Optimal. Leaf size=39 \[ \frac{1}{4} (d+e x)^4 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^5}{5 e^2} \]
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Rubi [A] time = 0.0171029, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {626, 43} \[ \frac{1}{4} (d+e x)^4 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^5}{5 e^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx &=\int (a e+c d x) (d+e x)^3 \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right ) (d+e x)^3}{e}+\frac{c d (d+e x)^4}{e}\right ) \, dx\\ &=\frac{1}{4} \left (a-\frac{c d^2}{e^2}\right ) (d+e x)^4+\frac{c d (d+e x)^5}{5 e^2}\\ \end{align*}
Mathematica [A] time = 0.0195682, size = 73, normalized size = 1.87 \[ \frac{1}{20} x \left (5 a e \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+c d x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 112, normalized size = 2.9 \begin{align*}{\frac{{e}^{3}dc{x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{2}{e}^{2}c+{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( c{d}^{3}e+2\,de \left ( a{e}^{2}+c{d}^{2} \right ) +ad{e}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ({d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +2\,a{d}^{2}{e}^{2} \right ){x}^{2}}{2}}+{d}^{3}aex \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1614, size = 101, normalized size = 2.59 \begin{align*} \frac{1}{5} \, c d e^{3} x^{5} + a d^{3} e x + \frac{1}{4} \,{\left (3 \, c d^{2} e^{2} + a e^{4}\right )} x^{4} +{\left (c d^{3} e + a d e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (c d^{4} + 3 \, a d^{2} e^{2}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.32734, size = 174, normalized size = 4.46 \begin{align*} \frac{1}{5} x^{5} e^{3} d c + \frac{3}{4} x^{4} e^{2} d^{2} c + \frac{1}{4} x^{4} e^{4} a + x^{3} e d^{3} c + x^{3} e^{3} d a + \frac{1}{2} x^{2} d^{4} c + \frac{3}{2} x^{2} e^{2} d^{2} a + x e d^{3} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.115237, size = 80, normalized size = 2.05 \begin{align*} a d^{3} e x + \frac{c d e^{3} x^{5}}{5} + x^{4} \left (\frac{a e^{4}}{4} + \frac{3 c d^{2} e^{2}}{4}\right ) + x^{3} \left (a d e^{3} + c d^{3} e\right ) + x^{2} \left (\frac{3 a d^{2} e^{2}}{2} + \frac{c d^{4}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24991, size = 101, normalized size = 2.59 \begin{align*} \frac{1}{5} \, c d x^{5} e^{3} + \frac{3}{4} \, c d^{2} x^{4} e^{2} + c d^{3} x^{3} e + \frac{1}{2} \, c d^{4} x^{2} + \frac{1}{4} \, a x^{4} e^{4} + a d x^{3} e^{3} + \frac{3}{2} \, a d^{2} x^{2} e^{2} + a d^{3} x e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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