Optimal. Leaf size=60 \[ \frac{8 a^4}{b (a-b x)^2}-\frac{32 a^3}{b (a-b x)}-\frac{24 a^2 \log (a-b x)}{b}-7 a x-\frac{b x^2}{2} \]
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Rubi [A] time = 0.0406074, antiderivative size = 60, 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} \[ \frac{8 a^4}{b (a-b x)^2}-\frac{32 a^3}{b (a-b x)}-\frac{24 a^2 \log (a-b x)}{b}-7 a x-\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^7}{\left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac{(a+b x)^4}{(a-b x)^3} \, dx\\ &=\int \left (-7 a-b x+\frac{16 a^4}{(a-b x)^3}-\frac{32 a^3}{(a-b x)^2}+\frac{24 a^2}{a-b x}\right ) \, dx\\ &=-7 a x-\frac{b x^2}{2}+\frac{8 a^4}{b (a-b x)^2}-\frac{32 a^3}{b (a-b x)}-\frac{24 a^2 \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0280393, size = 62, normalized size = 1.03 \[ \frac{8 a^4}{b (b x-a)^2}+\frac{32 a^3}{b (b x-a)}-\frac{24 a^2 \log (a-b x)}{b}-7 a x-\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 62, normalized size = 1. \begin{align*} -{\frac{b{x}^{2}}{2}}-7\,ax+8\,{\frac{{a}^{4}}{b \left ( bx-a \right ) ^{2}}}+32\,{\frac{{a}^{3}}{b \left ( bx-a \right ) }}-24\,{\frac{{a}^{2}\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 = 1.08461, size = 86, normalized size = 1.43 \begin{align*} -\frac{1}{2} \, b x^{2} - 7 \, a x - \frac{24 \, a^{2} \log \left (b x - a\right )}{b} + \frac{8 \,{\left (4 \, a^{3} b x - 3 \, a^{4}\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 = 1.62566, size = 203, normalized size = 3.38 \begin{align*} -\frac{b^{4} x^{4} + 12 \, a b^{3} x^{3} - 27 \, a^{2} b^{2} x^{2} - 50 \, a^{3} b x + 48 \, a^{4} + 48 \,{\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )} \log \left (b x - a\right )}{2 \,{\left (b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.495016, size = 58, normalized size = 0.97 \begin{align*} - \frac{24 a^{2} \log{\left (- a + b x \right )}}{b} - 7 a x - \frac{b x^{2}}{2} + \frac{- 24 a^{4} + 32 a^{3} 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 = 1.225, size = 88, normalized size = 1.47 \begin{align*} -\frac{24 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} + \frac{8 \,{\left (4 \, a^{3} b x - 3 \, a^{4}\right )}}{{\left (b x - a\right )}^{2} b} - \frac{b^{7} x^{2} + 14 \, a b^{6} x}{2 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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