Optimal. Leaf size=41 \[ -\frac{a^2}{2 b^3 (a+b x)^2}+\frac{2 a}{b^3 (a+b x)}+\frac{\log (a+b x)}{b^3} \]
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Rubi [A] time = 0.020299, antiderivative size = 41, 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} \[ -\frac{a^2}{2 b^3 (a+b x)^2}+\frac{2 a}{b^3 (a+b x)}+\frac{\log (a+b x)}{b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{(a+b x)^3} \, dx &=\int \left (\frac{a^2}{b^2 (a+b x)^3}-\frac{2 a}{b^2 (a+b x)^2}+\frac{1}{b^2 (a+b x)}\right ) \, dx\\ &=-\frac{a^2}{2 b^3 (a+b x)^2}+\frac{2 a}{b^3 (a+b x)}+\frac{\log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0194683, size = 33, normalized size = 0.8 \[ \frac{\frac{a (3 a+4 b x)}{(a+b x)^2}+2 \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 40, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+2\,{\frac{a}{{b}^{3} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01963, size = 65, normalized size = 1.59 \begin{align*} \frac{4 \, a b x + 3 \, a^{2}}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{\log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56722, size = 132, normalized size = 3.22 \begin{align*} \frac{4 \, a b x + 3 \, a^{2} + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.533647, size = 46, normalized size = 1.12 \begin{align*} \frac{3 a^{2} + 4 a b x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac{\log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14395, size = 50, normalized size = 1.22 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{4 \, a x + \frac{3 \, a^{2}}{b}}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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