Optimal. Leaf size=45 \[ a^2 d x+\frac{2}{3} a c d x^3+\frac{e \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} c^2 d x^5 \]
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Rubi [A] time = 0.0152334, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {641, 194} \[ a^2 d x+\frac{2}{3} a c d x^3+\frac{e \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} c^2 d x^5 \]
Antiderivative was successfully verified.
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Rule 641
Rule 194
Rubi steps
\begin{align*} \int (d+e x) \left (a+c x^2\right )^2 \, dx &=\frac{e \left (a+c x^2\right )^3}{6 c}+d \int \left (a+c x^2\right )^2 \, dx\\ &=\frac{e \left (a+c x^2\right )^3}{6 c}+d \int \left (a^2+2 a c x^2+c^2 x^4\right ) \, dx\\ &=a^2 d x+\frac{2}{3} a c d x^3+\frac{1}{5} c^2 d x^5+\frac{e \left (a+c x^2\right )^3}{6 c}\\ \end{align*}
Mathematica [A] time = 0.0021371, size = 60, normalized size = 1.33 \[ a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a c d x^3+\frac{1}{2} a c e x^4+\frac{1}{5} c^2 d x^5+\frac{1}{6} c^2 e x^6 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 51, normalized size = 1.1 \begin{align*}{\frac{{c}^{2}e{x}^{6}}{6}}+{\frac{{c}^{2}d{x}^{5}}{5}}+{\frac{ace{x}^{4}}{2}}+{\frac{2\,acd{x}^{3}}{3}}+{\frac{{a}^{2}e{x}^{2}}{2}}+{a}^{2}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13861, size = 68, normalized size = 1.51 \begin{align*} \frac{1}{6} \, c^{2} e x^{6} + \frac{1}{5} \, c^{2} d x^{5} + \frac{1}{2} \, a c e x^{4} + \frac{2}{3} \, a c d x^{3} + \frac{1}{2} \, a^{2} e x^{2} + a^{2} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57441, size = 120, normalized size = 2.67 \begin{align*} \frac{1}{6} x^{6} e c^{2} + \frac{1}{5} x^{5} d c^{2} + \frac{1}{2} x^{4} e c a + \frac{2}{3} x^{3} d c a + \frac{1}{2} x^{2} e a^{2} + x d a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.123217, size = 58, normalized size = 1.29 \begin{align*} a^{2} d x + \frac{a^{2} e x^{2}}{2} + \frac{2 a c d x^{3}}{3} + \frac{a c e x^{4}}{2} + \frac{c^{2} d x^{5}}{5} + \frac{c^{2} e x^{6}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24112, size = 72, normalized size = 1.6 \begin{align*} \frac{1}{6} \, c^{2} x^{6} e + \frac{1}{5} \, c^{2} d x^{5} + \frac{1}{2} \, a c x^{4} e + \frac{2}{3} \, a c d x^{3} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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