Optimal. Leaf size=34 \[ \frac{x^{12} (3 b+2 c x)^6}{279936}+\frac{b x^2}{2}+\frac{c x^3}{3} \]
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Rubi [A] time = 0.0329058, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {1591} \[ \frac{x^{12} (3 b+2 c x)^6}{279936}+\frac{b x^2}{2}+\frac{c x^3}{3} \]
Antiderivative was successfully verified.
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Rule 1591
Rubi steps
\begin{align*} \int \left (b x+c x^2\right ) \left (1+\left (\frac{b x^2}{2}+\frac{c x^3}{3}\right )^5\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^5\right ) \, dx,x,\frac{b x^2}{2}+\frac{c x^3}{3}\right )\\ &=\frac{b x^2}{2}+\frac{c x^3}{3}+\frac{x^{12} (3 b+2 c x)^6}{279936}\\ \end{align*}
Mathematica [B] time = 0.0077667, size = 98, normalized size = 2.88 \[ \frac{5}{648} b^2 c^4 x^{16}+\frac{5}{324} b^3 c^3 x^{15}+\frac{5}{288} b^4 c^2 x^{14}+\frac{1}{96} b^5 c x^{13}+\frac{b^6 x^{12}}{384}+\frac{1}{486} b c^5 x^{17}+\frac{b x^2}{2}+\frac{c^6 x^{18}}{4374}+\frac{c x^3}{3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 81, normalized size = 2.4 \begin{align*}{\frac{{c}^{6}{x}^{18}}{4374}}+{\frac{b{c}^{5}{x}^{17}}{486}}+{\frac{5\,{b}^{2}{c}^{4}{x}^{16}}{648}}+{\frac{5\,{b}^{3}{c}^{3}{x}^{15}}{324}}+{\frac{5\,{b}^{4}{c}^{2}{x}^{14}}{288}}+{\frac{{b}^{5}c{x}^{13}}{96}}+{\frac{{b}^{6}{x}^{12}}{384}}+{\frac{c{x}^{3}}{3}}+{\frac{b{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.975224, size = 108, normalized size = 3.18 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{486} \, b c^{5} x^{17} + \frac{5}{648} \, b^{2} c^{4} x^{16} + \frac{5}{324} \, b^{3} c^{3} x^{15} + \frac{5}{288} \, b^{4} c^{2} x^{14} + \frac{1}{96} \, b^{5} c x^{13} + \frac{1}{384} \, b^{6} x^{12} + \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.22939, size = 213, normalized size = 6.26 \begin{align*} \frac{1}{4374} x^{18} c^{6} + \frac{1}{486} x^{17} c^{5} b + \frac{5}{648} x^{16} c^{4} b^{2} + \frac{5}{324} x^{15} c^{3} b^{3} + \frac{5}{288} x^{14} c^{2} b^{4} + \frac{1}{96} x^{13} c b^{5} + \frac{1}{384} x^{12} b^{6} + \frac{1}{3} x^{3} c + \frac{1}{2} x^{2} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.094561, size = 90, normalized size = 2.65 \begin{align*} \frac{b^{6} x^{12}}{384} + \frac{b^{5} c x^{13}}{96} + \frac{5 b^{4} c^{2} x^{14}}{288} + \frac{5 b^{3} c^{3} x^{15}}{324} + \frac{5 b^{2} c^{4} x^{16}}{648} + \frac{b c^{5} x^{17}}{486} + \frac{b x^{2}}{2} + \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 = 1.17155, size = 108, normalized size = 3.18 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{486} \, b c^{5} x^{17} + \frac{5}{648} \, b^{2} c^{4} x^{16} + \frac{5}{324} \, b^{3} c^{3} x^{15} + \frac{5}{288} \, b^{4} c^{2} x^{14} + \frac{1}{96} \, b^{5} c x^{13} + \frac{1}{384} \, b^{6} x^{12} + \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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