Optimal. Leaf size=52 \[ -\frac{1}{4 a^2 b (a+b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^3 b}-\frac{1}{4 a b (a+b x)^2} \]
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Rubi [A] time = 0.0373936, antiderivative size = 52, 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} \[ -\frac{1}{4 a^2 b (a+b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^3 b}-\frac{1}{4 a b (a+b x)^2} \]
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 )} \, dx &=\int \frac{1}{(a-b x) (a+b x)^3} \, dx\\ &=\int \left (\frac{1}{2 a (a+b x)^3}+\frac{1}{4 a^2 (a+b x)^2}+\frac{1}{4 a^2 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=-\frac{1}{4 a b (a+b x)^2}-\frac{1}{4 a^2 b (a+b x)}+\frac{\int \frac{1}{a^2-b^2 x^2} \, dx}{4 a^2}\\ &=-\frac{1}{4 a b (a+b x)^2}-\frac{1}{4 a^2 b (a+b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^3 b}\\ \end{align*}
Mathematica [A] time = 0.0168035, size = 58, normalized size = 1.12 \[ \frac{-2 a (2 a+b x)+(a+b x)^2 (-\log (a-b x))+(a+b x)^2 \log (a+b x)}{8 a^3 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 62, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{8\,b{a}^{3}}}-{\frac{1}{4\,b{a}^{2} \left ( bx+a \right ) }}-{\frac{1}{4\,ab \left ( bx+a \right ) ^{2}}}-{\frac{\ln \left ( bx-a \right ) }{8\,b{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14737, size = 90, normalized size = 1.73 \begin{align*} -\frac{b x + 2 \, a}{4 \,{\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac{\log \left (b x + a\right )}{8 \, a^{3} b} - \frac{\log \left (b x - a\right )}{8 \, a^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85799, size = 192, normalized size = 3.69 \begin{align*} -\frac{2 \, a b x + 4 \, a^{2} -{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{8 \,{\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.492546, size = 58, normalized size = 1.12 \begin{align*} - \frac{2 a + b x}{4 a^{4} b + 8 a^{3} b^{2} x + 4 a^{2} b^{3} x^{2}} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{8} - \frac{\log{\left (\frac{a}{b} + x \right )}}{8}}{a^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20897, size = 69, normalized size = 1.33 \begin{align*} -\frac{\frac{b}{b x + a} + \frac{a b}{{\left (b x + a\right )}^{2}}}{4 \, a^{2} b^{2}} - \frac{\log \left ({\left | -\frac{2 \, a}{b x + a} + 1 \right |}\right )}{8 \, a^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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