Optimal. Leaf size=57 \[ \frac{(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac{c (d+e x)^5}{5 e^3}-\frac{c d (d+e x)^4}{2 e^3} \]
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Rubi [A] time = 0.0320099, 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} \[ \frac{(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac{c (d+e x)^5}{5 e^3}-\frac{c d (d+e x)^4}{2 e^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a+c x^2\right ) \, dx &=\int \left (\frac{\left (c d^2+a e^2\right ) (d+e x)^2}{e^2}-\frac{2 c d (d+e x)^3}{e^2}+\frac{c (d+e x)^4}{e^2}\right ) \, dx\\ &=\frac{\left (c d^2+a e^2\right ) (d+e x)^3}{3 e^3}-\frac{c d (d+e x)^4}{2 e^3}+\frac{c (d+e x)^5}{5 e^3}\\ \end{align*}
Mathematica [A] time = 0.0083728, size = 53, normalized size = 0.93 \[ \frac{1}{3} x^3 \left (a e^2+c d^2\right )+a d^2 x+a d e x^2+\frac{1}{2} c d e x^4+\frac{1}{5} c e^2 x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 48, normalized size = 0.8 \begin{align*}{\frac{c{e}^{2}{x}^{5}}{5}}+{\frac{cde{x}^{4}}{2}}+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ){x}^{3}}{3}}+ade{x}^{2}+a{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.29003, size = 63, normalized size = 1.11 \begin{align*} \frac{1}{5} \, c e^{2} x^{5} + \frac{1}{2} \, c d e x^{4} + a d e x^{2} + a d^{2} x + \frac{1}{3} \,{\left (c d^{2} + a e^{2}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71241, size = 115, normalized size = 2.02 \begin{align*} \frac{1}{5} x^{5} e^{2} c + \frac{1}{2} x^{4} e d c + \frac{1}{3} x^{3} d^{2} c + \frac{1}{3} x^{3} e^{2} a + x^{2} e d a + x d^{2} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.109106, size = 51, normalized size = 0.89 \begin{align*} a d^{2} x + a d e x^{2} + \frac{c d e x^{4}}{2} + \frac{c e^{2} x^{5}}{5} + x^{3} \left (\frac{a e^{2}}{3} + \frac{c d^{2}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25979, size = 66, normalized size = 1.16 \begin{align*} \frac{1}{5} \, c x^{5} e^{2} + \frac{1}{2} \, c d x^{4} e + \frac{1}{3} \, c d^{2} x^{3} + \frac{1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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