Optimal. Leaf size=78 \[ \frac{1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+\frac{1}{2} a d^2 x^2+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{6} c e^2 x^6 \]
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Rubi [A] time = 0.0734819, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {771} \[ \frac{1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{3} d x^3 (2 a e+b d)+\frac{1}{2} a d^2 x^2+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{6} c e^2 x^6 \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int x (d+e x)^2 \left (a+b x+c x^2\right ) \, dx &=\int \left (a d^2 x+d (b d+2 a e) x^2+\left (c d^2+e (2 b d+a e)\right ) x^3+e (2 c d+b e) x^4+c e^2 x^5\right ) \, dx\\ &=\frac{1}{2} a d^2 x^2+\frac{1}{3} d (b d+2 a e) x^3+\frac{1}{4} \left (c d^2+e (2 b d+a e)\right ) x^4+\frac{1}{5} e (2 c d+b e) x^5+\frac{1}{6} c e^2 x^6\\ \end{align*}
Mathematica [A] time = 0.0255852, size = 70, normalized size = 0.9 \[ \frac{1}{60} x^2 \left (15 x^2 \left (e (a e+2 b d)+c d^2\right )+20 d x (2 a e+b d)+30 a d^2+12 e x^3 (b e+2 c d)+10 c e^2 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 73, normalized size = 0.9 \begin{align*}{\frac{c{e}^{2}{x}^{6}}{6}}+{\frac{ \left ( b{e}^{2}+2\,cde \right ){x}^{5}}{5}}+{\frac{ \left ( a{e}^{2}+2\,bde+c{d}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,ade+b{d}^{2} \right ){x}^{3}}{3}}+{\frac{a{d}^{2}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01368, size = 97, normalized size = 1.24 \begin{align*} \frac{1}{6} \, c e^{2} x^{6} + \frac{1}{5} \,{\left (2 \, c d e + b e^{2}\right )} x^{5} + \frac{1}{2} \, a d^{2} x^{2} + \frac{1}{4} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (b d^{2} + 2 \, a d e\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0888, size = 193, normalized size = 2.47 \begin{align*} \frac{1}{6} x^{6} e^{2} c + \frac{2}{5} x^{5} e d c + \frac{1}{5} x^{5} e^{2} b + \frac{1}{4} x^{4} d^{2} c + \frac{1}{2} x^{4} e d b + \frac{1}{4} x^{4} e^{2} a + \frac{1}{3} x^{3} d^{2} b + \frac{2}{3} x^{3} e d a + \frac{1}{2} x^{2} d^{2} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.073932, size = 80, normalized size = 1.03 \begin{align*} \frac{a d^{2} x^{2}}{2} + \frac{c e^{2} x^{6}}{6} + x^{5} \left (\frac{b e^{2}}{5} + \frac{2 c d e}{5}\right ) + x^{4} \left (\frac{a e^{2}}{4} + \frac{b d e}{2} + \frac{c d^{2}}{4}\right ) + x^{3} \left (\frac{2 a d e}{3} + \frac{b d^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07128, size = 107, normalized size = 1.37 \begin{align*} \frac{1}{6} \, c x^{6} e^{2} + \frac{2}{5} \, c d x^{5} e + \frac{1}{4} \, c d^{2} x^{4} + \frac{1}{5} \, b x^{5} e^{2} + \frac{1}{2} \, b d x^{4} e + \frac{1}{3} \, b d^{2} x^{3} + \frac{1}{4} \, a x^{4} e^{2} + \frac{2}{3} \, a d x^{3} e + \frac{1}{2} \, a d^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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