Optimal. Leaf size=87 \[ \frac{1}{16 a^4 b (a-b x)}-\frac{3}{16 a^4 b (a+b x)}-\frac{1}{8 a^3 b (a+b x)^2}-\frac{1}{12 a^2 b (a+b x)^3}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^5 b} \]
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Rubi [A] time = 0.0527658, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {627, 44, 208} \[ \frac{1}{16 a^4 b (a-b x)}-\frac{3}{16 a^4 b (a+b x)}-\frac{1}{8 a^3 b (a+b x)^2}-\frac{1}{12 a^2 b (a+b x)^3}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^5 b} \]
Antiderivative was successfully verified.
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Rule 627
Rule 44
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^2 \left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a-b x)^2 (a+b x)^4} \, dx\\ &=\int \left (\frac{1}{16 a^4 (a-b x)^2}+\frac{1}{4 a^2 (a+b x)^4}+\frac{1}{4 a^3 (a+b x)^3}+\frac{3}{16 a^4 (a+b x)^2}+\frac{1}{4 a^4 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{16 a^4 b (a-b x)}-\frac{1}{12 a^2 b (a+b x)^3}-\frac{1}{8 a^3 b (a+b x)^2}-\frac{3}{16 a^4 b (a+b x)}+\frac{\int \frac{1}{a^2-b^2 x^2} \, dx}{4 a^4}\\ &=\frac{1}{16 a^4 b (a-b x)}-\frac{1}{12 a^2 b (a+b x)^3}-\frac{1}{8 a^3 b (a+b x)^2}-\frac{3}{16 a^4 b (a+b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^5 b}\\ \end{align*}
Mathematica [A] time = 0.040318, size = 75, normalized size = 0.86 \[ \frac{\frac{2 a \left (a^2 b x-4 a^3+6 a b^2 x^2+3 b^3 x^3\right )}{(a-b x) (a+b x)^3}-3 \log (a-b x)+3 \log (a+b x)}{24 a^5 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 94, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{8\,{a}^{5}b}}-{\frac{3}{16\,{a}^{4}b \left ( bx+a \right ) }}-{\frac{1}{8\,{a}^{3}b \left ( bx+a \right ) ^{2}}}-{\frac{1}{12\,b{a}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{\ln \left ( bx-a \right ) }{8\,{a}^{5}b}}-{\frac{1}{16\,{a}^{4}b \left ( bx-a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01817, size = 136, normalized size = 1.56 \begin{align*} -\frac{3 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + a^{2} b x - 4 \, a^{3}}{12 \,{\left (a^{4} b^{5} x^{4} + 2 \, a^{5} b^{4} x^{3} - 2 \, a^{7} b^{2} x - a^{8} b\right )}} + \frac{\log \left (b x + a\right )}{8 \, a^{5} b} - \frac{\log \left (b x - a\right )}{8 \, a^{5} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79541, size = 308, normalized size = 3.54 \begin{align*} -\frac{6 \, a b^{3} x^{3} + 12 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b x - 8 \, a^{4} - 3 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} - 2 \, a^{3} b x - a^{4}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} - 2 \, a^{3} b x - a^{4}\right )} \log \left (b x - a\right )}{24 \,{\left (a^{5} b^{5} x^{4} + 2 \, a^{6} b^{4} x^{3} - 2 \, a^{8} b^{2} x - a^{9} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.668692, size = 92, normalized size = 1.06 \begin{align*} - \frac{- 4 a^{3} + a^{2} b x + 6 a b^{2} x^{2} + 3 b^{3} x^{3}}{- 12 a^{8} b - 24 a^{7} b^{2} x + 24 a^{5} b^{4} x^{3} + 12 a^{4} b^{5} x^{4}} + \frac{- \frac{\log{\left (- \frac{a}{b} + x \right )}}{8} + \frac{\log{\left (\frac{a}{b} + x \right )}}{8}}{a^{5} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19928, size = 134, normalized size = 1.54 \begin{align*} -\frac{\log \left ({\left | -\frac{2 \, a}{b x + a} + 1 \right |}\right )}{8 \, a^{5} b} + \frac{1}{32 \, a^{5} b{\left (\frac{2 \, a}{b x + a} - 1\right )}} - \frac{\frac{9 \, a^{2} b^{5}}{b x + a} + \frac{6 \, a^{3} b^{5}}{{\left (b x + a\right )}^{2}} + \frac{4 \, a^{4} b^{5}}{{\left (b x + a\right )}^{3}}}{48 \, a^{6} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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