Integrand size = 24, antiderivative size = 171 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\frac {B d-A e}{2 d (c d-b e) (d+e x)^2}+\frac {B c d^2-A e (2 c d-b e)}{d^2 (c d-b e)^2 (d+e x)}+\frac {A \log (x)}{b d^3}+\frac {c^2 (b B-A c) \log (b+c x)}{b (c d-b e)^3}-\frac {\left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \log (d+e x)}{d^3 (c d-b e)^3} \]
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Time = 0.16 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=-\frac {\log (d+e x) \left (B c^2 d^3-A e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )\right )}{d^3 (c d-b e)^3}+\frac {c^2 (b B-A c) \log (b+c x)}{b (c d-b e)^3}+\frac {B c d^2-A e (2 c d-b e)}{d^2 (d+e x) (c d-b e)^2}+\frac {B d-A e}{2 d (d+e x)^2 (c d-b e)}+\frac {A \log (x)}{b d^3} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A}{b d^3 x}-\frac {c^3 (b B-A c)}{b (-c d+b e)^3 (b+c x)}-\frac {e (B d-A e)}{d (c d-b e) (d+e x)^3}+\frac {e \left (-B c d^2+A e (2 c d-b e)\right )}{d^2 (c d-b e)^2 (d+e x)^2}+\frac {e \left (-B c^2 d^3+A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right )}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx \\ & = \frac {B d-A e}{2 d (c d-b e) (d+e x)^2}+\frac {B c d^2-A e (2 c d-b e)}{d^2 (c d-b e)^2 (d+e x)}+\frac {A \log (x)}{b d^3}+\frac {c^2 (b B-A c) \log (b+c x)}{b (c d-b e)^3}-\frac {\left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \log (d+e x)}{d^3 (c d-b e)^3} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\frac {B d-A e}{2 d (c d-b e) (d+e x)^2}+\frac {B c d^2+A e (-2 c d+b e)}{d^2 (c d-b e)^2 (d+e x)}+\frac {A \log (x)}{b d^3}+\frac {c^2 (-b B+A c) \log (b+c x)}{b (-c d+b e)^3}-\frac {\left (B c^2 d^3-A e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \log (d+e x)}{d^3 (c d-b e)^3} \]
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Time = 0.31 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {A \ln \left (x \right )}{b \,d^{3}}+\frac {\left (A c -B b \right ) c^{2} \ln \left (c x +b \right )}{b \left (b e -c d \right )^{3}}+\frac {A b \,e^{2}-2 A c d e +B c \,d^{2}}{d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )}-\frac {\left (A \,b^{2} e^{3}-3 A b c d \,e^{2}+3 A \,c^{2} d^{2} e -B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{d^{3} \left (b e -c d \right )^{3}}+\frac {A e -B d}{2 d \left (b e -c d \right ) \left (e x +d \right )^{2}}\) | \(171\) |
norman | \(\frac {-\frac {e^{2} \left (3 A b \,e^{2}-5 A c d e -B b d e +3 B c \,d^{2}\right ) x^{2}}{2 d^{3} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {\left (2 A b \,e^{2}-3 A c d e -B b d e +2 B c \,d^{2}\right ) e x}{d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}}{\left (e x +d \right )^{2}}+\frac {A \ln \left (x \right )}{b \,d^{3}}+\frac {c^{2} \left (A c -B b \right ) \ln \left (c x +b \right )}{\left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) b}-\frac {\left (A \,b^{2} e^{3}-3 A b c d \,e^{2}+3 A \,c^{2} d^{2} e -B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\) | \(284\) |
risch | \(\frac {\frac {e \left (A b \,e^{2}-2 A c d e +B c \,d^{2}\right ) x}{d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {3 A b \,e^{2}-5 A c d e -B b d e +3 B c \,d^{2}}{2 d \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}}{\left (e x +d \right )^{2}}-\frac {\ln \left (-e x -d \right ) A \,b^{2} e^{3}}{d^{3} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {3 \ln \left (-e x -d \right ) A b c \,e^{2}}{d^{2} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {3 \ln \left (-e x -d \right ) A \,c^{2} e}{d \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {\ln \left (-e x -d \right ) B \,c^{2}}{b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}}+\frac {c^{3} \ln \left (c x +b \right ) A}{\left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) b}-\frac {c^{2} \ln \left (c x +b \right ) B}{b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}}+\frac {A \ln \left (-x \right )}{d^{3} b}\) | \(454\) |
parallelrisch | \(\frac {-4 A d \,b^{3} e^{4} x -2 A \ln \left (x \right ) x^{2} c^{3} d^{3} e^{2}+2 A \ln \left (c x +b \right ) x^{2} c^{3} d^{3} e^{2}+4 A \ln \left (x \right ) x \,b^{3} d \,e^{4}-4 A \ln \left (x \right ) x \,c^{3} d^{4} e +4 A \ln \left (c x +b \right ) x \,c^{3} d^{4} e -4 A \ln \left (e x +d \right ) x \,b^{3} d \,e^{4}-6 A \ln \left (x \right ) b^{2} c \,d^{3} e^{2}+6 A \ln \left (x \right ) b \,c^{2} d^{4} e +6 A \ln \left (e x +d \right ) b^{2} c \,d^{3} e^{2}-6 A \ln \left (e x +d \right ) b \,c^{2} d^{4} e -2 A \ln \left (x \right ) c^{3} d^{5}+2 A \ln \left (c x +b \right ) c^{3} d^{5}-3 A \,b^{3} e^{5} x^{2}-6 B \,b^{2} c \,d^{3} e^{2} x +4 B b \,c^{2} d^{4} e x +3 B b \,c^{2} d^{3} e^{2} x^{2}+10 A \,b^{2} c \,d^{2} e^{3} x -6 A b \,c^{2} d^{3} e^{2} x -4 B \,b^{2} c \,d^{2} e^{3} x^{2}+8 A \,b^{2} c d \,e^{4} x^{2}-5 A b \,c^{2} d^{2} e^{3} x^{2}+B \,b^{3} d \,e^{4} x^{2}+2 B \,b^{3} d^{2} e^{3} x -6 A \ln \left (x \right ) x^{2} b^{2} c d \,e^{4}+6 A \ln \left (x \right ) x^{2} b \,c^{2} d^{2} e^{3}+6 A \ln \left (e x +d \right ) x^{2} b^{2} c d \,e^{4}-6 A \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{2} e^{3}-2 B \ln \left (c x +b \right ) x^{2} b \,c^{2} d^{3} e^{2}+2 B \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{3} e^{2}-12 A \ln \left (x \right ) x \,b^{2} c \,d^{2} e^{3}+12 A \ln \left (x \right ) x b \,c^{2} d^{3} e^{2}+12 A \ln \left (e x +d \right ) x \,b^{2} c \,d^{2} e^{3}-12 A \ln \left (e x +d \right ) x b \,c^{2} d^{3} e^{2}-4 B \ln \left (c x +b \right ) x b \,c^{2} d^{4} e +4 B \ln \left (e x +d \right ) x b \,c^{2} d^{4} e +2 B \ln \left (e x +d \right ) b \,c^{2} d^{5}+2 A \ln \left (x \right ) x^{2} b^{3} e^{5}-2 A \ln \left (e x +d \right ) x^{2} b^{3} e^{5}+2 A \ln \left (x \right ) b^{3} d^{2} e^{3}-2 A \ln \left (e x +d \right ) b^{3} d^{2} e^{3}-2 B \ln \left (c x +b \right ) b \,c^{2} d^{5}}{2 \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) b \left (e x +d \right )^{2} d^{3}}\) | \(731\) |
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Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (169) = 338\).
Time = 24.50 (sec) , antiderivative size = 644, normalized size of antiderivative = 3.77 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\frac {3 \, B b c^{2} d^{5} - 3 \, A b^{3} d^{2} e^{3} - {\left (4 \, B b^{2} c + 5 \, A b c^{2}\right )} d^{4} e + {\left (B b^{3} + 8 \, A b^{2} c\right )} d^{3} e^{2} + 2 \, {\left (B b c^{2} d^{4} e + 3 \, A b^{2} c d^{2} e^{3} - A b^{3} d e^{4} - {\left (B b^{2} c + 2 \, A b c^{2}\right )} d^{3} e^{2}\right )} x + 2 \, {\left ({\left (B b c^{2} - A c^{3}\right )} d^{3} e^{2} x^{2} + 2 \, {\left (B b c^{2} - A c^{3}\right )} d^{4} e x + {\left (B b c^{2} - A c^{3}\right )} d^{5}\right )} \log \left (c x + b\right ) - 2 \, {\left (B b c^{2} d^{5} - 3 \, A b c^{2} d^{4} e + 3 \, A b^{2} c d^{3} e^{2} - A b^{3} d^{2} e^{3} + {\left (B b c^{2} d^{3} e^{2} - 3 \, A b c^{2} d^{2} e^{3} + 3 \, A b^{2} c d e^{4} - A b^{3} e^{5}\right )} x^{2} + 2 \, {\left (B b c^{2} d^{4} e - 3 \, A b c^{2} d^{3} e^{2} + 3 \, A b^{2} c d^{2} e^{3} - A b^{3} d e^{4}\right )} x\right )} \log \left (e x + d\right ) + 2 \, {\left (A c^{3} d^{5} - 3 \, A b c^{2} d^{4} e + 3 \, A b^{2} c d^{3} e^{2} - A b^{3} d^{2} e^{3} + {\left (A c^{3} d^{3} e^{2} - 3 \, A b c^{2} d^{2} e^{3} + 3 \, A b^{2} c d e^{4} - A b^{3} e^{5}\right )} x^{2} + 2 \, {\left (A c^{3} d^{4} e - 3 \, A b c^{2} d^{3} e^{2} + 3 \, A b^{2} c d^{2} e^{3} - A b^{3} d e^{4}\right )} x\right )} \log \left (x\right )}{2 \, {\left (b c^{3} d^{8} - 3 \, b^{2} c^{2} d^{7} e + 3 \, b^{3} c d^{6} e^{2} - b^{4} d^{5} e^{3} + {\left (b c^{3} d^{6} e^{2} - 3 \, b^{2} c^{2} d^{5} e^{3} + 3 \, b^{3} c d^{4} e^{4} - b^{4} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (b c^{3} d^{7} e - 3 \, b^{2} c^{2} d^{6} e^{2} + 3 \, b^{3} c d^{5} e^{3} - b^{4} d^{4} e^{4}\right )} x\right )}} \]
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Timed out. \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.82 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\frac {{\left (B b c^{2} - A c^{3}\right )} \log \left (c x + b\right )}{b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}} - \frac {{\left (B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, A b c d e^{2} - A b^{2} e^{3}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} + \frac {3 \, B c d^{3} + 3 \, A b d e^{2} - {\left (B b + 5 \, A c\right )} d^{2} e + 2 \, {\left (B c d^{2} e - 2 \, A c d e^{2} + A b e^{3}\right )} x}{2 \, {\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} + {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}} + \frac {A \log \left (x\right )}{b d^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\frac {{\left (B b c^{3} - A c^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{4} d^{3} - 3 \, b^{2} c^{3} d^{2} e + 3 \, b^{3} c^{2} d e^{2} - b^{4} c e^{3}} - \frac {{\left (B c^{2} d^{3} e - 3 \, A c^{2} d^{2} e^{2} + 3 \, A b c d e^{3} - A b^{2} e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} - b^{3} d^{3} e^{4}} + \frac {A \log \left ({\left | x \right |}\right )}{b d^{3}} + \frac {3 \, B c^{2} d^{5} - 4 \, B b c d^{4} e - 5 \, A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 8 \, A b c d^{3} e^{2} - 3 \, A b^{2} d^{2} e^{3} + 2 \, {\left (B c^{2} d^{4} e - B b c d^{3} e^{2} - 2 \, A c^{2} d^{3} e^{2} + 3 \, A b c d^{2} e^{3} - A b^{2} d e^{4}\right )} x}{2 \, {\left (c d - b e\right )}^{3} {\left (e x + d\right )}^{2} d^{3}} \]
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Time = 10.83 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.66 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )} \, dx=\frac {\frac {3\,A\,b\,e^2+3\,B\,c\,d^2-5\,A\,c\,d\,e-B\,b\,d\,e}{2\,d\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x\,\left (B\,c\,d^2\,e-2\,A\,c\,d\,e^2+A\,b\,e^3\right )}{d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{d^2+2\,d\,e\,x+e^2\,x^2}-\frac {\ln \left (d+e\,x\right )\,\left (c^2\,\left (B\,d^3-3\,A\,d^2\,e\right )-A\,b^2\,e^3+3\,A\,b\,c\,d\,e^2\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}+\frac {\ln \left (b+c\,x\right )\,\left (A\,c^3-B\,b\,c^2\right )}{b^4\,e^3-3\,b^3\,c\,d\,e^2+3\,b^2\,c^2\,d^2\,e-b\,c^3\,d^3}+\frac {A\,\ln \left (x\right )}{b\,d^3} \]
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