Integrand size = 19, antiderivative size = 41 \[ \int \frac {(a+b x)^2}{(a c-b c x)^2} \, dx=\frac {x}{c^2}+\frac {4 a^2}{b c^2 (a-b x)}+\frac {4 a \log (a-b x)}{b c^2} \]
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Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {45} \[ \int \frac {(a+b x)^2}{(a c-b c x)^2} \, dx=\frac {4 a^2}{b c^2 (a-b x)}+\frac {4 a \log (a-b x)}{b c^2}+\frac {x}{c^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{c^2}+\frac {4 a^2}{c^2 (a-b x)^2}-\frac {4 a}{c^2 (a-b x)}\right ) \, dx \\ & = \frac {x}{c^2}+\frac {4 a^2}{b c^2 (a-b x)}+\frac {4 a \log (a-b x)}{b c^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^2}{(a c-b c x)^2} \, dx=\frac {x+\frac {4 a^2}{b (a-b x)}+\frac {4 a \log (a-b x)}{b}}{c^2} \]
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Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {x +\frac {4 a \ln \left (-b x +a \right )}{b}+\frac {4 a^{2}}{b \left (-b x +a \right )}}{c^{2}}\) | \(36\) |
risch | \(\frac {x}{c^{2}}+\frac {4 a^{2}}{b \,c^{2} \left (-b x +a \right )}+\frac {4 a \ln \left (-b x +a \right )}{b \,c^{2}}\) | \(42\) |
norman | \(\frac {\frac {5 a^{2}}{b c}-\frac {b \,x^{2}}{c}}{c \left (-b x +a \right )}+\frac {4 a \ln \left (-b x +a \right )}{b \,c^{2}}\) | \(51\) |
parallelrisch | \(\frac {4 \ln \left (b x -a \right ) x a b +b^{2} x^{2}-4 a^{2} \ln \left (b x -a \right )-5 a^{2}}{b \,c^{2} \left (b x -a \right )}\) | \(56\) |
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none
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^2}{(a c-b c x)^2} \, dx=\frac {b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \, {\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} c^{2} x - a b c^{2}} \]
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Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^2}{(a c-b c x)^2} \, dx=- \frac {4 a^{2}}{- a b c^{2} + b^{2} c^{2} x} + \frac {4 a \log {\left (- a + b x \right )}}{b c^{2}} + \frac {x}{c^{2}} \]
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none
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^2}{(a c-b c x)^2} \, dx=-\frac {4 \, a^{2}}{b^{2} c^{2} x - a b c^{2}} + \frac {x}{c^{2}} + \frac {4 \, a \log \left (b x - a\right )}{b c^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.93 \[ \int \frac {(a+b x)^2}{(a c-b c x)^2} \, dx=-\frac {4 \, a^{2}}{{\left (b c x - a c\right )} b c} - \frac {4 \, a \log \left (\frac {{\left | b c x - a c \right |}}{{\left (b c x - a c\right )}^{2} {\left | b \right |} {\left | c \right |}}\right )}{b c^{2}} + \frac {b c x - a c}{b c^{3}} \]
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Time = 0.34 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^2}{(a c-b c x)^2} \, dx=\frac {x}{c^2}+\frac {4\,a^2}{b\,\left (a\,c^2-b\,c^2\,x\right )}+\frac {4\,a\,\ln \left (b\,x-a\right )}{b\,c^2} \]
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