Integrand size = 18, antiderivative size = 55 \[ \int \frac {a+b x+c x^2}{(d+e x)^2} \, dx=\frac {c x}{e^2}-\frac {c d^2-b d e+a e^2}{e^3 (d+e x)}-\frac {(2 c d-b e) \log (d+e x)}{e^3} \]
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Time = 0.03 (sec) , antiderivative size = 55, 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 = {712} \[ \int \frac {a+b x+c x^2}{(d+e x)^2} \, dx=-\frac {a e^2-b d e+c d^2}{e^3 (d+e x)}-\frac {(2 c d-b e) \log (d+e x)}{e^3}+\frac {c x}{e^2} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c}{e^2}+\frac {c d^2-b d e+a e^2}{e^2 (d+e x)^2}+\frac {-2 c d+b e}{e^2 (d+e x)}\right ) \, dx \\ & = \frac {c x}{e^2}-\frac {c d^2-b d e+a e^2}{e^3 (d+e x)}-\frac {(2 c d-b e) \log (d+e x)}{e^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x+c x^2}{(d+e x)^2} \, dx=\frac {c e x-\frac {c d^2-b d e+a e^2}{d+e x}+(-2 c d+b e) \log (d+e x)}{e^3} \]
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Time = 4.56 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {c x}{e^{2}}-\frac {e^{2} a -b d e +c \,d^{2}}{e^{3} \left (e x +d \right )}+\frac {\left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{3}}\) | \(54\) |
norman | \(\frac {\frac {c \,x^{2}}{e}-\frac {e^{2} a -b d e +2 c \,d^{2}}{e^{3}}}{e x +d}+\frac {\left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{3}}\) | \(59\) |
risch | \(\frac {c x}{e^{2}}-\frac {a}{e \left (e x +d \right )}+\frac {b d}{e^{2} \left (e x +d \right )}-\frac {c \,d^{2}}{e^{3} \left (e x +d \right )}+\frac {\ln \left (e x +d \right ) b}{e^{2}}-\frac {2 c d \ln \left (e x +d \right )}{e^{3}}\) | \(74\) |
parallelrisch | \(\frac {\ln \left (e x +d \right ) x b \,e^{2}-2 \ln \left (e x +d \right ) x c d e +c \,x^{2} e^{2}+\ln \left (e x +d \right ) b d e -2 \ln \left (e x +d \right ) c \,d^{2}-e^{2} a +b d e -2 c \,d^{2}}{e^{3} \left (e x +d \right )}\) | \(83\) |
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Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.42 \[ \int \frac {a+b x+c x^2}{(d+e x)^2} \, dx=\frac {c e^{2} x^{2} + c d e x - c d^{2} + b d e - a e^{2} - {\left (2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \]
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Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x+c x^2}{(d+e x)^2} \, dx=\frac {c x}{e^{2}} + \frac {- a e^{2} + b d e - c d^{2}}{d e^{3} + e^{4} x} + \frac {\left (b e - 2 c d\right ) \log {\left (d + e x \right )}}{e^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.05 \[ \int \frac {a+b x+c x^2}{(d+e x)^2} \, dx=-\frac {c d^{2} - b d e + a e^{2}}{e^{4} x + d e^{3}} + \frac {c x}{e^{2}} - \frac {{\left (2 \, c d - b e\right )} \log \left (e x + d\right )}{e^{3}} \]
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Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.98 \[ \int \frac {a+b x+c x^2}{(d+e x)^2} \, dx=c {\left (\frac {2 \, d \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{3}} + \frac {e x + d}{e^{3}} - \frac {d^{2}}{{\left (e x + d\right )} e^{3}}\right )} - \frac {b {\left (\frac {\log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e} - \frac {d}{{\left (e x + d\right )} e}\right )}}{e} - \frac {a}{{\left (e x + d\right )} e} \]
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Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07 \[ \int \frac {a+b x+c x^2}{(d+e x)^2} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (b\,e-2\,c\,d\right )}{e^3}-\frac {c\,d^2-b\,d\,e+a\,e^2}{e\,\left (x\,e^3+d\,e^2\right )}+\frac {c\,x}{e^2} \]
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