Optimal. Leaf size=72 \[ \frac{3 a^2 \left (a+b x^2\right )^{10}}{20 b^4}-\frac{a^3 \left (a+b x^2\right )^9}{18 b^4}+\frac{\left (a+b x^2\right )^{12}}{24 b^4}-\frac{3 a \left (a+b x^2\right )^{11}}{22 b^4} \]
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Rubi [A] time = 0.115799, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{3 a^2 \left (a+b x^2\right )^{10}}{20 b^4}-\frac{a^3 \left (a+b x^2\right )^9}{18 b^4}+\frac{\left (a+b x^2\right )^{12}}{24 b^4}-\frac{3 a \left (a+b x^2\right )^{11}}{22 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^7 \left (a+b x^2\right )^8 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^8}{b^3}+\frac{3 a^2 (a+b x)^9}{b^3}-\frac{3 a (a+b x)^{10}}{b^3}+\frac{(a+b x)^{11}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3 \left (a+b x^2\right )^9}{18 b^4}+\frac{3 a^2 \left (a+b x^2\right )^{10}}{20 b^4}-\frac{3 a \left (a+b x^2\right )^{11}}{22 b^4}+\frac{\left (a+b x^2\right )^{12}}{24 b^4}\\ \end{align*}
Mathematica [A] time = 0.0028108, size = 106, normalized size = 1.47 \[ \frac{7}{5} a^2 b^6 x^{20}+\frac{28}{9} a^3 b^5 x^{18}+\frac{35}{8} a^4 b^4 x^{16}+4 a^5 b^3 x^{14}+\frac{7}{3} a^6 b^2 x^{12}+\frac{4}{5} a^7 b x^{10}+\frac{a^8 x^8}{8}+\frac{4}{11} a b^7 x^{22}+\frac{b^8 x^{24}}{24} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 91, normalized size = 1.3 \begin{align*}{\frac{{b}^{8}{x}^{24}}{24}}+{\frac{4\,a{b}^{7}{x}^{22}}{11}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{20}}{5}}+{\frac{28\,{a}^{3}{b}^{5}{x}^{18}}{9}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{16}}{8}}+4\,{a}^{5}{b}^{3}{x}^{14}+{\frac{7\,{a}^{6}{b}^{2}{x}^{12}}{3}}+{\frac{4\,{a}^{7}b{x}^{10}}{5}}+{\frac{{a}^{8}{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59132, size = 122, normalized size = 1.69 \begin{align*} \frac{1}{24} \, b^{8} x^{24} + \frac{4}{11} \, a b^{7} x^{22} + \frac{7}{5} \, a^{2} b^{6} x^{20} + \frac{28}{9} \, a^{3} b^{5} x^{18} + \frac{35}{8} \, a^{4} b^{4} x^{16} + 4 \, a^{5} b^{3} x^{14} + \frac{7}{3} \, a^{6} b^{2} x^{12} + \frac{4}{5} \, a^{7} b x^{10} + \frac{1}{8} \, a^{8} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37431, size = 215, normalized size = 2.99 \begin{align*} \frac{1}{24} x^{24} b^{8} + \frac{4}{11} x^{22} b^{7} a + \frac{7}{5} x^{20} b^{6} a^{2} + \frac{28}{9} x^{18} b^{5} a^{3} + \frac{35}{8} x^{16} b^{4} a^{4} + 4 x^{14} b^{3} a^{5} + \frac{7}{3} x^{12} b^{2} a^{6} + \frac{4}{5} x^{10} b a^{7} + \frac{1}{8} x^{8} a^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.082515, size = 105, normalized size = 1.46 \begin{align*} \frac{a^{8} x^{8}}{8} + \frac{4 a^{7} b x^{10}}{5} + \frac{7 a^{6} b^{2} x^{12}}{3} + 4 a^{5} b^{3} x^{14} + \frac{35 a^{4} b^{4} x^{16}}{8} + \frac{28 a^{3} b^{5} x^{18}}{9} + \frac{7 a^{2} b^{6} x^{20}}{5} + \frac{4 a b^{7} x^{22}}{11} + \frac{b^{8} x^{24}}{24} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.10331, size = 122, normalized size = 1.69 \begin{align*} \frac{1}{24} \, b^{8} x^{24} + \frac{4}{11} \, a b^{7} x^{22} + \frac{7}{5} \, a^{2} b^{6} x^{20} + \frac{28}{9} \, a^{3} b^{5} x^{18} + \frac{35}{8} \, a^{4} b^{4} x^{16} + 4 \, a^{5} b^{3} x^{14} + \frac{7}{3} \, a^{6} b^{2} x^{12} + \frac{4}{5} \, a^{7} b x^{10} + \frac{1}{8} \, a^{8} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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