Optimal. Leaf size=62 \[ -\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{d (d+e x)^4 (c d-b e)}{4 e^3}+\frac{c (d+e x)^6}{6 e^3} \]
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Rubi [A] time = 0.0488904, antiderivative size = 62, normalized size of antiderivative = 1., 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} \[ -\frac{(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac{d (d+e x)^4 (c d-b e)}{4 e^3}+\frac{c (d+e x)^6}{6 e^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int (d+e x)^3 \left (b x+c x^2\right ) \, dx &=\int \left (\frac{d (c d-b e) (d+e x)^3}{e^2}+\frac{(-2 c d+b e) (d+e x)^4}{e^2}+\frac{c (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac{d (c d-b e) (d+e x)^4}{4 e^3}-\frac{(2 c d-b e) (d+e x)^5}{5 e^3}+\frac{c (d+e x)^6}{6 e^3}\\ \end{align*}
Mathematica [A] time = 0.0162647, size = 67, normalized size = 1.08 \[ \frac{1}{60} x^2 \left (20 d^2 x (3 b e+c d)+12 e^2 x^3 (b e+3 c d)+45 d e x^2 (b e+c d)+30 b d^3+10 c e^3 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 76, normalized size = 1.2 \begin{align*}{\frac{{e}^{3}c{x}^{6}}{6}}+{\frac{ \left ({e}^{3}b+3\,d{e}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,d{e}^{2}b+3\,{d}^{2}ec \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{d}^{2}eb+{d}^{3}c \right ){x}^{3}}{3}}+{\frac{{d}^{3}b{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1356, size = 99, normalized size = 1.6 \begin{align*} \frac{1}{6} \, c e^{3} x^{6} + \frac{1}{2} \, b d^{3} x^{2} + \frac{1}{5} \,{\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + \frac{3}{4} \,{\left (c d^{2} e + b d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{3} + 3 \, b d^{2} e\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42969, size = 177, normalized size = 2.85 \begin{align*} \frac{1}{6} x^{6} e^{3} c + \frac{3}{5} x^{5} e^{2} d c + \frac{1}{5} x^{5} e^{3} b + \frac{3}{4} x^{4} e d^{2} c + \frac{3}{4} x^{4} e^{2} d b + \frac{1}{3} x^{3} d^{3} c + x^{3} e d^{2} b + \frac{1}{2} x^{2} d^{3} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.288298, size = 80, normalized size = 1.29 \begin{align*} \frac{b d^{3} x^{2}}{2} + \frac{c e^{3} x^{6}}{6} + x^{5} \left (\frac{b e^{3}}{5} + \frac{3 c d e^{2}}{5}\right ) + x^{4} \left (\frac{3 b d e^{2}}{4} + \frac{3 c d^{2} e}{4}\right ) + x^{3} \left (b d^{2} e + \frac{c d^{3}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29579, size = 100, normalized size = 1.61 \begin{align*} \frac{1}{6} \, c x^{6} e^{3} + \frac{3}{5} \, c d x^{5} e^{2} + \frac{3}{4} \, c d^{2} x^{4} e + \frac{1}{3} \, c d^{3} x^{3} + \frac{1}{5} \, b x^{5} e^{3} + \frac{3}{4} \, b d x^{4} e^{2} + b d^{2} x^{3} e + \frac{1}{2} \, b d^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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