Optimal. Leaf size=46 \[ a^2 x+\frac {1}{3} x^3 \left (2 a c+b^2\right )+a b x^2+\frac {1}{2} b c x^4+\frac {c^2 x^5}{5} \]
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Rubi [A] time = 0.02, antiderivative size = 46, 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} \begin {gather*} a^2 x+\frac {1}{3} x^3 \left (2 a c+b^2\right )+a b x^2+\frac {1}{2} b c x^4+\frac {c^2 x^5}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 611
Rubi steps
\begin {align*} \int \left (a+b x+c x^2\right )^2 \, dx &=\int \left (a^2+2 a b x+b^2 \left (1+\frac {2 a c}{b^2}\right ) x^2+2 b c x^3+c^2 x^4\right ) \, dx\\ &=a^2 x+a b x^2+\frac {1}{3} \left (b^2+2 a c\right ) x^3+\frac {1}{2} b c x^4+\frac {c^2 x^5}{5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.00 \begin {gather*} a^2 x+\frac {1}{3} x^3 \left (2 a c+b^2\right )+a b x^2+\frac {1}{2} b c x^4+\frac {c^2 x^5}{5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b x+c x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.34, size = 42, normalized size = 0.91 \begin {gather*} \frac {1}{5} x^{5} c^{2} + \frac {1}{2} x^{4} c b + \frac {1}{3} x^{3} b^{2} + \frac {2}{3} x^{3} c a + x^{2} b a + x a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 42, normalized size = 0.91 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {1}{2} \, b c x^{4} + \frac {1}{3} \, b^{2} x^{3} + \frac {2}{3} \, a c x^{3} + a b x^{2} + a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 41, normalized size = 0.89 \begin {gather*} \frac {c^{2} x^{5}}{5}+\frac {b c \,x^{4}}{2}+a b \,x^{2}+a^{2} x +\frac {\left (2 a c +b^{2}\right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 45, normalized size = 0.98 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {1}{2} \, b c x^{4} + \frac {1}{3} \, b^{2} x^{3} + a^{2} x + \frac {1}{3} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 41, normalized size = 0.89 \begin {gather*} a^2\,x+x^3\,\left (\frac {b^2}{3}+\frac {2\,a\,c}{3}\right )+\frac {c^2\,x^5}{5}+a\,b\,x^2+\frac {b\,c\,x^4}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 42, normalized size = 0.91 \begin {gather*} a^{2} x + a b x^{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} + x^{3} \left (\frac {2 a c}{3} + \frac {b^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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