Integrand size = 18, antiderivative size = 87 \[ \int \frac {1}{x (a+b x) (c+d x)^2} \, dx=-\frac {d}{c (b c-a d) (c+d x)}+\frac {\log (x)}{a c^2}-\frac {b^2 \log (a+b x)}{a (b c-a d)^2}+\frac {d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2} \]
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Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {84} \[ \int \frac {1}{x (a+b x) (c+d x)^2} \, dx=-\frac {b^2 \log (a+b x)}{a (b c-a d)^2}+\frac {d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2}-\frac {d}{c (c+d x) (b c-a d)}+\frac {\log (x)}{a c^2} \]
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Rule 84
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a c^2 x}-\frac {b^3}{a (-b c+a d)^2 (a+b x)}+\frac {d^2}{c (b c-a d) (c+d x)^2}+\frac {d^2 (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)}\right ) \, dx \\ & = -\frac {d}{c (b c-a d) (c+d x)}+\frac {\log (x)}{a c^2}-\frac {b^2 \log (a+b x)}{a (b c-a d)^2}+\frac {d (2 b c-a d) \log (c+d x)}{c^2 (b c-a d)^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x (a+b x) (c+d x)^2} \, dx=\frac {\log (x)+\frac {-b^2 c^2 (c+d x) \log (a+b x)+a d (c (-b c+a d)+(2 b c-a d) (c+d x) \log (c+d x))}{(b c-a d)^2 (c+d x)}}{a c^2} \]
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Time = 1.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\ln \left (x \right )}{a \,c^{2}}+\frac {d}{c \left (a d -b c \right ) \left (d x +c \right )}-\frac {d \left (a d -2 b c \right ) \ln \left (d x +c \right )}{c^{2} \left (a d -b c \right )^{2}}-\frac {b^{2} \ln \left (b x +a \right )}{a \left (a d -b c \right )^{2}}\) | \(87\) |
norman | \(-\frac {d^{2} x}{c^{2} \left (a d -b c \right ) \left (d x +c \right )}+\frac {\ln \left (x \right )}{a \,c^{2}}-\frac {b^{2} \ln \left (b x +a \right )}{a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {d \left (a d -2 b c \right ) \ln \left (d x +c \right )}{c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(117\) |
risch | \(\frac {d}{c \left (a d -b c \right ) \left (d x +c \right )}+\frac {\ln \left (-x \right )}{a \,c^{2}}-\frac {d^{2} \ln \left (-d x -c \right ) a}{c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 d \ln \left (-d x -c \right ) b}{c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {b^{2} \ln \left (b x +a \right )}{a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(152\) |
parallelrisch | \(\frac {\ln \left (x \right ) x \,a^{2} d^{4}-2 \ln \left (x \right ) x a b c \,d^{3}+\ln \left (x \right ) x \,b^{2} c^{2} d^{2}-\ln \left (b x +a \right ) x \,b^{2} c^{2} d^{2}-\ln \left (d x +c \right ) x \,a^{2} d^{4}+2 \ln \left (d x +c \right ) x a b c \,d^{3}+\ln \left (x \right ) a^{2} c \,d^{3}-2 \ln \left (x \right ) a b \,c^{2} d^{2}+\ln \left (x \right ) b^{2} c^{3} d -\ln \left (b x +a \right ) b^{2} c^{3} d -\ln \left (d x +c \right ) a^{2} c \,d^{3}+2 \ln \left (d x +c \right ) a b \,c^{2} d^{2}+a^{2} c \,d^{3}-a b \,c^{2} d^{2}}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a \left (d x +c \right ) c^{2} d}\) | \(220\) |
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Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (87) = 174\).
Time = 0.61 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.38 \[ \int \frac {1}{x (a+b x) (c+d x)^2} \, dx=-\frac {a b c^{2} d - a^{2} c d^{2} + {\left (b^{2} c^{2} d x + b^{2} c^{3}\right )} \log \left (b x + a\right ) - {\left (2 \, a b c^{2} d - a^{2} c d^{2} + {\left (2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \log \left (d x + c\right ) - {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \log \left (x\right )}{a b^{2} c^{5} - 2 \, a^{2} b c^{4} d + a^{3} c^{3} d^{2} + {\left (a b^{2} c^{4} d - 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )} x} \]
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Timed out. \[ \int \frac {1}{x (a+b x) (c+d x)^2} \, dx=\text {Timed out} \]
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none
Time = 0.21 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x (a+b x) (c+d x)^2} \, dx=-\frac {b^{2} \log \left (b x + a\right )}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}} + \frac {{\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}} - \frac {d}{b c^{3} - a c^{2} d + {\left (b c^{2} d - a c d^{2}\right )} x} + \frac {\log \left (x\right )}{a c^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (87) = 174\).
Time = 0.28 (sec) , antiderivative size = 282, normalized size of antiderivative = 3.24 \[ \int \frac {1}{x (a+b x) (c+d x)^2} \, dx=-\frac {1}{2} \, d {\left (\frac {{\left (2 \, b c - a d\right )} \log \left ({\left | -b + \frac {2 \, b c}{d x + c} - \frac {b c^{2}}{{\left (d x + c\right )}^{2}} - \frac {a d}{d x + c} + \frac {a c d}{{\left (d x + c\right )}^{2}} \right |}\right )}{b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}} + \frac {2 \, d^{2}}{{\left (b c^{2} d^{2} - a c d^{3}\right )} {\left (d x + c\right )}} + \frac {{\left (2 \, b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (\frac {{\left | -2 \, b c d + \frac {2 \, b c^{2} d}{d x + c} + a d^{2} - \frac {2 \, a c d^{2}}{d x + c} - d^{2} {\left | a \right |} \right |}}{{\left | -2 \, b c d + \frac {2 \, b c^{2} d}{d x + c} + a d^{2} - \frac {2 \, a c d^{2}}{d x + c} + d^{2} {\left | a \right |} \right |}}\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} d^{2} {\left | a \right |}}\right )} \]
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Time = 0.73 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x (a+b x) (c+d x)^2} \, dx=\frac {\ln \left (x\right )}{a\,c^2}-\frac {\ln \left (c+d\,x\right )\,\left (a\,d^2-2\,b\,c\,d\right )}{a^2\,c^2\,d^2-2\,a\,b\,c^3\,d+b^2\,c^4}-\frac {b^2\,\ln \left (a+b\,x\right )}{a^3\,d^2-2\,a^2\,b\,c\,d+a\,b^2\,c^2}+\frac {d}{c\,\left (a\,d-b\,c\right )\,\left (c+d\,x\right )} \]
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