Optimal. Leaf size=48 \[ 3 a^2 b^2 x^2+4 a^3 b \log (x)-\frac{a^4}{2 x^2}+a b^3 x^4+\frac{b^4 x^6}{6} \]
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Rubi [A] time = 0.0372027, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ 3 a^2 b^2 x^2+4 a^3 b \log (x)-\frac{a^4}{2 x^2}+a b^3 x^4+\frac{b^4 x^6}{6} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^3} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{x^3} \, dx}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^4}{x^2} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (6 a^2 b^6+\frac{a^4 b^4}{x^2}+\frac{4 a^3 b^5}{x}+4 a b^7 x+b^8 x^2\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac{a^4}{2 x^2}+3 a^2 b^2 x^2+a b^3 x^4+\frac{b^4 x^6}{6}+4 a^3 b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0047694, size = 48, normalized size = 1. \[ 3 a^2 b^2 x^2+4 a^3 b \log (x)-\frac{a^4}{2 x^2}+a b^3 x^4+\frac{b^4 x^6}{6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 45, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{2\,{x}^{2}}}+3\,{a}^{2}{b}^{2}{x}^{2}+a{b}^{3}{x}^{4}+{\frac{{b}^{4}{x}^{6}}{6}}+4\,{a}^{3}b\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.996465, size = 62, normalized size = 1.29 \begin{align*} \frac{1}{6} \, b^{4} x^{6} + a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b \log \left (x^{2}\right ) - \frac{a^{4}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75956, size = 108, normalized size = 2.25 \begin{align*} \frac{b^{4} x^{8} + 6 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x^{2} \log \left (x\right ) - 3 \, a^{4}}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.311597, size = 46, normalized size = 0.96 \begin{align*} - \frac{a^{4}}{2 x^{2}} + 4 a^{3} b \log{\left (x \right )} + 3 a^{2} b^{2} x^{2} + a b^{3} x^{4} + \frac{b^{4} x^{6}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13619, size = 76, normalized size = 1.58 \begin{align*} \frac{1}{6} \, b^{4} x^{6} + a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b \log \left (x^{2}\right ) - \frac{4 \, a^{3} b x^{2} + a^{4}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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