Optimal. Leaf size=55 \[ \frac {1}{4} e x^4 (b e+2 c d)+\frac {1}{3} d x^3 (2 b e+c d)+\frac {1}{2} b d^2 x^2+\frac {1}{5} c e^2 x^5 \]
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Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {698} \begin {gather*} \frac {1}{4} e x^4 (b e+2 c d)+\frac {1}{3} d x^3 (2 b e+c d)+\frac {1}{2} b d^2 x^2+\frac {1}{5} c e^2 x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int (d+e x)^2 \left (b x+c x^2\right ) \, dx &=\int \left (b d^2 x+d (c d+2 b e) x^2+e (2 c d+b e) x^3+c e^2 x^4\right ) \, dx\\ &=\frac {1}{2} b d^2 x^2+\frac {1}{3} d (c d+2 b e) x^3+\frac {1}{4} e (2 c d+b e) x^4+\frac {1}{5} c e^2 x^5\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 0.89 \begin {gather*} \frac {1}{60} x^2 \left (15 e x^2 (b e+2 c d)+20 d x (2 b e+c d)+30 b d^2+12 c e^2 x^3\right ) \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)^2 \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 = 53, normalized size = 0.96 \begin {gather*} \frac {1}{5} x^{5} e^{2} c + \frac {1}{2} x^{4} e d c + \frac {1}{4} x^{4} e^{2} b + \frac {1}{3} x^{3} d^{2} c + \frac {2}{3} x^{3} e d b + \frac {1}{2} x^{2} d^{2} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{5} \, c x^{5} e^{2} + \frac {1}{2} \, c d x^{4} e + \frac {1}{3} \, c d^{2} x^{3} + \frac {1}{4} \, b x^{4} e^{2} + \frac {2}{3} \, b d x^{3} e + \frac {1}{2} \, b d^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 52, normalized size = 0.95 \begin {gather*} \frac {c \,e^{2} x^{5}}{5}+\frac {b \,d^{2} x^{2}}{2}+\frac {\left (b \,e^{2}+2 d e c \right ) x^{4}}{4}+\frac {\left (2 b d e +c \,d^{2}\right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{5} \, c e^{2} x^{5} + \frac {1}{2} \, b d^{2} x^{2} + \frac {1}{4} \, {\left (2 \, c d e + b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{2} + 2 \, b d e\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 0.93 \begin {gather*} x^3\,\left (\frac {c\,d^2}{3}+\frac {2\,b\,e\,d}{3}\right )+x^4\,\left (\frac {b\,e^2}{4}+\frac {c\,d\,e}{2}\right )+\frac {b\,d^2\,x^2}{2}+\frac {c\,e^2\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 54, normalized size = 0.98 \begin {gather*} \frac {b d^{2} x^{2}}{2} + \frac {c e^{2} x^{5}}{5} + x^{4} \left (\frac {b e^{2}}{4} + \frac {c d e}{2}\right ) + x^{3} \left (\frac {2 b d e}{3} + \frac {c d^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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