Optimal. Leaf size=42 \[ \frac {1}{2} x^2 (a B+A b)+a A x+\frac {1}{3} x^3 (A c+b B)+\frac {1}{4} B c x^4 \]
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Rubi [A] time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {631} \begin {gather*} \frac {1}{2} x^2 (a B+A b)+a A x+\frac {1}{3} x^3 (A c+b B)+\frac {1}{4} B c x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin {align*} \int (A+B x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a A+(A b+a B) x+(b B+A c) x^2+B c x^3\right ) \, dx\\ &=a A x+\frac {1}{2} (A b+a B) x^2+\frac {1}{3} (b B+A c) x^3+\frac {1}{4} B c x^4\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 (a B+A b)+a A x+\frac {1}{3} x^3 (A c+b B)+\frac {1}{4} B c x^4 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) \left (a+b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.34, size = 40, normalized size = 0.95 \begin {gather*} \frac {1}{4} x^{4} c B + \frac {1}{3} x^{3} b B + \frac {1}{3} x^{3} c A + \frac {1}{2} x^{2} a B + \frac {1}{2} x^{2} b A + x a A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 40, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, B c x^{4} + \frac {1}{3} \, B b x^{3} + \frac {1}{3} \, A c x^{3} + \frac {1}{2} \, B a x^{2} + \frac {1}{2} \, A b x^{2} + A a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 37, normalized size = 0.88 \begin {gather*} \frac {B c \,x^{4}}{4}+A a x +\frac {\left (A c +b B \right ) x^{3}}{3}+\frac {\left (A b +B a \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 36, normalized size = 0.86 \begin {gather*} \frac {1}{4} \, B c x^{4} + \frac {1}{3} \, {\left (B b + A c\right )} x^{3} + A a x + \frac {1}{2} \, {\left (B a + A b\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 38, normalized size = 0.90 \begin {gather*} \frac {B\,c\,x^4}{4}+\left (\frac {A\,c}{3}+\frac {B\,b}{3}\right )\,x^3+\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x^2+A\,a\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 39, normalized size = 0.93 \begin {gather*} A a x + \frac {B c x^{4}}{4} + x^{3} \left (\frac {A c}{3} + \frac {B b}{3}\right ) + x^{2} \left (\frac {A b}{2} + \frac {B a}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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