Optimal. Leaf size=34 \[ \frac{\left (a+b x^2\right )^5}{10 b^2}-\frac{a \left (a+b x^2\right )^4}{8 b^2} \]
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Rubi [A] time = 0.0316579, antiderivative size = 34, 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} \[ \frac{\left (a+b x^2\right )^5}{10 b^2}-\frac{a \left (a+b x^2\right )^4}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (a+b x^2\right )^3 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^3 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac{a \left (a+b x^2\right )^4}{8 b^2}+\frac{\left (a+b x^2\right )^5}{10 b^2}\\ \end{align*}
Mathematica [A] time = 0.0016854, size = 43, normalized size = 1.26 \[ \frac{1}{2} a^2 b x^6+\frac{a^3 x^4}{4}+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{10}}{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 36, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}{x}^{10}}{10}}+{\frac{3\,a{b}^{2}{x}^{8}}{8}}+{\frac{{a}^{2}b{x}^{6}}{2}}+{\frac{{a}^{3}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.84903, size = 47, normalized size = 1.38 \begin{align*} \frac{1}{10} \, b^{3} x^{10} + \frac{3}{8} \, a b^{2} x^{8} + \frac{1}{2} \, a^{2} b x^{6} + \frac{1}{4} \, a^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78459, size = 82, normalized size = 2.41 \begin{align*} \frac{1}{10} x^{10} b^{3} + \frac{3}{8} x^{8} b^{2} a + \frac{1}{2} x^{6} b a^{2} + \frac{1}{4} x^{4} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.063058, size = 37, normalized size = 1.09 \begin{align*} \frac{a^{3} x^{4}}{4} + \frac{a^{2} b x^{6}}{2} + \frac{3 a b^{2} x^{8}}{8} + \frac{b^{3} x^{10}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.59351, size = 47, normalized size = 1.38 \begin{align*} \frac{1}{10} \, b^{3} x^{10} + \frac{3}{8} \, a b^{2} x^{8} + \frac{1}{2} \, a^{2} b x^{6} + \frac{1}{4} \, a^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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