Integrand size = 22, antiderivative size = 49 \[ \int \frac {(a+b x)^6}{\left (a^2-b^2 x^2\right )^3} \, dx=-x+\frac {4 a^3}{b (a-b x)^2}-\frac {12 a^2}{b (a-b x)}-\frac {6 a \log (a-b x)}{b} \]
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Time = 0.02 (sec) , antiderivative size = 49, 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} \[ \int \frac {(a+b x)^6}{\left (a^2-b^2 x^2\right )^3} \, dx=\frac {4 a^3}{b (a-b x)^2}-\frac {12 a^2}{b (a-b x)}-\frac {6 a \log (a-b x)}{b}-x \]
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Rule 45
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^3}{(a-b x)^3} \, dx \\ & = \int \left (-1+\frac {8 a^3}{(a-b x)^3}-\frac {12 a^2}{(a-b x)^2}+\frac {6 a}{a-b x}\right ) \, dx \\ & = -x+\frac {4 a^3}{b (a-b x)^2}-\frac {12 a^2}{b (a-b x)}-\frac {6 a \log (a-b x)}{b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^6}{\left (a^2-b^2 x^2\right )^3} \, dx=-x+\frac {4 a^2 (-2 a+3 b x)}{b (a-b x)^2}-\frac {6 a \log (a-b x)}{b} \]
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Time = 2.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86
method | result | size |
risch | \(-x +\frac {12 a^{2} x -\frac {8 a^{3}}{b}}{\left (-b x +a \right )^{2}}-\frac {6 a \ln \left (-b x +a \right )}{b}\) | \(42\) |
default | \(-x +\frac {4 a^{3}}{b \left (-b x +a \right )^{2}}-\frac {12 a^{2}}{b \left (-b x +a \right )}-\frac {6 a \ln \left (-b x +a \right )}{b}\) | \(50\) |
norman | \(\frac {-5 a^{4} x -b^{4} x^{5}+14 a^{2} b^{2} x^{3}-\frac {8 a^{5}}{b}+16 a^{3} b \,x^{2}}{\left (-b^{2} x^{2}+a^{2}\right )^{2}}-\frac {6 a \ln \left (-b x +a \right )}{b}\) | \(73\) |
parallelrisch | \(-\frac {6 \ln \left (b x -a \right ) x^{2} a \,b^{3}+b^{4} x^{3}-12 \ln \left (b x -a \right ) x \,a^{2} b^{2}+6 \ln \left (b x -a \right ) a^{3} b -15 a^{2} b^{2} x +10 a^{3} b}{b^{2} \left (b x -a \right )^{2}}\) | \(86\) |
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Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.67 \[ \int \frac {(a+b x)^6}{\left (a^2-b^2 x^2\right )^3} \, dx=-\frac {b^{3} x^{3} - 2 \, a b^{2} x^{2} - 11 \, a^{2} b x + 8 \, a^{3} + 6 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \log \left (b x - a\right )}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \]
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Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^6}{\left (a^2-b^2 x^2\right )^3} \, dx=- \frac {6 a \log {\left (- a + b x \right )}}{b} - x - \frac {8 a^{3} - 12 a^{2} b x}{a^{2} b - 2 a b^{2} x + b^{3} x^{2}} \]
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Time = 0.22 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^6}{\left (a^2-b^2 x^2\right )^3} \, dx=-x - \frac {6 \, a \log \left (b x - a\right )}{b} + \frac {4 \, {\left (3 \, a^{2} b x - 2 \, a^{3}\right )}}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \]
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Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^6}{\left (a^2-b^2 x^2\right )^3} \, dx=-x - \frac {6 \, a \log \left ({\left | b x - a \right |}\right )}{b} + \frac {4 \, {\left (3 \, a^{2} b x - 2 \, a^{3}\right )}}{{\left (b x - a\right )}^{2} b} \]
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Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b x)^6}{\left (a^2-b^2 x^2\right )^3} \, dx=\frac {12\,a^2\,x-\frac {8\,a^3}{b}}{a^2-2\,a\,b\,x+b^2\,x^2}-x-\frac {6\,a\,\ln \left (b\,x-a\right )}{b} \]
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