Optimal. Leaf size=29 \[ \frac{1}{7} a^2 c x^7+\frac{1}{3} a c x^6+\frac{c x^5}{5} \]
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Rubi [A] time = 0.0546501, 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} \[ \frac{1}{7} a^2 c x^7+\frac{1}{3} a c x^6+\frac{c x^5}{5} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x^4 \left (c-a^2 c x^2\right ) \, dx &=c \int x^4 (1+a x)^2 \, dx\\ &=c \int \left (x^4+2 a x^5+a^2 x^6\right ) \, dx\\ &=\frac{c x^5}{5}+\frac{1}{3} a c x^6+\frac{1}{7} a^2 c x^7\\ \end{align*}
Mathematica [A] time = 0.0167055, size = 22, normalized size = 0.76 \[ \frac{1}{105} c x^5 \left (15 a^2 x^2+35 a x+21\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 23, normalized size = 0.8 \begin{align*} c \left ({\frac{{a}^{2}{x}^{7}}{7}}+{\frac{{x}^{6}a}{3}}+{\frac{{x}^{5}}{5}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946904, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{7} \, a^{2} c x^{7} + \frac{1}{3} \, a c x^{6} + \frac{1}{5} \, c x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86128, size = 55, normalized size = 1.9 \begin{align*} \frac{1}{7} \, a^{2} c x^{7} + \frac{1}{3} \, a c x^{6} + \frac{1}{5} \, c x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.077603, size = 24, normalized size = 0.83 \begin{align*} \frac{a^{2} c x^{7}}{7} + \frac{a c x^{6}}{3} + \frac{c x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18084, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{7} \, a^{2} c x^{7} + \frac{1}{3} \, a c x^{6} + \frac{1}{5} \, c x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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