Optimal. Leaf size=63 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{4 a^3 b c^3}+\frac {1}{4 a^2 b c^3 (a-b x)}+\frac {1}{4 a b c^3 (a-b x)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
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*}
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Mathematica [A] time = 0.02, size = 65, normalized size = 1.03 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a+b x) (a c-b c x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.49, size = 98, normalized size = 1.56 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 69, normalized size = 1.10 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 78, normalized size = 1.24 \begin {gather*} \frac {1}{4 \left (b x -a \right )^{2} a b \,c^{3}}-\frac {1}{4 \left (b x -a \right ) a^{2} b \,c^{3}}-\frac {\ln \left (b x -a \right )}{8 a^{3} b \,c^{3}}+\frac {\ln \left (b x +a \right )}{8 a^{3} b \,c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 82, normalized size = 1.30 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 64, normalized size = 1.02 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{4\,a^3\,b\,c^3}-\frac {\frac {x}{4\,a^2}-\frac {1}{2\,a\,b}}{a^2\,c^3-2\,a\,b\,c^3\,x+b^2\,c^3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 71, normalized size = 1.13 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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