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 = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 43
Rule 6150
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.57, size = 23, normalized size = 0.79 \[ \frac {1}{4} \, a^{2} c x^{4} + \frac {2}{3} \, a c x^{3} + \frac {1}{2} \, c x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 23, normalized size = 0.79 \[ \frac {1}{4} \, a^{2} c x^{4} + \frac {2}{3} \, a c x^{3} + \frac {1}{2} \, c x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 23, normalized size = 0.79 \[ c \left (\frac {1}{4} a^{2} x^{4}+\frac {2}{3} x^{3} a +\frac {1}{2} x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 23, normalized size = 0.79 \[ \frac {1}{4} \, a^{2} c x^{4} + \frac {2}{3} \, a c x^{3} + \frac {1}{2} \, c x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 20, normalized size = 0.69 \[ \frac {c\,x^2\,\left (3\,a^2\,x^2+8\,a\,x+6\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 26, normalized size = 0.90 \[ \frac {a^{2} c x^{4}}{4} + \frac {2 a c x^{3}}{3} + \frac {c x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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