Optimal. Leaf size=63 \[ \frac{1}{4 a^2 b c^3 (a-b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^3 b c^3}+\frac{1}{4 a b c^3 (a-b x)^2} \]
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Rubi [A] time = 0.0385831, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {44, 208} \[ \frac{1}{4 a^2 b c^3 (a-b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^3 b c^3}+\frac{1}{4 a b c^3 (a-b x)^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) (a c-b c x)^3} \, dx &=\int \left (\frac{1}{2 a c^3 (a-b x)^3}+\frac{1}{4 a^2 c^3 (a-b x)^2}+\frac{1}{4 a^2 c^3 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{4 a b c^3 (a-b x)^2}+\frac{1}{4 a^2 b c^3 (a-b x)}+\frac{\int \frac{1}{a^2-b^2 x^2} \, dx}{4 a^2 c^3}\\ &=\frac{1}{4 a b c^3 (a-b x)^2}+\frac{1}{4 a^2 b c^3 (a-b x)}+\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^3 b c^3}\\ \end{align*}
Mathematica [A] time = 0.0205471, size = 65, normalized size = 1.03 \[ \frac{2 a (2 a-b x)+(a-b x)^2 (-\log (a-b x))+(a-b x)^2 \log (a+b x)}{8 a^3 b c^3 (a-b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 78, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{8\,{c}^{3}{a}^{3}b}}-{\frac{\ln \left ( bx-a \right ) }{8\,{c}^{3}{a}^{3}b}}-{\frac{1}{4\,{c}^{3}{a}^{2}b \left ( bx-a \right ) }}+{\frac{1}{4\,{c}^{3}ba \left ( bx-a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02586, size = 111, normalized size = 1.76 \begin{align*} -\frac{b x - 2 \, a}{4 \,{\left (a^{2} b^{3} c^{3} x^{2} - 2 \, a^{3} b^{2} c^{3} x + a^{4} b c^{3}\right )}} + \frac{\log \left (b x + a\right )}{8 \, a^{3} b c^{3}} - \frac{\log \left (b x - a\right )}{8 \, a^{3} b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51312, size = 208, normalized size = 3.3 \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} c^{3} x^{2} - 2 \, a^{4} b^{2} c^{3} x + a^{5} b c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.521837, size = 71, normalized size = 1.13 \begin{align*} - \frac{- 2 a + b x}{4 a^{4} b c^{3} - 8 a^{3} b^{2} c^{3} x + 4 a^{2} b^{3} c^{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 c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07019, size = 93, normalized size = 1.48 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{8 \, a^{3} b c^{3}} - \frac{\log \left ({\left | b x - a \right |}\right )}{8 \, a^{3} b c^{3}} - \frac{a b x - 2 \, a^{2}}{4 \,{\left (b x - a\right )}^{2} a^{3} b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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