Optimal. Leaf size=83 \[ \frac{1}{4 a^3 b c^3 (a-b x)}-\frac{1}{8 a^3 b c^3 (a+b x)}+\frac{1}{8 a^2 b c^3 (a-b x)^2}+\frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b c^3} \]
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Rubi [A] time = 0.0495295, antiderivative size = 83, 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} \[ \frac{1}{4 a^3 b c^3 (a-b x)}-\frac{1}{8 a^3 b c^3 (a+b x)}+\frac{1}{8 a^2 b c^3 (a-b x)^2}+\frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b c^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^2 (a c-b c x)^3} \, dx &=\int \left (\frac{1}{4 a^2 c^3 (a-b x)^3}+\frac{1}{4 a^3 c^3 (a-b x)^2}+\frac{1}{8 a^3 c^3 (a+b x)^2}+\frac{3}{8 a^3 c^3 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{8 a^2 b c^3 (a-b x)^2}+\frac{1}{4 a^3 b c^3 (a-b x)}-\frac{1}{8 a^3 b c^3 (a+b x)}+\frac{3 \int \frac{1}{a^2-b^2 x^2} \, dx}{8 a^3 c^3}\\ &=\frac{1}{8 a^2 b c^3 (a-b x)^2}+\frac{1}{4 a^3 b c^3 (a-b x)}-\frac{1}{8 a^3 b c^3 (a+b x)}+\frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b c^3}\\ \end{align*}
Mathematica [A] time = 0.0355216, size = 68, normalized size = 0.82 \[ \frac{\frac{2 a \left (2 a^2+3 a b x-3 b^2 x^2\right )}{(a-b x)^2 (a+b x)}-3 \log (a-b x)+3 \log (a+b x)}{16 a^4 b c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 96, normalized size = 1.2 \begin{align*}{\frac{3\,\ln \left ( bx+a \right ) }{16\,{c}^{3}{a}^{4}b}}-{\frac{1}{8\,{a}^{3}b{c}^{3} \left ( bx+a \right ) }}-{\frac{3\,\ln \left ( bx-a \right ) }{16\,{c}^{3}{a}^{4}b}}-{\frac{1}{4\,{a}^{3}b{c}^{3} \left ( bx-a \right ) }}+{\frac{1}{8\,{c}^{3}{a}^{2}b \left ( bx-a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07872, size = 146, normalized size = 1.76 \begin{align*} -\frac{3 \, b^{2} x^{2} - 3 \, a b x - 2 \, a^{2}}{8 \,{\left (a^{3} b^{4} c^{3} x^{3} - a^{4} b^{3} c^{3} x^{2} - a^{5} b^{2} c^{3} x + a^{6} b c^{3}\right )}} + \frac{3 \, \log \left (b x + a\right )}{16 \, a^{4} b c^{3}} - \frac{3 \, \log \left (b x - a\right )}{16 \, a^{4} b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66517, size = 290, normalized size = 3.49 \begin{align*} -\frac{6 \, a b^{2} x^{2} - 6 \, a^{2} b x - 4 \, a^{3} - 3 \,{\left (b^{3} x^{3} - a b^{2} x^{2} - a^{2} b x + a^{3}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{3} x^{3} - a b^{2} x^{2} - a^{2} b x + a^{3}\right )} \log \left (b x - a\right )}{16 \,{\left (a^{4} b^{4} c^{3} x^{3} - a^{5} b^{3} c^{3} x^{2} - a^{6} b^{2} c^{3} x + a^{7} b c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.655645, size = 104, normalized size = 1.25 \begin{align*} - \frac{- 2 a^{2} - 3 a b x + 3 b^{2} x^{2}}{8 a^{6} b c^{3} - 8 a^{5} b^{2} c^{3} x - 8 a^{4} b^{3} c^{3} x^{2} + 8 a^{3} b^{4} c^{3} x^{3}} - \frac{\frac{3 \log{\left (- \frac{a}{b} + x \right )}}{16} - \frac{3 \log{\left (\frac{a}{b} + x \right )}}{16}}{a^{4} b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06237, size = 109, normalized size = 1.31 \begin{align*} -\frac{3 \, \log \left ({\left | -\frac{2 \, a}{b x + a} + 1 \right |}\right )}{16 \, a^{4} b c^{3}} - \frac{1}{8 \,{\left (b x + a\right )} a^{3} b c^{3}} + \frac{\frac{12 \, a}{b x + a} - 5}{32 \, a^{4} b c^{3}{\left (\frac{2 \, a}{b x + a} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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