Optimal. Leaf size=33 \[ -\frac{a^2}{b^3 (a+b x)}-\frac{2 a \log (a+b x)}{b^3}+\frac{x}{b^2} \]
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Rubi [A] time = 0.023916, antiderivative size = 33, 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} \[ -\frac{a^2}{b^3 (a+b x)}-\frac{2 a \log (a+b x)}{b^3}+\frac{x}{b^2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 43
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a x^2+b x^3\right )^2} \, dx &=\int \frac{x^2}{(a+b x)^2} \, dx\\ &=\int \left (\frac{1}{b^2}+\frac{a^2}{b^2 (a+b x)^2}-\frac{2 a}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{x}{b^2}-\frac{a^2}{b^3 (a+b x)}-\frac{2 a \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0122328, size = 29, normalized size = 0.88 \[ \frac{-\frac{a^2}{a+b x}-2 a \log (a+b x)+b x}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 34, normalized size = 1. \begin{align*}{\frac{x}{{b}^{2}}}-{\frac{{a}^{2}}{{b}^{3} \left ( bx+a \right ) }}-2\,{\frac{a\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 = 0.991994, size = 49, normalized size = 1.48 \begin{align*} -\frac{a^{2}}{b^{4} x + a b^{3}} + \frac{x}{b^{2}} - \frac{2 \, a \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 = 0.780113, size = 97, normalized size = 2.94 \begin{align*} \frac{b^{2} x^{2} + a b x - a^{2} - 2 \,{\left (a b x + a^{2}\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.558101, size = 31, normalized size = 0.94 \begin{align*} - \frac{a^{2}}{a b^{3} + b^{4} x} - \frac{2 a \log{\left (a + b x \right )}}{b^{3}} + \frac{x}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16743, size = 46, normalized size = 1.39 \begin{align*} \frac{x}{b^{2}} - \frac{2 \, a \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac{a^{2}}{{\left (b x + a\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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