Optimal. Leaf size=29 \[ \frac{1}{4} a^2 c x^4+\frac{2}{3} a c x^3+\frac{c x^2}{2} \]
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Rubi [A] time = 0.040211, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {6150, 43} \[ \frac{1}{4} a^2 c x^4+\frac{2}{3} a c x^3+\frac{c x^2}{2} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right ) \, dx &=c \int x (1+a x)^2 \, dx\\ &=c \int \left (x+2 a x^2+a^2 x^3\right ) \, dx\\ &=\frac{c x^2}{2}+\frac{2}{3} a c x^3+\frac{1}{4} a^2 c x^4\\ \end{align*}
Mathematica [A] time = 0.014375, size = 22, normalized size = 0.76 \[ \frac{1}{12} c x^2 \left (3 a^2 x^2+8 a x+6\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 23, normalized size = 0.8 \begin{align*} c \left ({\frac{{a}^{2}{x}^{4}}{4}}+{\frac{2\,{x}^{3}a}{3}}+{\frac{{x}^{2}}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954782, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{4} \, a^{2} c x^{4} + \frac{2}{3} \, a c x^{3} + \frac{1}{2} \, c x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91731, size = 55, normalized size = 1.9 \begin{align*} \frac{1}{4} \, a^{2} c x^{4} + \frac{2}{3} \, a c x^{3} + \frac{1}{2} \, c x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.076318, size = 26, normalized size = 0.9 \begin{align*} \frac{a^{2} c x^{4}}{4} + \frac{2 a c x^{3}}{3} + \frac{c x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13828, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{4} \, a^{2} c x^{4} + \frac{2}{3} \, a c x^{3} + \frac{1}{2} \, c x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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