Optimal. Leaf size=42 \[ -\frac {2 b \log (x)}{a^3}+\frac {2 b \log (a+b x)}{a^3}-\frac {b}{a^2 (a+b x)}-\frac {1}{a^2 x} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, 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, 44} \[ -\frac {b}{a^2 (a+b x)}-\frac {2 b \log (x)}{a^3}+\frac {2 b \log (a+b x)}{a^3}-\frac {1}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 44
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a x^2+b x^3\right )^2} \, dx &=\int \frac {1}{x^2 (a+b x)^2} \, dx\\ &=\int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac {1}{a^2 x}-\frac {b}{a^2 (a+b x)}-\frac {2 b \log (x)}{a^3}+\frac {2 b \log (a+b x)}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 35, normalized size = 0.83 \[ -\frac {a \left (\frac {b}{a+b x}+\frac {1}{x}\right )-2 b \log (a+b x)+2 b \log (x)}{a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 63, normalized size = 1.50 \[ -\frac {2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \relax (x)}{a^{3} b x^{2} + a^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 45, normalized size = 1.07 \[ \frac {2 \, b \log \left ({\left | b x + a \right |}\right )}{a^{3}} - \frac {2 \, b \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {2 \, b x + a}{{\left (b x^{2} + a x\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 43, normalized size = 1.02 \[ -\frac {b}{\left (b x +a \right ) a^{2}}-\frac {2 b \ln \relax (x )}{a^{3}}+\frac {2 b \ln \left (b x +a \right )}{a^{3}}-\frac {1}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 45, normalized size = 1.07 \[ -\frac {2 \, b x + a}{a^{2} b x^{2} + a^{3} x} + \frac {2 \, b \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b \log \relax (x)}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.34, size = 41, normalized size = 0.98 \[ \frac {4\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^3}-\frac {\frac {1}{a}+\frac {2\,b\,x}{a^2}}{b\,x^2+a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 37, normalized size = 0.88 \[ \frac {- a - 2 b x}{a^{3} x + a^{2} b x^{2}} + \frac {2 b \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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