Integrand size = 22, antiderivative size = 44 \[ \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^2} \, dx=5 a x+\frac {b x^2}{2}+\frac {8 a^3}{b (a-b x)}+\frac {12 a^2 \log (a-b x)}{b} \]
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Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {641, 45} \[ \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^2} \, dx=\frac {8 a^3}{b (a-b x)}+\frac {12 a^2 \log (a-b x)}{b}+5 a x+\frac {b x^2}{2} \]
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Rule 45
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^3}{(a-b x)^2} \, dx \\ & = \int \left (5 a+b x+\frac {8 a^3}{(a-b x)^2}-\frac {12 a^2}{a-b x}\right ) \, dx \\ & = 5 a x+\frac {b x^2}{2}+\frac {8 a^3}{b (a-b x)}+\frac {12 a^2 \log (a-b x)}{b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^2} \, dx=5 a x+\frac {b x^2}{2}-\frac {8 a^3}{b (-a+b x)}+\frac {12 a^2 \log (a-b x)}{b} \]
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Time = 2.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98
method | result | size |
default | \(5 a x +\frac {b \,x^{2}}{2}+\frac {8 a^{3}}{b \left (-b x +a \right )}+\frac {12 a^{2} \ln \left (-b x +a \right )}{b}\) | \(43\) |
risch | \(5 a x +\frac {b \,x^{2}}{2}+\frac {8 a^{3}}{b \left (-b x +a \right )}+\frac {12 a^{2} \ln \left (-b x +a \right )}{b}\) | \(43\) |
norman | \(\frac {13 a^{3} x -\frac {b^{3} x^{4}}{2}-5 a \,b^{2} x^{3}+\frac {17 a^{4}}{2 b}}{-b^{2} x^{2}+a^{2}}+\frac {12 a^{2} \ln \left (-b x +a \right )}{b}\) | \(64\) |
parallelrisch | \(\frac {b^{3} x^{3}+24 \ln \left (b x -a \right ) x \,a^{2} b +9 a \,b^{2} x^{2}-24 a^{3} \ln \left (b x -a \right )-26 a^{3}}{2 \left (b x -a \right ) b}\) | \(65\) |
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Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^2} \, dx=\frac {b^{3} x^{3} + 9 \, a b^{2} x^{2} - 10 \, a^{2} b x - 16 \, a^{3} + 24 \, {\left (a^{2} b x - a^{3}\right )} \log \left (b x - a\right )}{2 \, {\left (b^{2} x - a b\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^2} \, dx=- \frac {8 a^{3}}{- a b + b^{2} x} + \frac {12 a^{2} \log {\left (- a + b x \right )}}{b} + 5 a x + \frac {b x^{2}}{2} \]
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Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^2} \, dx=\frac {1}{2} \, b x^{2} - \frac {8 \, a^{3}}{b^{2} x - a b} + 5 \, a x + \frac {12 \, a^{2} \log \left (b x - a\right )}{b} \]
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Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^2} \, dx=\frac {12 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {8 \, a^{3}}{{\left (b x - a\right )} b} + \frac {b^{5} x^{2} + 10 \, a b^{4} x}{2 \, b^{4}} \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^2} \, dx=5\,a\,x+\frac {b\,x^2}{2}+\frac {8\,a^3}{b\,\left (a-b\,x\right )}+\frac {12\,a^2\,\ln \left (a-b\,x\right )}{b} \]
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