Optimal. Leaf size=77 \[ \frac{5}{2} a^2 b^4 x^6+5 a^3 b^3 x^4+\frac{15}{2} a^4 b^2 x^2+6 a^5 b \log (x)-\frac{a^6}{2 x^2}+\frac{3}{4} a b^5 x^8+\frac{b^6 x^{10}}{10} \]
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Rubi [A] time = 0.0562668, antiderivative size = 77, 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} \[ \frac{5}{2} a^2 b^4 x^6+5 a^3 b^3 x^4+\frac{15}{2} a^4 b^2 x^2+6 a^5 b \log (x)-\frac{a^6}{2 x^2}+\frac{3}{4} a b^5 x^8+\frac{b^6 x^{10}}{10} \]
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 )^3}{x^3} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^3} \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^2} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (15 a^4 b^8+\frac{a^6 b^6}{x^2}+\frac{6 a^5 b^7}{x}+20 a^3 b^9 x+15 a^2 b^{10} x^2+6 a b^{11} x^3+b^{12} x^4\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac{a^6}{2 x^2}+\frac{15}{2} a^4 b^2 x^2+5 a^3 b^3 x^4+\frac{5}{2} a^2 b^4 x^6+\frac{3}{4} a b^5 x^8+\frac{b^6 x^{10}}{10}+6 a^5 b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0075471, size = 77, normalized size = 1. \[ \frac{5}{2} a^2 b^4 x^6+5 a^3 b^3 x^4+\frac{15}{2} a^4 b^2 x^2+6 a^5 b \log (x)-\frac{a^6}{2 x^2}+\frac{3}{4} a b^5 x^8+\frac{b^6 x^{10}}{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 68, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{2\,{x}^{2}}}+{\frac{15\,{a}^{4}{b}^{2}{x}^{2}}{2}}+5\,{a}^{3}{b}^{3}{x}^{4}+{\frac{5\,{a}^{2}{b}^{4}{x}^{6}}{2}}+{\frac{3\,a{b}^{5}{x}^{8}}{4}}+{\frac{{b}^{6}{x}^{10}}{10}}+6\,{a}^{5}b\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998797, size = 93, normalized size = 1.21 \begin{align*} \frac{1}{10} \, b^{6} x^{10} + \frac{3}{4} \, a b^{5} x^{8} + \frac{5}{2} \, a^{2} b^{4} x^{6} + 5 \, a^{3} b^{3} x^{4} + \frac{15}{2} \, a^{4} b^{2} x^{2} + 3 \, a^{5} b \log \left (x^{2}\right ) - \frac{a^{6}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60346, size = 167, normalized size = 2.17 \begin{align*} \frac{2 \, b^{6} x^{12} + 15 \, a b^{5} x^{10} + 50 \, a^{2} b^{4} x^{8} + 100 \, a^{3} b^{3} x^{6} + 150 \, a^{4} b^{2} x^{4} + 120 \, a^{5} b x^{2} \log \left (x\right ) - 10 \, a^{6}}{20 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.342027, size = 76, normalized size = 0.99 \begin{align*} - \frac{a^{6}}{2 x^{2}} + 6 a^{5} b \log{\left (x \right )} + \frac{15 a^{4} b^{2} x^{2}}{2} + 5 a^{3} b^{3} x^{4} + \frac{5 a^{2} b^{4} x^{6}}{2} + \frac{3 a b^{5} x^{8}}{4} + \frac{b^{6} x^{10}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1932, size = 107, normalized size = 1.39 \begin{align*} \frac{1}{10} \, b^{6} x^{10} + \frac{3}{4} \, a b^{5} x^{8} + \frac{5}{2} \, a^{2} b^{4} x^{6} + 5 \, a^{3} b^{3} x^{4} + \frac{15}{2} \, a^{4} b^{2} x^{2} + 3 \, a^{5} b \log \left (x^{2}\right ) - \frac{6 \, a^{5} b x^{2} + a^{6}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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