Optimal. Leaf size=30 \[ \frac{1}{6} \left (a x+\frac{c x^3}{3}\right )^6+a x+\frac{c x^3}{3} \]
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Rubi [A] time = 0.0198891, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1591} \[ \frac{1}{6} \left (a x+\frac{c x^3}{3}\right )^6+a x+\frac{c x^3}{3} \]
Antiderivative was successfully verified.
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Rule 1591
Rubi steps
\begin{align*} \int \left (a+c x^2\right ) \left (1+\left (a x+\frac{c x^3}{3}\right )^5\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^5\right ) \, dx,x,a x+\frac{c x^3}{3}\right )\\ &=a x+\frac{c x^3}{3}+\frac{1}{6} \left (a x+\frac{c x^3}{3}\right )^6\\ \end{align*}
Mathematica [B] time = 0.0057631, size = 93, normalized size = 3.1 \[ \frac{5}{162} a^2 c^4 x^{14}+\frac{10}{81} a^3 c^3 x^{12}+\frac{5}{18} a^4 c^2 x^{10}+\frac{1}{3} a^5 c x^8+\frac{a^6 x^6}{6}+\frac{1}{243} a c^5 x^{16}+a x+\frac{c^6 x^{18}}{4374}+\frac{c x^3}{3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 78, normalized size = 2.6 \begin{align*}{\frac{{c}^{6}{x}^{18}}{4374}}+{\frac{{c}^{5}a{x}^{16}}{243}}+{\frac{5\,{c}^{4}{a}^{2}{x}^{14}}{162}}+{\frac{10\,{a}^{3}{c}^{3}{x}^{12}}{81}}+{\frac{5\,{a}^{4}{c}^{2}{x}^{10}}{18}}+{\frac{{a}^{5}c{x}^{8}}{3}}+{\frac{{a}^{6}{x}^{6}}{6}}+{\frac{c{x}^{3}}{3}}+ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00994, size = 104, normalized size = 3.47 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{243} \, a c^{5} x^{16} + \frac{5}{162} \, a^{2} c^{4} x^{14} + \frac{10}{81} \, a^{3} c^{3} x^{12} + \frac{5}{18} \, a^{4} c^{2} x^{10} + \frac{1}{3} \, a^{5} c x^{8} + \frac{1}{6} \, a^{6} x^{6} + \frac{1}{3} \, c x^{3} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.11426, size = 197, normalized size = 6.57 \begin{align*} \frac{1}{4374} x^{18} c^{6} + \frac{1}{243} x^{16} c^{5} a + \frac{5}{162} x^{14} c^{4} a^{2} + \frac{10}{81} x^{12} c^{3} a^{3} + \frac{5}{18} x^{10} c^{2} a^{4} + \frac{1}{3} x^{8} c a^{5} + \frac{1}{6} x^{6} a^{6} + \frac{1}{3} x^{3} c + x a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.090498, size = 87, normalized size = 2.9 \begin{align*} \frac{a^{6} x^{6}}{6} + \frac{a^{5} c x^{8}}{3} + \frac{5 a^{4} c^{2} x^{10}}{18} + \frac{10 a^{3} c^{3} x^{12}}{81} + \frac{5 a^{2} c^{4} x^{14}}{162} + \frac{a c^{5} x^{16}}{243} + a x + \frac{c^{6} x^{18}}{4374} + \frac{c x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22165, size = 104, normalized size = 3.47 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{243} \, a c^{5} x^{16} + \frac{5}{162} \, a^{2} c^{4} x^{14} + \frac{10}{81} \, a^{3} c^{3} x^{12} + \frac{5}{18} \, a^{4} c^{2} x^{10} + \frac{1}{3} \, a^{5} c x^{8} + \frac{1}{6} \, a^{6} x^{6} + \frac{1}{3} \, c x^{3} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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