Optimal. Leaf size=69 \[ -\frac {4 b^3 \log (x)}{a^5}+\frac {4 b^3 \log (a+b x)}{a^5}-\frac {b^3}{a^4 (a+b x)}-\frac {3 b^2}{a^4 x}+\frac {b}{a^3 x^2}-\frac {1}{3 a^2 x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 44
Rule 1593
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*}
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Mathematica [A] time = 0.07, size = 66, normalized size = 0.96 \[ -\frac {\frac {a \left (a^3-2 a^2 b x+6 a b^2 x^2+12 b^3 x^3\right )}{x^3 (a+b x)}-12 b^3 \log (a+b x)+12 b^3 \log (x)}{3 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 95, normalized size = 1.38 \[ -\frac {12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4} - 12 \, {\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \relax (x)}{3 \, {\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 73, normalized size = 1.06 \[ \frac {4 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{5}} - \frac {4 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}}{3 \, {\left (b x + a\right )} a^{5} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 68, normalized size = 0.99 \[ -\frac {b^{3}}{\left (b x +a \right ) a^{4}}-\frac {4 b^{3} \ln \relax (x )}{a^{5}}+\frac {4 b^{3} \ln \left (b x +a \right )}{a^{5}}-\frac {3 b^{2}}{a^{4} x}+\frac {b}{a^{3} x^{2}}-\frac {1}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 73, normalized size = 1.06 \[ -\frac {12 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}}{3 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac {4 \, b^{3} \log \left (b x + a\right )}{a^{5}} - \frac {4 \, b^{3} \log \relax (x)}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 69, normalized size = 1.00 \[ \frac {8\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^5}-\frac {\frac {1}{3\,a}+\frac {2\,b^2\,x^2}{a^3}+\frac {4\,b^3\,x^3}{a^4}-\frac {2\,b\,x}{3\,a^2}}{b\,x^4+a\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 66, normalized size = 0.96 \[ \frac {- a^{3} + 2 a^{2} b x - 6 a b^{2} x^{2} - 12 b^{3} x^{3}}{3 a^{5} x^{3} + 3 a^{4} b x^{4}} + \frac {4 b^{3} \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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