Integrand size = 26, antiderivative size = 57 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {1}{(b d-a e) (a+b x)}-\frac {e \log (a+b x)}{(b d-a e)^2}+\frac {e \log (d+e x)}{(b d-a e)^2} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 46} \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {1}{(a+b x) (b d-a e)}-\frac {e \log (a+b x)}{(b d-a e)^2}+\frac {e \log (d+e x)}{(b d-a e)^2} \]
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Rule 27
Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^2 (d+e x)} \, dx \\ & = \int \left (\frac {b}{(b d-a e) (a+b x)^2}-\frac {b e}{(b d-a e)^2 (a+b x)}+\frac {e^2}{(b d-a e)^2 (d+e x)}\right ) \, dx \\ & = -\frac {1}{(b d-a e) (a+b x)}-\frac {e \log (a+b x)}{(b d-a e)^2}+\frac {e \log (d+e x)}{(b d-a e)^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {-b d+a e-e (a+b x) \log (a+b x)+e (a+b x) \log (d+e x)}{(b d-a e)^2 (a+b x)} \]
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Time = 2.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {1}{\left (a e -b d \right ) \left (b x +a \right )}-\frac {e \ln \left (b x +a \right )}{\left (a e -b d \right )^{2}}+\frac {e \ln \left (e x +d \right )}{\left (a e -b d \right )^{2}}\) | \(57\) |
risch | \(\frac {1}{\left (a e -b d \right ) \left (b x +a \right )}-\frac {e \ln \left (b x +a \right )}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}+\frac {e \ln \left (-e x -d \right )}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}\) | \(86\) |
norman | \(-\frac {b x}{a \left (a e -b d \right ) \left (b x +a \right )}+\frac {e \ln \left (e x +d \right )}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}-\frac {e \ln \left (b x +a \right )}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}\) | \(89\) |
parallelrisch | \(-\frac {\ln \left (b x +a \right ) x a b e -\ln \left (e x +d \right ) x a b e +\ln \left (b x +a \right ) a^{2} e -\ln \left (e x +d \right ) a^{2} e +a b e x -b^{2} d x}{\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (b x +a \right ) a}\) | \(95\) |
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Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {b d - a e + {\left (b e x + a e\right )} \log \left (b x + a\right ) - {\left (b e x + a e\right )} \log \left (e x + d\right )}{a b^{2} d^{2} - 2 \, a^{2} b d e + a^{3} e^{2} + {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (46) = 92\).
Time = 0.37 (sec) , antiderivative size = 233, normalized size of antiderivative = 4.09 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {e \log {\left (x + \frac {- \frac {a^{3} e^{4}}{\left (a e - b d\right )^{2}} + \frac {3 a^{2} b d e^{3}}{\left (a e - b d\right )^{2}} - \frac {3 a b^{2} d^{2} e^{2}}{\left (a e - b d\right )^{2}} + a e^{2} + \frac {b^{3} d^{3} e}{\left (a e - b d\right )^{2}} + b d e}{2 b e^{2}} \right )}}{\left (a e - b d\right )^{2}} - \frac {e \log {\left (x + \frac {\frac {a^{3} e^{4}}{\left (a e - b d\right )^{2}} - \frac {3 a^{2} b d e^{3}}{\left (a e - b d\right )^{2}} + \frac {3 a b^{2} d^{2} e^{2}}{\left (a e - b d\right )^{2}} + a e^{2} - \frac {b^{3} d^{3} e}{\left (a e - b d\right )^{2}} + b d e}{2 b e^{2}} \right )}}{\left (a e - b d\right )^{2}} + \frac {1}{a^{2} e - a b d + x \left (a b e - b^{2} d\right )} \]
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Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.61 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {e \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac {e \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac {1}{a b d - a^{2} e + {\left (b^{2} d - a b e\right )} x} \]
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Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {b e \log \left ({\left | b x + a \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} + \frac {e^{2} \log \left ({\left | e x + d \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} - \frac {1}{{\left (b d - a e\right )} {\left (b x + a\right )}} \]
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Time = 10.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {1}{\left (a\,e-b\,d\right )\,\left (a+b\,x\right )}-\frac {2\,e\,\mathrm {atanh}\left (\frac {a^2\,e^2-b^2\,d^2}{{\left (a\,e-b\,d\right )}^2}+\frac {2\,b\,e\,x}{a\,e-b\,d}\right )}{{\left (a\,e-b\,d\right )}^2} \]
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