Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^2\right )^7}{14 b^3}+\frac{\left (a+b x^2\right )^9}{18 b^3}-\frac{a \left (a+b x^2\right )^8}{8 b^3} \]
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Rubi [A] time = 0.0853567, antiderivative size = 53, 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} \[ \frac{a^2 \left (a+b x^2\right )^7}{14 b^3}+\frac{\left (a+b x^2\right )^9}{18 b^3}-\frac{a \left (a+b x^2\right )^8}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac{\int x^5 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int x^2 \left (a b+b^2 x\right )^6 \, dx,x,x^2\right )}{2 b^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2 \left (a b+b^2 x\right )^6}{b^2}-\frac{2 a \left (a b+b^2 x\right )^7}{b^3}+\frac{\left (a b+b^2 x\right )^8}{b^4}\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=\frac{a^2 \left (a+b x^2\right )^7}{14 b^3}-\frac{a \left (a+b x^2\right )^8}{8 b^3}+\frac{\left (a+b x^2\right )^9}{18 b^3}\\ \end{align*}
Mathematica [A] time = 0.0026915, size = 82, normalized size = 1.55 \[ \frac{15}{14} a^2 b^4 x^{14}+\frac{5}{3} a^3 b^3 x^{12}+\frac{3}{2} a^4 b^2 x^{10}+\frac{3}{4} a^5 b x^8+\frac{a^6 x^6}{6}+\frac{3}{8} a b^5 x^{16}+\frac{b^6 x^{18}}{18} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 69, normalized size = 1.3 \begin{align*}{\frac{{b}^{6}{x}^{18}}{18}}+{\frac{3\,a{b}^{5}{x}^{16}}{8}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{14}}{14}}+{\frac{5\,{a}^{3}{b}^{3}{x}^{12}}{3}}+{\frac{3\,{a}^{4}{b}^{2}{x}^{10}}{2}}+{\frac{3\,{a}^{5}b{x}^{8}}{4}}+{\frac{{a}^{6}{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980579, size = 92, normalized size = 1.74 \begin{align*} \frac{1}{18} \, b^{6} x^{18} + \frac{3}{8} \, a b^{5} x^{16} + \frac{15}{14} \, a^{2} b^{4} x^{14} + \frac{5}{3} \, a^{3} b^{3} x^{12} + \frac{3}{2} \, a^{4} b^{2} x^{10} + \frac{3}{4} \, a^{5} b x^{8} + \frac{1}{6} \, a^{6} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51299, size = 163, normalized size = 3.08 \begin{align*} \frac{1}{18} x^{18} b^{6} + \frac{3}{8} x^{16} b^{5} a + \frac{15}{14} x^{14} b^{4} a^{2} + \frac{5}{3} x^{12} b^{3} a^{3} + \frac{3}{2} x^{10} b^{2} a^{4} + \frac{3}{4} x^{8} b a^{5} + \frac{1}{6} x^{6} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.081546, size = 80, normalized size = 1.51 \begin{align*} \frac{a^{6} x^{6}}{6} + \frac{3 a^{5} b x^{8}}{4} + \frac{3 a^{4} b^{2} x^{10}}{2} + \frac{5 a^{3} b^{3} x^{12}}{3} + \frac{15 a^{2} b^{4} x^{14}}{14} + \frac{3 a b^{5} x^{16}}{8} + \frac{b^{6} x^{18}}{18} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1859, size = 92, normalized size = 1.74 \begin{align*} \frac{1}{18} \, b^{6} x^{18} + \frac{3}{8} \, a b^{5} x^{16} + \frac{15}{14} \, a^{2} b^{4} x^{14} + \frac{5}{3} \, a^{3} b^{3} x^{12} + \frac{3}{2} \, a^{4} b^{2} x^{10} + \frac{3}{4} \, a^{5} b x^{8} + \frac{1}{6} \, a^{6} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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