Optimal. Leaf size=72 \[ \frac{3 a^2 \left (a+b x^2\right )^8}{16 b^4}-\frac{a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac{\left (a+b x^2\right )^{10}}{20 b^4}-\frac{a \left (a+b x^2\right )^9}{6 b^4} \]
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Rubi [A] time = 0.116991, 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} \[ \frac{3 a^2 \left (a+b x^2\right )^8}{16 b^4}-\frac{a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac{\left (a+b x^2\right )^{10}}{20 b^4}-\frac{a \left (a+b x^2\right )^9}{6 b^4} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac{\int x^7 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int x^3 \left (a b+b^2 x\right )^6 \, dx,x,x^2\right )}{2 b^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^3 \left (a b+b^2 x\right )^6}{b^3}+\frac{3 a^2 \left (a b+b^2 x\right )^7}{b^4}-\frac{3 a \left (a b+b^2 x\right )^8}{b^5}+\frac{\left (a b+b^2 x\right )^9}{b^6}\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac{a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac{3 a^2 \left (a+b x^2\right )^8}{16 b^4}-\frac{a \left (a+b x^2\right )^9}{6 b^4}+\frac{\left (a+b x^2\right )^{10}}{20 b^4}\\ \end{align*}
Mathematica [A] time = 0.0024967, size = 82, normalized size = 1.14 \[ \frac{15}{16} a^2 b^4 x^{16}+\frac{10}{7} a^3 b^3 x^{14}+\frac{5}{4} a^4 b^2 x^{12}+\frac{3}{5} a^5 b x^{10}+\frac{a^6 x^8}{8}+\frac{1}{3} a b^5 x^{18}+\frac{b^6 x^{20}}{20} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 69, normalized size = 1. \begin{align*}{\frac{{b}^{6}{x}^{20}}{20}}+{\frac{a{b}^{5}{x}^{18}}{3}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{16}}{16}}+{\frac{10\,{a}^{3}{b}^{3}{x}^{14}}{7}}+{\frac{5\,{a}^{4}{b}^{2}{x}^{12}}{4}}+{\frac{3\,{a}^{5}b{x}^{10}}{5}}+{\frac{{a}^{6}{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984808, size = 92, normalized size = 1.28 \begin{align*} \frac{1}{20} \, b^{6} x^{20} + \frac{1}{3} \, a b^{5} x^{18} + \frac{15}{16} \, a^{2} b^{4} x^{16} + \frac{10}{7} \, a^{3} b^{3} x^{14} + \frac{5}{4} \, a^{4} b^{2} x^{12} + \frac{3}{5} \, a^{5} b x^{10} + \frac{1}{8} \, a^{6} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71961, size = 166, normalized size = 2.31 \begin{align*} \frac{1}{20} x^{20} b^{6} + \frac{1}{3} x^{18} b^{5} a + \frac{15}{16} x^{16} b^{4} a^{2} + \frac{10}{7} x^{14} b^{3} a^{3} + \frac{5}{4} x^{12} b^{2} a^{4} + \frac{3}{5} x^{10} b a^{5} + \frac{1}{8} x^{8} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.080959, size = 78, normalized size = 1.08 \begin{align*} \frac{a^{6} x^{8}}{8} + \frac{3 a^{5} b x^{10}}{5} + \frac{5 a^{4} b^{2} x^{12}}{4} + \frac{10 a^{3} b^{3} x^{14}}{7} + \frac{15 a^{2} b^{4} x^{16}}{16} + \frac{a b^{5} x^{18}}{3} + \frac{b^{6} x^{20}}{20} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15233, size = 92, normalized size = 1.28 \begin{align*} \frac{1}{20} \, b^{6} x^{20} + \frac{1}{3} \, a b^{5} x^{18} + \frac{15}{16} \, a^{2} b^{4} x^{16} + \frac{10}{7} \, a^{3} b^{3} x^{14} + \frac{5}{4} \, a^{4} b^{2} x^{12} + \frac{3}{5} \, a^{5} b x^{10} + \frac{1}{8} \, a^{6} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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