Optimal. Leaf size=81 \[ a^3 x+\frac {3}{2} a^2 b x^2+\frac {3}{5} c x^5 \left (a c+b^2\right )+\frac {1}{4} b x^4 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac {1}{2} b c^2 x^6+\frac {c^3 x^7}{7} \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 81, normalized size = 1.00 \[ a^3 x+\frac {3}{2} a^2 b x^2+\frac {3}{5} c x^5 \left (a c+b^2\right )+\frac {1}{4} b x^4 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac {1}{2} b c^2 x^6+\frac {c^3 x^7}{7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 82, normalized size = 1.01 \[ \frac {1}{7} x^{7} c^{3} + \frac {1}{2} x^{6} c^{2} b + \frac {3}{5} x^{5} c b^{2} + \frac {3}{5} x^{5} c^{2} a + \frac {1}{4} x^{4} b^{3} + \frac {3}{2} x^{4} c b a + x^{3} b^{2} a + x^{3} c a^{2} + \frac {3}{2} x^{2} b a^{2} + x a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 82, normalized size = 1.01 \[ \frac {1}{7} \, c^{3} x^{7} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{5} \, b^{2} c x^{5} + \frac {3}{5} \, a c^{2} x^{5} + \frac {1}{4} \, b^{3} x^{4} + \frac {3}{2} \, a b c x^{4} + a b^{2} x^{3} + a^{2} c x^{3} + \frac {3}{2} \, a^{2} b x^{2} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 108, normalized size = 1.33 \[ \frac {c^{3} x^{7}}{7}+\frac {b \,c^{2} x^{6}}{2}+\frac {3 a^{2} b \,x^{2}}{2}+\frac {\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) x^{5}}{5}+a^{3} x +\frac {\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) x^{4}}{4}+\frac {\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 85, normalized size = 1.05 \[ \frac {1}{7} \, c^{3} x^{7} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{5} \, b^{2} c x^{5} + \frac {1}{4} \, b^{3} x^{4} + a^{3} x + \frac {1}{2} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} a^{2} + \frac {1}{10} \, {\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 72, normalized size = 0.89 \[ x^4\,\left (\frac {b^3}{4}+\frac {3\,a\,c\,b}{2}\right )+a^3\,x+\frac {c^3\,x^7}{7}+\frac {3\,a^2\,b\,x^2}{2}+\frac {b\,c^2\,x^6}{2}+a\,x^3\,\left (b^2+a\,c\right )+\frac {3\,c\,x^5\,\left (b^2+a\,c\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 85, normalized size = 1.05 \[ a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + x^{5} \left (\frac {3 a c^{2}}{5} + \frac {3 b^{2} c}{5}\right ) + x^{4} \left (\frac {3 a b c}{2} + \frac {b^{3}}{4}\right ) + x^{3} \left (a^{2} c + a b^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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