Integrand size = 19, antiderivative size = 52 \[ \int \frac {(a+b x)^2}{(a c-b c x)^3} \, dx=\frac {2 a^2}{b c^3 (a-b x)^2}-\frac {4 a}{b c^3 (a-b x)}-\frac {\log (a-b x)}{b c^3} \]
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Time = 0.02 (sec) , antiderivative size = 52, 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} \[ \int \frac {(a+b x)^2}{(a c-b c x)^3} \, dx=\frac {2 a^2}{b c^3 (a-b x)^2}-\frac {4 a}{b c^3 (a-b x)}-\frac {\log (a-b x)}{b c^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 a^2}{c^3 (a-b x)^3}-\frac {4 a}{c^3 (a-b x)^2}+\frac {1}{c^3 (a-b x)}\right ) \, dx \\ & = \frac {2 a^2}{b c^3 (a-b x)^2}-\frac {4 a}{b c^3 (a-b x)}-\frac {\log (a-b x)}{b c^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.63 \[ \int \frac {(a+b x)^2}{(a c-b c x)^3} \, dx=-\frac {\frac {2 a (a-2 b x)}{(a-b x)^2}+\log (a-b x)}{b c^3} \]
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Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {4 a x -\frac {2 a^{2}}{b}}{c^{3} \left (-b x +a \right )^{2}}-\frac {\ln \left (-b x +a \right )}{b \,c^{3}}\) | \(42\) |
default | \(\frac {-\frac {\ln \left (-b x +a \right )}{b}+\frac {2 a^{2}}{b \left (-b x +a \right )^{2}}-\frac {4 a}{b \left (-b x +a \right )}}{c^{3}}\) | \(48\) |
norman | \(\frac {-\frac {2 a^{2}}{b c}+\frac {4 a x}{c}}{c^{2} \left (-b x +a \right )^{2}}-\frac {\ln \left (-b x +a \right )}{b \,c^{3}}\) | \(48\) |
parallelrisch | \(\frac {-\ln \left (b x -a \right ) x^{2} a^{2} b^{2}+2 \ln \left (b x -a \right ) x \,a^{3} b +2 a^{2} b^{2} x^{2}-\ln \left (b x -a \right ) a^{4}}{a^{2} c^{3} \left (b x -a \right )^{2} b}\) | \(79\) |
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Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b x)^2}{(a c-b c x)^3} \, dx=\frac {4 \, a b x - 2 \, a^{2} - {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{3} x + a^{2} b c^{3}} \]
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Time = 0.40 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^2}{(a c-b c x)^3} \, dx=- \frac {2 a^{2} - 4 a b x}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} - \frac {\log {\left (- a + b x \right )}}{b c^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^2}{(a c-b c x)^3} \, dx=\frac {2 \, {\left (2 \, a b x - a^{2}\right )}}{b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{3} x + a^{2} b c^{3}} - \frac {\log \left (b x - a\right )}{b c^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^2}{(a c-b c x)^3} \, dx=-\frac {\log \left ({\left | b x - a \right |}\right )}{b c^{3}} + \frac {2 \, {\left (2 \, a b x - a^{2}\right )}}{{\left (b x - a\right )}^{2} b c^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^2}{(a c-b c x)^3} \, dx=\frac {4\,a\,x-\frac {2\,a^2}{b}}{a^2\,c^3-2\,a\,b\,c^3\,x+b^2\,c^3\,x^2}-\frac {\ln \left (b\,x-a\right )}{b\,c^3} \]
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