Optimal. Leaf size=29 \[ \frac{1}{5} a^2 c x^5+\frac{1}{2} a c x^4+\frac{c x^3}{3} \]
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Rubi [A] time = 0.0534224, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {6150, 43} \[ \frac{1}{5} a^2 c x^5+\frac{1}{2} a c x^4+\frac{c x^3}{3} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x^2 \left (c-a^2 c x^2\right ) \, dx &=c \int x^2 (1+a x)^2 \, dx\\ &=c \int \left (x^2+2 a x^3+a^2 x^4\right ) \, dx\\ &=\frac{c x^3}{3}+\frac{1}{2} a c x^4+\frac{1}{5} a^2 c x^5\\ \end{align*}
Mathematica [A] time = 0.014817, size = 22, normalized size = 0.76 \[ \frac{1}{30} c x^3 \left (6 a^2 x^2+15 a x+10\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 23, normalized size = 0.8 \begin{align*} c \left ({\frac{{x}^{5}{a}^{2}}{5}}+{\frac{{x}^{4}a}{2}}+{\frac{{x}^{3}}{3}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.939426, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{5} \, a^{2} c x^{5} + \frac{1}{2} \, a c x^{4} + \frac{1}{3} \, c x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94123, size = 55, normalized size = 1.9 \begin{align*} \frac{1}{5} \, a^{2} c x^{5} + \frac{1}{2} \, a c x^{4} + \frac{1}{3} \, c x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.077207, size = 24, normalized size = 0.83 \begin{align*} \frac{a^{2} c x^{5}}{5} + \frac{a c x^{4}}{2} + \frac{c x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12715, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{5} \, a^{2} c x^{5} + \frac{1}{2} \, a c x^{4} + \frac{1}{3} \, c x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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