Optimal. Leaf size=79 \[ \frac{1}{3} x^3 \left (2 a c e+b^2 e+3 b c d\right )+\frac{1}{2} x^2 \left (a b e+2 a c d+b^2 d\right )+a b d x+\frac{1}{4} c x^4 (3 b e+2 c d)+\frac{2}{5} c^2 e x^5 \]
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Rubi [A] time = 0.0642936, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {771} \[ \frac{1}{3} x^3 \left (2 a c e+b^2 e+3 b c d\right )+\frac{1}{2} x^2 \left (a b e+2 a c d+b^2 d\right )+a b d x+\frac{1}{4} c x^4 (3 b e+2 c d)+\frac{2}{5} c^2 e x^5 \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a b d+\left (b^2 d+2 a c d+a b e\right ) x+\left (3 b c d+b^2 e+2 a c e\right ) x^2+c (2 c d+3 b e) x^3+2 c^2 e x^4\right ) \, dx\\ &=a b d x+\frac{1}{2} \left (b^2 d+2 a c d+a b e\right ) x^2+\frac{1}{3} \left (3 b c d+b^2 e+2 a c e\right ) x^3+\frac{1}{4} c (2 c d+3 b e) x^4+\frac{2}{5} c^2 e x^5\\ \end{align*}
Mathematica [A] time = 0.0181632, size = 79, normalized size = 1. \[ \frac{1}{3} x^3 \left (2 a c e+b^2 e+3 b c d\right )+\frac{1}{2} x^2 \left (a b e+2 a c d+b^2 d\right )+a b d x+\frac{1}{4} c x^4 (3 b e+2 c d)+\frac{2}{5} c^2 e x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 83, normalized size = 1.1 \begin{align*}{\frac{2\,{c}^{2}e{x}^{5}}{5}}+{\frac{ \left ( \left ( be+2\,cd \right ) c+2\,ceb \right ){x}^{4}}{4}}+{\frac{ \left ( bcd+ \left ( be+2\,cd \right ) b+2\,ace \right ){x}^{3}}{3}}+{\frac{ \left ({b}^{2}d+ \left ( be+2\,cd \right ) a \right ){x}^{2}}{2}}+abdx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969976, size = 99, normalized size = 1.25 \begin{align*} \frac{2}{5} \, c^{2} e x^{5} + \frac{1}{4} \,{\left (2 \, c^{2} d + 3 \, b c e\right )} x^{4} + a b d x + \frac{1}{3} \,{\left (3 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41207, size = 196, normalized size = 2.48 \begin{align*} \frac{2}{5} x^{5} e c^{2} + \frac{1}{2} x^{4} d c^{2} + \frac{3}{4} x^{4} e c b + x^{3} d c b + \frac{1}{3} x^{3} e b^{2} + \frac{2}{3} x^{3} e c a + \frac{1}{2} x^{2} d b^{2} + x^{2} d c a + \frac{1}{2} x^{2} e b a + x d b a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.073326, size = 82, normalized size = 1.04 \begin{align*} a b d x + \frac{2 c^{2} e x^{5}}{5} + x^{4} \left (\frac{3 b c e}{4} + \frac{c^{2} d}{2}\right ) + x^{3} \left (\frac{2 a c e}{3} + \frac{b^{2} e}{3} + b c d\right ) + x^{2} \left (\frac{a b e}{2} + a c d + \frac{b^{2} d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18525, size = 115, normalized size = 1.46 \begin{align*} \frac{2}{5} \, c^{2} x^{5} e + \frac{1}{2} \, c^{2} d x^{4} + \frac{3}{4} \, b c x^{4} e + b c d x^{3} + \frac{1}{3} \, b^{2} x^{3} e + \frac{2}{3} \, a c x^{3} e + \frac{1}{2} \, b^{2} d x^{2} + a c d x^{2} + \frac{1}{2} \, a b x^{2} e + a b d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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