Integrand size = 22, antiderivative size = 69 \[ \int \frac {1}{(a+b x)^3 \left (a^2-b^2 x^2\right )} \, dx=-\frac {1}{6 a b (a+b x)^3}-\frac {1}{8 a^2 b (a+b x)^2}-\frac {1}{8 a^3 b (a+b x)}+\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{8 a^4 b} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {641, 46, 214} \[ \int \frac {1}{(a+b x)^3 \left (a^2-b^2 x^2\right )} \, dx=\frac {\text {arctanh}\left (\frac {b x}{a}\right )}{8 a^4 b}-\frac {1}{8 a^3 b (a+b x)}-\frac {1}{8 a^2 b (a+b x)^2}-\frac {1}{6 a b (a+b x)^3} \]
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Rule 46
Rule 214
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a-b x) (a+b x)^4} \, dx \\ & = \int \left (\frac {1}{2 a (a+b x)^4}+\frac {1}{4 a^2 (a+b x)^3}+\frac {1}{8 a^3 (a+b x)^2}+\frac {1}{8 a^3 \left (a^2-b^2 x^2\right )}\right ) \, dx \\ & = -\frac {1}{6 a b (a+b x)^3}-\frac {1}{8 a^2 b (a+b x)^2}-\frac {1}{8 a^3 b (a+b x)}+\frac {\int \frac {1}{a^2-b^2 x^2} \, dx}{8 a^3} \\ & = -\frac {1}{6 a b (a+b x)^3}-\frac {1}{8 a^2 b (a+b x)^2}-\frac {1}{8 a^3 b (a+b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{8 a^4 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(a+b x)^3 \left (a^2-b^2 x^2\right )} \, dx=\frac {-2 a \left (10 a^2+9 a b x+3 b^2 x^2\right )-3 (a+b x)^3 \log (a-b x)+3 (a+b x)^3 \log (a+b x)}{48 a^4 b (a+b x)^3} \]
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Time = 2.33 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91
method | result | size |
norman | \(\frac {-\frac {5}{12 b a}-\frac {3 x}{8 a^{2}}-\frac {b \,x^{2}}{8 a^{3}}}{\left (b x +a \right )^{3}}-\frac {\ln \left (-b x +a \right )}{16 a^{4} b}+\frac {\ln \left (b x +a \right )}{16 a^{4} b}\) | \(63\) |
risch | \(\frac {-\frac {5}{12 b a}-\frac {3 x}{8 a^{2}}-\frac {b \,x^{2}}{8 a^{3}}}{\left (b x +a \right )^{3}}-\frac {\ln \left (-b x +a \right )}{16 a^{4} b}+\frac {\ln \left (b x +a \right )}{16 a^{4} b}\) | \(63\) |
default | \(\frac {\ln \left (b x +a \right )}{16 a^{4} b}-\frac {1}{8 a^{3} b \left (b x +a \right )}-\frac {1}{8 a^{2} b \left (b x +a \right )^{2}}-\frac {1}{6 a b \left (b x +a \right )^{3}}-\frac {\ln \left (-b x +a \right )}{16 a^{4} b}\) | \(76\) |
parallelrisch | \(-\frac {3 \ln \left (b x -a \right ) x^{3} b^{5}-3 \ln \left (b x +a \right ) x^{3} b^{5}+9 \ln \left (b x -a \right ) x^{2} a \,b^{4}-9 \ln \left (b x +a \right ) x^{2} a \,b^{4}+9 \ln \left (b x -a \right ) x \,a^{2} b^{3}-9 \ln \left (b x +a \right ) x \,a^{2} b^{3}+6 b^{4} x^{2} a +3 \ln \left (b x -a \right ) a^{3} b^{2}-3 \ln \left (b x +a \right ) a^{3} b^{2}+18 a^{2} b^{3} x +20 a^{3} b^{2}}{48 a^{4} b^{3} \left (b x +a \right )^{3}}\) | \(167\) |
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (61) = 122\).
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(a+b x)^3 \left (a^2-b^2 x^2\right )} \, dx=-\frac {6 \, a b^{2} x^{2} + 18 \, a^{2} b x + 20 \, a^{3} - 3 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right ) + 3 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x - a\right )}{48 \, {\left (a^{4} b^{4} x^{3} + 3 \, a^{5} b^{3} x^{2} + 3 \, a^{6} b^{2} x + a^{7} b\right )}} \]
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Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(a+b x)^3 \left (a^2-b^2 x^2\right )} \, dx=- \frac {10 a^{2} + 9 a b x + 3 b^{2} x^{2}}{24 a^{6} b + 72 a^{5} b^{2} x + 72 a^{4} b^{3} x^{2} + 24 a^{3} b^{4} x^{3}} - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{16} - \frac {\log {\left (\frac {a}{b} + x \right )}}{16}}{a^{4} b} \]
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none
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(a+b x)^3 \left (a^2-b^2 x^2\right )} \, dx=-\frac {3 \, b^{2} x^{2} + 9 \, a b x + 10 \, a^{2}}{24 \, {\left (a^{3} b^{4} x^{3} + 3 \, a^{4} b^{3} x^{2} + 3 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {\log \left (b x + a\right )}{16 \, a^{4} b} - \frac {\log \left (b x - a\right )}{16 \, a^{4} b} \]
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Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(a+b x)^3 \left (a^2-b^2 x^2\right )} \, dx=\frac {\log \left ({\left | b x + a \right |}\right )}{16 \, a^{4} b} - \frac {\log \left ({\left | b x - a \right |}\right )}{16 \, a^{4} b} - \frac {3 \, a b^{2} x^{2} + 9 \, a^{2} b x + 10 \, a^{3}}{24 \, {\left (b x + a\right )}^{3} a^{4} b} \]
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Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(a+b x)^3 \left (a^2-b^2 x^2\right )} \, dx=\frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{8\,a^4\,b}-\frac {\frac {3\,x}{8\,a^2}+\frac {5}{12\,a\,b}+\frac {b\,x^2}{8\,a^3}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]
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