Optimal. Leaf size=50 \[ \frac{a^3}{2 b^4 (a+b x)^2}-\frac{3 a^2}{b^4 (a+b x)}-\frac{3 a \log (a+b x)}{b^4}+\frac{x}{b^3} \]
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Rubi [A] time = 0.0262389, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{a^3}{2 b^4 (a+b x)^2}-\frac{3 a^2}{b^4 (a+b x)}-\frac{3 a \log (a+b x)}{b^4}+\frac{x}{b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{(a+b x)^3} \, dx &=\int \left (\frac{1}{b^3}-\frac{a^3}{b^3 (a+b x)^3}+\frac{3 a^2}{b^3 (a+b x)^2}-\frac{3 a}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{x}{b^3}+\frac{a^3}{2 b^4 (a+b x)^2}-\frac{3 a^2}{b^4 (a+b x)}-\frac{3 a \log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0613447, size = 40, normalized size = 0.8 \[ -\frac{\frac{a^2 (5 a+6 b x)}{(a+b x)^2}+6 a \log (a+b x)-2 b x}{2 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 49, normalized size = 1. \begin{align*}{\frac{x}{{b}^{3}}}+{\frac{{a}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-3\,{\frac{{a}^{2}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{a\ln \left ( bx+a \right ) }{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0621, size = 77, normalized size = 1.54 \begin{align*} -\frac{6 \, a^{2} b x + 5 \, a^{3}}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac{x}{b^{3}} - \frac{3 \, a \log \left (b x + a\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51677, size = 176, normalized size = 3.52 \begin{align*} \frac{2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} - 4 \, a^{2} b x - 5 \, a^{3} - 6 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.600725, size = 56, normalized size = 1.12 \begin{align*} - \frac{3 a \log{\left (a + b x \right )}}{b^{4}} - \frac{5 a^{3} + 6 a^{2} b x}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac{x}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15452, size = 59, normalized size = 1.18 \begin{align*} \frac{x}{b^{3}} - \frac{3 \, a \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{6 \, a^{2} b x + 5 \, a^{3}}{2 \,{\left (b x + a\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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