Optimal. Leaf size=49 \[ \frac{4 a^3}{b (a-b x)^2}-\frac{12 a^2}{b (a-b x)}-\frac{6 a \log (a-b x)}{b}-x \]
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Rubi [A] time = 0.0295678, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {627, 43} \[ \frac{4 a^3}{b (a-b x)^2}-\frac{12 a^2}{b (a-b x)}-\frac{6 a \log (a-b x)}{b}-x \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^6}{\left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac{(a+b x)^3}{(a-b x)^3} \, dx\\ &=\int \left (-1+\frac{8 a^3}{(a-b x)^3}-\frac{12 a^2}{(a-b x)^2}+\frac{6 a}{a-b x}\right ) \, dx\\ &=-x+\frac{4 a^3}{b (a-b x)^2}-\frac{12 a^2}{b (a-b x)}-\frac{6 a \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.044352, size = 41, normalized size = 0.84 \[ \frac{4 a^2 (3 b x-2 a)}{b (a-b x)^2}-\frac{6 a \log (a-b x)}{b}-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 53, normalized size = 1.1 \begin{align*} -x+4\,{\frac{{a}^{3}}{b \left ( bx-a \right ) ^{2}}}+12\,{\frac{{a}^{2}}{b \left ( bx-a \right ) }}-6\,{\frac{a\ln \left ( bx-a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03819, size = 74, normalized size = 1.51 \begin{align*} -x - \frac{6 \, a \log \left (b x - a\right )}{b} + \frac{4 \,{\left (3 \, a^{2} b x - 2 \, a^{3}\right )}}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74147, size = 167, normalized size = 3.41 \begin{align*} -\frac{b^{3} x^{3} - 2 \, a b^{2} x^{2} - 11 \, a^{2} b x + 8 \, a^{3} + 6 \,{\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \log \left (b x - a\right )}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.426304, size = 46, normalized size = 0.94 \begin{align*} - \frac{6 a \log{\left (- a + b x \right )}}{b} - x + \frac{- 8 a^{3} + 12 a^{2} b x}{a^{2} b - 2 a b^{2} x + b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17542, size = 62, normalized size = 1.27 \begin{align*} -x - \frac{6 \, a \log \left ({\left | b x - a \right |}\right )}{b} + \frac{4 \,{\left (3 \, a^{2} b x - 2 \, a^{3}\right )}}{{\left (b x - a\right )}^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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