \,c^{3}-24 A \ln \left (c x +b \right ) x^{3} b \,c^{3}-18 A \,c^{4} x^{4}-12 B \ln \left (x \right ) x^{3} b^{2} c^{2}+12 B \ln \left (c x +b \right ) x^{3} b^{2} c^{2}+9 B b \,c^{3} x^{4}+12 A \ln \left (x \right ) x^{2} b^{2} c^{2}-12 A \ln \left (c x +b \right ) x^{2} b^{2} c^{2}-24 A b \,c^{3} x^{3}-6 B \ln \left (x \right ) x^{2} b^{3} c +6 B \ln \left (c x +b \right ) x^{2} b^{3} c +12 B \,b^{2} c^{2} x^{3}+4 A \,b^{3} c x -2 b^{4} B x -A \,b^{4}}{2 b^{5} x^{2} \left (c x +b \right )^{2}}\) | \(253\) |
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Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (105) = 210\).
Time = 0.29 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.06 \[ \int \frac {A+B x}{\left (b x+c x^2\right )^3} \, dx=-\frac {A b^{4} + 6 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} + 9 \, {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2} + 2 \, {\left (B b^{4} - 2 \, A b^{3} c\right )} x - 6 \, {\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} x^{4} + 2 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} + {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \, {\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} x^{4} + 2 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} + {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{2} x^{4} + 2 \, b^{6} c x^{3} + b^{7} x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (104) = 208\).
Time = 0.35 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.01 \[ \int \frac {A+B x}{\left (b x+c x^2\right )^3} \, dx=\frac {- A b^{3} + x^{3} \cdot \left (12 A c^{3} - 6 B b c^{2}\right ) + x^{2} \cdot \left (18 A b c^{2} - 9 B b^{2} c\right ) + x \left (4 A b^{2} c - 2 B b^{3}\right )}{2 b^{6} x^{2} + 4 b^{5} c x^{3} + 2 b^{4} c^{2} x^{4}} - \frac {3 c \left (- 2 A c + B b\right ) \log {\left (x + \frac {- 6 A b c^{2} + 3 B b^{2} c - 3 b c \left (- 2 A c + B b\right )}{- 12 A c^{3} + 6 B b c^{2}} \right )}}{b^{5}} + \frac {3 c \left (- 2 A c + B b\right ) \log {\left (x + \frac {- 6 A b c^{2} + 3 B b^{2} c + 3 b c \left (- 2 A c + B b\right )}{- 12 A c^{3} + 6 B b c^{2}} \right )}}{b^{5}} \]
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Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.20 \[ \int \frac {A+B x}{\left (b x+c x^2\right )^3} \, dx=-\frac {A b^{3} + 6 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} x^{3} + 9 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} x^{2} + 2 \, {\left (B b^{3} - 2 \, A b^{2} c\right )} x}{2 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} + \frac {3 \, {\left (B b c - 2 \, A c^{2}\right )} \log \left (c x + b\right )}{b^{5}} - \frac {3 \, {\left (B b c - 2 \, A c^{2}\right )} \log \left (x\right )}{b^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x}{\left (b x+c x^2\right )^3} \, dx=-\frac {3 \, {\left (B b c - 2 \, A c^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {3 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac {6 \, B b c^{2} x^{3} - 12 \, A c^{3} x^{3} + 9 \, B b^{2} c x^{2} - 18 \, A b c^{2} x^{2} + 2 \, B b^{3} x - 4 \, A b^{2} c x + A b^{3}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4}} \]
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Time = 0.11 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{\left (b x+c x^2\right )^3} \, dx=\frac {\frac {x\,\left (2\,A\,c-B\,b\right )}{b^2}-\frac {A}{2\,b}+\frac {3\,c^2\,x^3\,\left (2\,A\,c-B\,b\right )}{b^4}+\frac {9\,c\,x^2\,\left (2\,A\,c-B\,b\right )}{2\,b^3}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {6\,c\,\mathrm {atanh}\left (\frac {3\,c\,\left (2\,A\,c-B\,b\right )\,\left (b+2\,c\,x\right )}{b\,\left (6\,A\,c^2-3\,B\,b\,c\right )}\right )\,\left (2\,A\,c-B\,b\right )}{b^5} \]
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