Optimal. Leaf size=29 \[ \frac{1}{6} a^2 c x^6+\frac{2}{5} a c x^5+\frac{c x^4}{4} \]
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Rubi [A] time = 0.0546217, 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} \[ \frac{1}{6} a^2 c x^6+\frac{2}{5} a c x^5+\frac{c x^4}{4} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx &=c \int x^3 (1+a x)^2 \, dx\\ &=c \int \left (x^3+2 a x^4+a^2 x^5\right ) \, dx\\ &=\frac{c x^4}{4}+\frac{2}{5} a c x^5+\frac{1}{6} a^2 c x^6\\ \end{align*}
Mathematica [A] time = 0.0165648, size = 22, normalized size = 0.76 \[ \frac{1}{60} c x^4 \left (10 a^2 x^2+24 a x+15\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 23, normalized size = 0.8 \begin{align*} c \left ({\frac{{x}^{6}{a}^{2}}{6}}+{\frac{2\,a{x}^{5}}{5}}+{\frac{{x}^{4}}{4}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.934618, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{6} \, a^{2} c x^{6} + \frac{2}{5} \, a c x^{5} + \frac{1}{4} \, c x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92134, size = 55, normalized size = 1.9 \begin{align*} \frac{1}{6} \, a^{2} c x^{6} + \frac{2}{5} \, a c x^{5} + \frac{1}{4} \, c x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.077675, size = 26, normalized size = 0.9 \begin{align*} \frac{a^{2} c x^{6}}{6} + \frac{2 a c x^{5}}{5} + \frac{c x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16305, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{6} \, a^{2} c x^{6} + \frac{2}{5} \, a c x^{5} + \frac{1}{4} \, c x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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