Optimal. Leaf size=56 \[ a^2 c d x^3+a^3 d x+\frac{3}{5} a c^2 d x^5+\frac{e \left (a+c x^2\right )^4}{8 c}+\frac{1}{7} c^3 d x^7 \]
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Rubi [A] time = 0.021189, antiderivative size = 56, 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} \[ a^2 c d x^3+a^3 d x+\frac{3}{5} a c^2 d x^5+\frac{e \left (a+c x^2\right )^4}{8 c}+\frac{1}{7} c^3 d x^7 \]
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 )^3 \, dx &=\frac{e \left (a+c x^2\right )^4}{8 c}+d \int \left (a+c x^2\right )^3 \, dx\\ &=\frac{e \left (a+c x^2\right )^4}{8 c}+d \int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx\\ &=a^3 d x+a^2 c d x^3+\frac{3}{5} a c^2 d x^5+\frac{1}{7} c^3 d x^7+\frac{e \left (a+c x^2\right )^4}{8 c}\\ \end{align*}
Mathematica [A] time = 0.0024985, size = 85, normalized size = 1.52 \[ a^2 c d x^3+\frac{3}{4} a^2 c e x^4+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{3}{5} a c^2 d x^5+\frac{1}{2} a c^2 e x^6+\frac{1}{7} c^3 d x^7+\frac{1}{8} c^3 e x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 74, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}e{x}^{8}}{8}}+{\frac{{c}^{3}d{x}^{7}}{7}}+{\frac{ea{c}^{2}{x}^{6}}{2}}+{\frac{3\,a{c}^{2}d{x}^{5}}{5}}+{\frac{3\,e{a}^{2}c{x}^{4}}{4}}+{a}^{2}cd{x}^{3}+{\frac{{a}^{3}e{x}^{2}}{2}}+{a}^{3}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17, size = 99, normalized size = 1.77 \begin{align*} \frac{1}{8} \, c^{3} e x^{8} + \frac{1}{7} \, c^{3} d x^{7} + \frac{1}{2} \, a c^{2} e x^{6} + \frac{3}{5} \, a c^{2} d x^{5} + \frac{3}{4} \, a^{2} c e x^{4} + a^{2} c d x^{3} + \frac{1}{2} \, a^{3} e x^{2} + a^{3} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81113, size = 169, normalized size = 3.02 \begin{align*} \frac{1}{8} x^{8} e c^{3} + \frac{1}{7} x^{7} d c^{3} + \frac{1}{2} x^{6} e c^{2} a + \frac{3}{5} x^{5} d c^{2} a + \frac{3}{4} x^{4} e c a^{2} + x^{3} d c a^{2} + \frac{1}{2} x^{2} e a^{3} + x d a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.087168, size = 85, normalized size = 1.52 \begin{align*} a^{3} d x + \frac{a^{3} e x^{2}}{2} + a^{2} c d x^{3} + \frac{3 a^{2} c e x^{4}}{4} + \frac{3 a c^{2} d x^{5}}{5} + \frac{a c^{2} e x^{6}}{2} + \frac{c^{3} d x^{7}}{7} + \frac{c^{3} e x^{8}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31346, size = 104, normalized size = 1.86 \begin{align*} \frac{1}{8} \, c^{3} x^{8} e + \frac{1}{7} \, c^{3} d x^{7} + \frac{1}{2} \, a c^{2} x^{6} e + \frac{3}{5} \, a c^{2} d x^{5} + \frac{3}{4} \, a^{2} c x^{4} e + a^{2} c d x^{3} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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