Optimal. Leaf size=58 \[ -\frac{a^4}{b^5 (a+b x)}+\frac{3 a^2 x}{b^4}-\frac{4 a^3 \log (a+b x)}{b^5}-\frac{a x^2}{b^3}+\frac{x^3}{3 b^2} \]
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Rubi [A] time = 0.040053, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 43} \[ -\frac{a^4}{b^5 (a+b x)}+\frac{3 a^2 x}{b^4}-\frac{4 a^3 \log (a+b x)}{b^5}-\frac{a x^2}{b^3}+\frac{x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 43
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a x^2+b x^3\right )^2} \, dx &=\int \frac{x^4}{(a+b x)^2} \, dx\\ &=\int \left (\frac{3 a^2}{b^4}-\frac{2 a x}{b^3}+\frac{x^2}{b^2}+\frac{a^4}{b^4 (a+b x)^2}-\frac{4 a^3}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{3 a^2 x}{b^4}-\frac{a x^2}{b^3}+\frac{x^3}{3 b^2}-\frac{a^4}{b^5 (a+b x)}-\frac{4 a^3 \log (a+b x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0200056, size = 54, normalized size = 0.93 \[ \frac{-\frac{3 a^4}{a+b x}+9 a^2 b x-12 a^3 \log (a+b x)-3 a b^2 x^2+b^3 x^3}{3 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 57, normalized size = 1. \begin{align*} 3\,{\frac{{a}^{2}x}{{b}^{4}}}-{\frac{a{x}^{2}}{{b}^{3}}}+{\frac{{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{4}}{{b}^{5} \left ( bx+a \right ) }}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976477, size = 80, normalized size = 1.38 \begin{align*} -\frac{a^{4}}{b^{6} x + a b^{5}} - \frac{4 \, a^{3} \log \left (b x + a\right )}{b^{5}} + \frac{b^{2} x^{3} - 3 \, a b x^{2} + 9 \, a^{2} x}{3 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.67851, size = 155, normalized size = 2.67 \begin{align*} \frac{b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \,{\left (a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{3 \,{\left (b^{6} x + a b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.19462, size = 54, normalized size = 0.93 \begin{align*} - \frac{a^{4}}{a b^{5} + b^{6} x} - \frac{4 a^{3} \log{\left (a + b x \right )}}{b^{5}} + \frac{3 a^{2} x}{b^{4}} - \frac{a x^{2}}{b^{3}} + \frac{x^{3}}{3 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15278, size = 84, normalized size = 1.45 \begin{align*} -\frac{4 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac{a^{4}}{{\left (b x + a\right )} b^{5}} + \frac{b^{4} x^{3} - 3 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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