Optimal. Leaf size=43 \[ \frac{3}{10} a^2 b x^{10}+\frac{a^3 x^8}{8}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{14}}{14} \]
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Rubi [A] time = 0.0268245, antiderivative size = 43, 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} \[ \frac{3}{10} a^2 b x^{10}+\frac{a^3 x^8}{8}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{14}}{14} \]
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 )^3 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^3 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^3 x^3+3 a^2 b x^4+3 a b^2 x^5+b^3 x^6\right ) \, dx,x,x^2\right )\\ &=\frac{a^3 x^8}{8}+\frac{3}{10} a^2 b x^{10}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{14}}{14}\\ \end{align*}
Mathematica [A] time = 0.0015993, size = 43, normalized size = 1. \[ \frac{3}{10} a^2 b x^{10}+\frac{a^3 x^8}{8}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{14}}{14} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{8}}{8}}+{\frac{3\,{a}^{2}b{x}^{10}}{10}}+{\frac{a{b}^{2}{x}^{12}}{4}}+{\frac{{b}^{3}{x}^{14}}{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19351, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{14} \, b^{3} x^{14} + \frac{1}{4} \, a b^{2} x^{12} + \frac{3}{10} \, a^{2} b x^{10} + \frac{1}{8} \, a^{3} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31896, size = 86, normalized size = 2. \begin{align*} \frac{1}{14} x^{14} b^{3} + \frac{1}{4} x^{12} b^{2} a + \frac{3}{10} x^{10} b a^{2} + \frac{1}{8} x^{8} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.064422, size = 37, normalized size = 0.86 \begin{align*} \frac{a^{3} x^{8}}{8} + \frac{3 a^{2} b x^{10}}{10} + \frac{a b^{2} x^{12}}{4} + \frac{b^{3} x^{14}}{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.01414, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{14} \, b^{3} x^{14} + \frac{1}{4} \, a b^{2} x^{12} + \frac{3}{10} \, a^{2} b x^{10} + \frac{1}{8} \, a^{3} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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