Optimal. Leaf size=69 \[ -\frac{b^3}{a^4 (a+b x)}-\frac{3 b^2}{a^4 x}-\frac{4 b^3 \log (x)}{a^5}+\frac{4 b^3 \log (a+b x)}{a^5}+\frac{b}{a^3 x^2}-\frac{1}{3 a^2 x^3} \]
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Rubi [A] time = 0.0389185, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1593, 44} \[ -\frac{b^3}{a^4 (a+b x)}-\frac{3 b^2}{a^4 x}-\frac{4 b^3 \log (x)}{a^5}+\frac{4 b^3 \log (a+b x)}{a^5}+\frac{b}{a^3 x^2}-\frac{1}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a x^2+b x^3\right )^2} \, dx &=\int \frac{1}{x^4 (a+b x)^2} \, dx\\ &=\int \left (\frac{1}{a^2 x^4}-\frac{2 b}{a^3 x^3}+\frac{3 b^2}{a^4 x^2}-\frac{4 b^3}{a^5 x}+\frac{b^4}{a^4 (a+b x)^2}+\frac{4 b^4}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{1}{3 a^2 x^3}+\frac{b}{a^3 x^2}-\frac{3 b^2}{a^4 x}-\frac{b^3}{a^4 (a+b x)}-\frac{4 b^3 \log (x)}{a^5}+\frac{4 b^3 \log (a+b x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0528533, size = 66, normalized size = 0.96 \[ -\frac{\frac{a \left (-2 a^2 b x+a^3+6 a b^2 x^2+12 b^3 x^3\right )}{x^3 (a+b x)}-12 b^3 \log (a+b x)+12 b^3 \log (x)}{3 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 68, normalized size = 1. \begin{align*} -{\frac{1}{3\,{x}^{3}{a}^{2}}}+{\frac{b}{{x}^{2}{a}^{3}}}-3\,{\frac{{b}^{2}}{{a}^{4}x}}-{\frac{{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}-4\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{5}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986677, size = 99, normalized size = 1.43 \begin{align*} -\frac{12 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}}{3 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac{4 \, b^{3} \log \left (b x + a\right )}{a^{5}} - \frac{4 \, b^{3} \log \left (x\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.8104, size = 204, normalized size = 2.96 \begin{align*} -\frac{12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4} - 12 \,{\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.76697, size = 66, normalized size = 0.96 \begin{align*} - \frac{a^{3} - 2 a^{2} b x + 6 a b^{2} x^{2} + 12 b^{3} x^{3}}{3 a^{5} x^{3} + 3 a^{4} b x^{4}} + \frac{4 b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16329, size = 99, normalized size = 1.43 \begin{align*} \frac{4 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{5}} - \frac{4 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac{12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}}{3 \,{\left (b x + a\right )} a^{5} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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