Optimal. Leaf size=72 \[ \frac{15}{4} a^2 b^4 x^4+10 a^3 b^3 x^2+15 a^4 b^2 \log (x)-\frac{3 a^5 b}{x^2}-\frac{a^6}{4 x^4}+a b^5 x^6+\frac{b^6 x^8}{8} \]
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Rubi [A] time = 0.052438, antiderivative size = 72, 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} \[ \frac{15}{4} a^2 b^4 x^4+10 a^3 b^3 x^2+15 a^4 b^2 \log (x)-\frac{3 a^5 b}{x^2}-\frac{a^6}{4 x^4}+a b^5 x^6+\frac{b^6 x^8}{8} \]
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^5} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^5} \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^3} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (20 a^3 b^9+\frac{a^6 b^6}{x^3}+\frac{6 a^5 b^7}{x^2}+\frac{15 a^4 b^8}{x}+15 a^2 b^{10} x+6 a b^{11} x^2+b^{12} x^3\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac{a^6}{4 x^4}-\frac{3 a^5 b}{x^2}+10 a^3 b^3 x^2+\frac{15}{4} a^2 b^4 x^4+a b^5 x^6+\frac{b^6 x^8}{8}+15 a^4 b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0048738, size = 72, normalized size = 1. \[ \frac{15}{4} a^2 b^4 x^4+10 a^3 b^3 x^2+15 a^4 b^2 \log (x)-\frac{3 a^5 b}{x^2}-\frac{a^6}{4 x^4}+a b^5 x^6+\frac{b^6 x^8}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{4\,{x}^{4}}}-3\,{\frac{{a}^{5}b}{{x}^{2}}}+10\,{x}^{2}{a}^{3}{b}^{3}+{\frac{15\,{a}^{2}{b}^{4}{x}^{4}}{4}}+a{b}^{5}{x}^{6}+{\frac{{b}^{6}{x}^{8}}{8}}+15\,{a}^{4}{b}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01768, size = 93, normalized size = 1.29 \begin{align*} \frac{1}{8} \, b^{6} x^{8} + a b^{5} x^{6} + \frac{15}{4} \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{2} + \frac{15}{2} \, a^{4} b^{2} \log \left (x^{2}\right ) - \frac{12 \, a^{5} b x^{2} + a^{6}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71567, size = 158, normalized size = 2.19 \begin{align*} \frac{b^{6} x^{12} + 8 \, a b^{5} x^{10} + 30 \, a^{2} b^{4} x^{8} + 80 \, a^{3} b^{3} x^{6} + 120 \, a^{4} b^{2} x^{4} \log \left (x\right ) - 24 \, a^{5} b x^{2} - 2 \, a^{6}}{8 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.395663, size = 71, normalized size = 0.99 \begin{align*} 15 a^{4} b^{2} \log{\left (x \right )} + 10 a^{3} b^{3} x^{2} + \frac{15 a^{2} b^{4} x^{4}}{4} + a b^{5} x^{6} + \frac{b^{6} x^{8}}{8} - \frac{a^{6} + 12 a^{5} b x^{2}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14632, size = 108, normalized size = 1.5 \begin{align*} \frac{1}{8} \, b^{6} x^{8} + a b^{5} x^{6} + \frac{15}{4} \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{2} + \frac{15}{2} \, a^{4} b^{2} \log \left (x^{2}\right ) - \frac{45 \, a^{4} b^{2} x^{4} + 12 \, a^{5} b x^{2} + a^{6}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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