Integrand size = 27, antiderivative size = 25 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx=\frac {d x}{b}+\frac {(b c-a d) \log (a+b x)}{b^2} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {24, 45} \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx=\frac {(b c-a d) \log (a+b x)}{b^2}+\frac {d x}{b} \]
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Rule 24
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {b^2 c+b^2 d x}{a+b x} \, dx}{b^2} \\ & = \frac {\int \left (b d+\frac {b (b c-a d)}{a+b x}\right ) \, dx}{b^2} \\ & = \frac {d x}{b}+\frac {(b c-a d) \log (a+b x)}{b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx=\frac {d x}{b}+\frac {(b c-a d) \log (a+b x)}{b^2} \]
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Time = 2.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {d x}{b}+\frac {\left (-a d +b c \right ) \ln \left (b x +a \right )}{b^{2}}\) | \(26\) |
parallelrisch | \(-\frac {\ln \left (b x +a \right ) a d -\ln \left (b x +a \right ) b c -b d x}{b^{2}}\) | \(31\) |
risch | \(\frac {d x}{b}-\frac {\ln \left (b x +a \right ) a d}{b^{2}}+\frac {c \ln \left (b x +a \right )}{b}\) | \(32\) |
norman | \(\frac {d \,x^{2}-\frac {a^{2} d}{b^{2}}}{b x +a}-\frac {\left (a d -b c \right ) \ln \left (b x +a \right )}{b^{2}}\) | \(44\) |
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx=\frac {b d x + {\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx=\frac {d x}{b} - \frac {\left (a d - b c\right ) \log {\left (a + b x \right )}}{b^{2}} \]
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Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx=\frac {d x}{b} + \frac {{\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 4.68 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx=b d {\left (\frac {2 \, a \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} + \frac {b x + a}{b^{3}} - \frac {a^{2}}{{\left (b x + a\right )} b^{3}}\right )} - \frac {{\left (b c + a d\right )} {\left (\frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x + a\right )} b}\right )}}{b} - \frac {a c}{{\left (b x + a\right )} b} \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx=\frac {d\,x}{b}-\frac {\ln \left (a+b\,x\right )\,\left (a\,d-b\,c\right )}{b^2} \]
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