Optimal. Leaf size=57 \[ \frac{(d+e x)^4 \left (a e^2+c d^2\right )}{4 e^3}+\frac{c (d+e x)^6}{6 e^3}-\frac{2 c d (d+e x)^5}{5 e^3} \]
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Rubi [A] time = 0.0455637, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {697} \[ \frac{(d+e x)^4 \left (a e^2+c d^2\right )}{4 e^3}+\frac{c (d+e x)^6}{6 e^3}-\frac{2 c d (d+e x)^5}{5 e^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+c x^2\right ) \, dx &=\int \left (\frac{\left (c d^2+a e^2\right ) (d+e x)^3}{e^2}-\frac{2 c d (d+e x)^4}{e^2}+\frac{c (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right ) (d+e x)^4}{4 e^3}-\frac{2 c d (d+e x)^5}{5 e^3}+\frac{c (d+e x)^6}{6 e^3}\\ \end{align*}
Mathematica [A] time = 0.013662, size = 74, normalized size = 1.3 \[ \frac{1}{4} a x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+\frac{1}{60} c x^3 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 73, normalized size = 1.3 \begin{align*}{\frac{{e}^{3}c{x}^{6}}{6}}+{\frac{3\,d{e}^{2}c{x}^{5}}{5}}+{\frac{ \left ({e}^{3}a+3\,{d}^{2}ec \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}a+{d}^{3}c \right ){x}^{3}}{3}}+{\frac{3\,{d}^{2}ea{x}^{2}}{2}}+{d}^{3}ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15385, size = 97, normalized size = 1.7 \begin{align*} \frac{1}{6} \, c e^{3} x^{6} + \frac{3}{5} \, c d e^{2} x^{5} + \frac{3}{2} \, a d^{2} e x^{2} + a d^{3} x + \frac{1}{4} \,{\left (3 \, c d^{2} e + a e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{3} + 3 \, a d e^{2}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60084, size = 169, normalized size = 2.96 \begin{align*} \frac{1}{6} x^{6} e^{3} c + \frac{3}{5} x^{5} e^{2} d c + \frac{3}{4} x^{4} e d^{2} c + \frac{1}{4} x^{4} e^{3} a + \frac{1}{3} x^{3} d^{3} c + x^{3} e^{2} d a + \frac{3}{2} x^{2} e d^{2} a + x d^{3} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.125414, size = 80, normalized size = 1.4 \begin{align*} a d^{3} x + \frac{3 a d^{2} e x^{2}}{2} + \frac{3 c d e^{2} x^{5}}{5} + \frac{c e^{3} x^{6}}{6} + x^{4} \left (\frac{a e^{3}}{4} + \frac{3 c d^{2} e}{4}\right ) + x^{3} \left (a d e^{2} + \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 = 1.31064, size = 96, normalized size = 1.68 \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}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac{3}{2} \, a d^{2} x^{2} e + a d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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