Optimal. Leaf size=81 \[ \frac{3}{2} a^2 b x^2+a^3 x+\frac{3}{5} c x^5 \left (a c+b^2\right )+\frac{1}{4} b x^4 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7} \]
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Rubi [A] time = 0.0557333, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {611} \[ \frac{3}{2} a^2 b x^2+a^3 x+\frac{3}{5} c x^5 \left (a c+b^2\right )+\frac{1}{4} b x^4 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7} \]
Antiderivative was successfully verified.
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Rule 611
Rubi steps
\begin{align*} \int \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3+3 a^2 b x+3 a b^2 \left (1+\frac{a c}{b^2}\right ) x^2+b^3 \left (1+\frac{6 a c}{b^2}\right ) x^3+3 b^2 c \left (1+\frac{a c}{b^2}\right ) x^4+3 b c^2 x^5+c^3 x^6\right ) \, dx\\ &=a^3 x+\frac{3}{2} a^2 b x^2+a \left (b^2+a c\right ) x^3+\frac{1}{4} b \left (b^2+6 a c\right ) x^4+\frac{3}{5} c \left (b^2+a c\right ) x^5+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7}\\ \end{align*}
Mathematica [A] time = 0.0128007, size = 81, normalized size = 1. \[ \frac{3}{2} a^2 b x^2+a^3 x+\frac{3}{5} c x^5 \left (a c+b^2\right )+\frac{1}{4} b x^4 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac{1}{2} b c^2 x^6+\frac{c^3 x^7}{7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 108, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}{x}^{7}}{7}}+{\frac{b{c}^{2}{x}^{6}}{2}}+{\frac{ \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ){x}^{3}}{3}}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+x{a}^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01875, size = 115, normalized size = 1.42 \begin{align*} \frac{1}{7} \, c^{3} x^{7} + \frac{1}{2} \, b c^{2} x^{6} + \frac{3}{5} \, b^{2} c x^{5} + \frac{1}{4} \, b^{3} x^{4} + a^{3} x + \frac{1}{2} \,{\left (2 \, c x^{3} + 3 \, b x^{2}\right )} a^{2} + \frac{1}{10} \,{\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69686, size = 188, normalized size = 2.32 \begin{align*} \frac{1}{7} x^{7} c^{3} + \frac{1}{2} x^{6} c^{2} b + \frac{3}{5} x^{5} c b^{2} + \frac{3}{5} x^{5} c^{2} a + \frac{1}{4} x^{4} b^{3} + \frac{3}{2} x^{4} c b a + x^{3} b^{2} a + x^{3} c a^{2} + \frac{3}{2} x^{2} b a^{2} + x a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.078463, size = 85, normalized size = 1.05 \begin{align*} a^{3} x + \frac{3 a^{2} b x^{2}}{2} + \frac{b c^{2} x^{6}}{2} + \frac{c^{3} x^{7}}{7} + x^{5} \left (\frac{3 a c^{2}}{5} + \frac{3 b^{2} c}{5}\right ) + x^{4} \left (\frac{3 a b c}{2} + \frac{b^{3}}{4}\right ) + x^{3} \left (a^{2} c + a b^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07453, size = 111, normalized size = 1.37 \begin{align*} \frac{1}{7} \, c^{3} x^{7} + \frac{1}{2} \, b c^{2} x^{6} + \frac{3}{5} \, b^{2} c x^{5} + \frac{3}{5} \, a c^{2} x^{5} + \frac{1}{4} \, b^{3} x^{4} + \frac{3}{2} \, a b c x^{4} + a b^{2} x^{3} + a^{2} c x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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