Optimal. Leaf size=71 \[ \frac{16 a^5}{b (a-b x)^2}-\frac{80 a^4}{b (a-b x)}-\frac{80 a^3 \log (a-b x)}{b}-31 a^2 x-4 a b x^2-\frac{b^2 x^3}{3} \]
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Rubi [A] time = 0.0514683, antiderivative size = 71, 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} \[ \frac{16 a^5}{b (a-b x)^2}-\frac{80 a^4}{b (a-b x)}-\frac{80 a^3 \log (a-b x)}{b}-31 a^2 x-4 a b x^2-\frac{b^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^8}{\left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac{(a+b x)^5}{(a-b x)^3} \, dx\\ &=\int \left (-31 a^2-8 a b x-b^2 x^2+\frac{32 a^5}{(a-b x)^3}-\frac{80 a^4}{(a-b x)^2}+\frac{80 a^3}{a-b x}\right ) \, dx\\ &=-31 a^2 x-4 a b x^2-\frac{b^2 x^3}{3}+\frac{16 a^5}{b (a-b x)^2}-\frac{80 a^4}{b (a-b x)}-\frac{80 a^3 \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0330701, size = 73, normalized size = 1.03 \[ \frac{16 a^5}{b (b x-a)^2}+\frac{80 a^4}{b (b x-a)}-\frac{80 a^3 \log (a-b x)}{b}-31 a^2 x-4 a b x^2-\frac{b^2 x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 73, normalized size = 1. \begin{align*} -{\frac{{b}^{2}{x}^{3}}{3}}-4\,ab{x}^{2}-31\,{a}^{2}x+16\,{\frac{{a}^{5}}{b \left ( bx-a \right ) ^{2}}}+80\,{\frac{{a}^{4}}{b \left ( bx-a \right ) }}-80\,{\frac{{a}^{3}\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 = 1.08801, size = 101, normalized size = 1.42 \begin{align*} -\frac{1}{3} \, b^{2} x^{3} - 4 \, a b x^{2} - 31 \, a^{2} x - \frac{80 \, a^{3} \log \left (b x - a\right )}{b} + \frac{16 \,{\left (5 \, a^{4} b x - 4 \, a^{5}\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 = 1.70623, size = 231, normalized size = 3.25 \begin{align*} -\frac{b^{5} x^{5} + 10 \, a b^{4} x^{4} + 70 \, a^{2} b^{3} x^{3} - 174 \, a^{3} b^{2} x^{2} - 147 \, a^{4} b x + 192 \, a^{5} + 240 \,{\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )} \log \left (b x - a\right )}{3 \,{\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 = 0.513292, size = 70, normalized size = 0.99 \begin{align*} - \frac{80 a^{3} \log{\left (- a + b x \right )}}{b} - 31 a^{2} x - 4 a b x^{2} - \frac{b^{2} x^{3}}{3} + \frac{- 64 a^{5} + 80 a^{4} 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 = 1.23523, size = 103, normalized size = 1.45 \begin{align*} -\frac{80 \, a^{3} \log \left ({\left | b x - a \right |}\right )}{b} + \frac{16 \,{\left (5 \, a^{4} b x - 4 \, a^{5}\right )}}{{\left (b x - a\right )}^{2} b} - \frac{b^{11} x^{3} + 12 \, a b^{10} x^{2} + 93 \, a^{2} b^{9} x}{3 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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