Optimal. Leaf size=39 \[ \frac{1}{5} (d+e x)^5 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^6}{6 e^2} \]
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Rubi [A] time = 0.0180514, 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} \[ \frac{1}{5} (d+e x)^5 \left (a-\frac{c d^2}{e^2}\right )+\frac{c d (d+e x)^6}{6 e^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^3 \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)^4 \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right ) (d+e x)^4}{e}+\frac{c d (d+e x)^5}{e}\right ) \, dx\\ &=\frac{1}{5} \left (a-\frac{c d^2}{e^2}\right ) (d+e x)^5+\frac{c d (d+e x)^6}{6 e^2}\\ \end{align*}
Mathematica [B] time = 0.0190197, size = 95, normalized size = 2.44 \[ \frac{1}{30} x \left (6 a e \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+c d x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 155, normalized size = 4. \begin{align*}{\frac{{e}^{4}dc{x}^{6}}{6}}+{\frac{ \left ( 3\,{d}^{2}{e}^{3}c+{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{d}^{3}{e}^{2}c+3\,d{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +{e}^{4}ad \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{4}ec+3\,{d}^{2}e \left ( a{e}^{2}+c{d}^{2} \right ) +3\,{d}^{2}{e}^{3}a \right ){x}^{3}}{3}}+{\frac{ \left ({d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) +3\,{d}^{3}{e}^{2}a \right ){x}^{2}}{2}}+{d}^{4}aex \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01701, size = 138, normalized size = 3.54 \begin{align*} \frac{1}{6} \, c d e^{4} x^{6} + a d^{4} e x + \frac{1}{5} \,{\left (4 \, c d^{2} e^{3} + a e^{5}\right )} x^{5} + \frac{1}{2} \,{\left (3 \, c d^{3} e^{2} + 2 \, a d e^{4}\right )} x^{4} + \frac{2}{3} \,{\left (2 \, c d^{4} e + 3 \, a d^{2} e^{3}\right )} x^{3} + \frac{1}{2} \,{\left (c d^{5} + 4 \, a d^{3} e^{2}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.27457, size = 228, normalized size = 5.85 \begin{align*} \frac{1}{6} x^{6} e^{4} d c + \frac{4}{5} x^{5} e^{3} d^{2} c + \frac{1}{5} x^{5} e^{5} a + \frac{3}{2} x^{4} e^{2} d^{3} c + x^{4} e^{4} d a + \frac{4}{3} x^{3} e d^{4} c + 2 x^{3} e^{3} d^{2} a + \frac{1}{2} x^{2} d^{5} c + 2 x^{2} e^{2} d^{3} a + x e d^{4} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.092746, size = 107, normalized size = 2.74 \begin{align*} a d^{4} e x + \frac{c d e^{4} x^{6}}{6} + x^{5} \left (\frac{a e^{5}}{5} + \frac{4 c d^{2} e^{3}}{5}\right ) + x^{4} \left (a d e^{4} + \frac{3 c d^{3} e^{2}}{2}\right ) + x^{3} \left (2 a d^{2} e^{3} + \frac{4 c d^{4} e}{3}\right ) + x^{2} \left (2 a d^{3} e^{2} + \frac{c d^{5}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23442, size = 132, normalized size = 3.38 \begin{align*} \frac{1}{6} \, c d x^{6} e^{4} + \frac{4}{5} \, c d^{2} x^{5} e^{3} + \frac{3}{2} \, c d^{3} x^{4} e^{2} + \frac{4}{3} \, c d^{4} x^{3} e + \frac{1}{2} \, c d^{5} x^{2} + \frac{1}{5} \, a x^{5} e^{5} + a d x^{4} e^{4} + 2 \, a d^{2} x^{3} e^{3} + 2 \, a d^{3} x^{2} e^{2} + a d^{4} x e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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