Optimal. Leaf size=33 \[ \frac {1}{3} x^3 (b e+c d)+\frac {1}{2} b d x^2+\frac {1}{4} c e x^4 \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {631} \begin {gather*} \frac {1}{3} x^3 (b e+c d)+\frac {1}{2} b d x^2+\frac {1}{4} c e x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin {align*} \int (d+e x) \left (b x+c x^2\right ) \, dx &=\int \left (b d x+(c d+b e) x^2+c e x^3\right ) \, dx\\ &=\frac {1}{2} b d x^2+\frac {1}{3} (c d+b e) x^3+\frac {1}{4} c e x^4\\ \end {align*}
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Mathematica [A] time = 0.00, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{12} x^2 (b (6 d+4 e x)+c x (4 d+3 e x)) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{4} x^{4} e c + \frac {1}{3} x^{3} d c + \frac {1}{3} x^{3} e b + \frac {1}{2} x^{2} d b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 31, normalized size = 0.94 \begin {gather*} \frac {1}{4} \, c x^{4} e + \frac {1}{3} \, c d x^{3} + \frac {1}{3} \, b x^{3} e + \frac {1}{2} \, b d x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 0.85 \begin {gather*} \frac {c e \,x^{4}}{4}+\frac {b d \,x^{2}}{2}+\frac {\left (b e +c d \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{4} \, c e x^{4} + \frac {1}{2} \, b d x^{2} + \frac {1}{3} \, {\left (c d + b e\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 28, normalized size = 0.85 \begin {gather*} \frac {c\,e\,x^4}{4}+\left (\frac {b\,e}{3}+\frac {c\,d}{3}\right )\,x^3+\frac {b\,d\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 29, normalized size = 0.88 \begin {gather*} \frac {b d x^{2}}{2} + \frac {c e x^{4}}{4} + x^{3} \left (\frac {b e}{3} + \frac {c d}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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