Optimal. Leaf size=76 \[ \frac{15}{8} a^2 b^4 x^8+\frac{10}{3} a^3 b^3 x^6+\frac{15}{4} a^4 b^2 x^4+3 a^5 b x^2+a^6 \log (x)+\frac{3}{5} a b^5 x^{10}+\frac{b^6 x^{12}}{12} \]
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Rubi [A] time = 0.0545137, antiderivative size = 76, 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} \[ \frac{15}{8} a^2 b^4 x^8+\frac{10}{3} a^3 b^3 x^6+\frac{15}{4} a^4 b^2 x^4+3 a^5 b x^2+a^6 \log (x)+\frac{3}{5} a b^5 x^{10}+\frac{b^6 x^{12}}{12} \]
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} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x} \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (6 a^5 b^7+\frac{a^6 b^6}{x}+15 a^4 b^8 x+20 a^3 b^9 x^2+15 a^2 b^{10} x^3+6 a b^{11} x^4+b^{12} x^5\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=3 a^5 b x^2+\frac{15}{4} a^4 b^2 x^4+\frac{10}{3} a^3 b^3 x^6+\frac{15}{8} a^2 b^4 x^8+\frac{3}{5} a b^5 x^{10}+\frac{b^6 x^{12}}{12}+a^6 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0044944, size = 76, normalized size = 1. \[ \frac{15}{8} a^2 b^4 x^8+\frac{10}{3} a^3 b^3 x^6+\frac{15}{4} a^4 b^2 x^4+3 a^5 b x^2+a^6 \log (x)+\frac{3}{5} a b^5 x^{10}+\frac{b^6 x^{12}}{12} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 67, normalized size = 0.9 \begin{align*} 3\,{a}^{5}b{x}^{2}+{\frac{15\,{a}^{4}{b}^{2}{x}^{4}}{4}}+{\frac{10\,{a}^{3}{b}^{3}{x}^{6}}{3}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{8}}{8}}+{\frac{3\,a{b}^{5}{x}^{10}}{5}}+{\frac{{b}^{6}{x}^{12}}{12}}+{a}^{6}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01517, size = 93, normalized size = 1.22 \begin{align*} \frac{1}{12} \, b^{6} x^{12} + \frac{3}{5} \, a b^{5} x^{10} + \frac{15}{8} \, a^{2} b^{4} x^{8} + \frac{10}{3} \, a^{3} b^{3} x^{6} + \frac{15}{4} \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + \frac{1}{2} \, a^{6} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75273, size = 157, normalized size = 2.07 \begin{align*} \frac{1}{12} \, b^{6} x^{12} + \frac{3}{5} \, a b^{5} x^{10} + \frac{15}{8} \, a^{2} b^{4} x^{8} + \frac{10}{3} \, a^{3} b^{3} x^{6} + \frac{15}{4} \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.30263, size = 76, normalized size = 1. \begin{align*} a^{6} \log{\left (x \right )} + 3 a^{5} b x^{2} + \frac{15 a^{4} b^{2} x^{4}}{4} + \frac{10 a^{3} b^{3} x^{6}}{3} + \frac{15 a^{2} b^{4} x^{8}}{8} + \frac{3 a b^{5} x^{10}}{5} + \frac{b^{6} x^{12}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12874, size = 93, normalized size = 1.22 \begin{align*} \frac{1}{12} \, b^{6} x^{12} + \frac{3}{5} \, a b^{5} x^{10} + \frac{15}{8} \, a^{2} b^{4} x^{8} + \frac{10}{3} \, a^{3} b^{3} x^{6} + \frac{15}{4} \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + \frac{1}{2} \, a^{6} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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