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 = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 43
Rule 6150
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.52, size = 23, normalized size = 0.79 \[ \frac {1}{5} \, a^{2} c x^{5} + \frac {1}{2} \, a c x^{4} + \frac {1}{3} \, c x^{3} \]
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 \[ \frac {1}{5} \, a^{2} c x^{5} + \frac {1}{2} \, a c x^{4} + \frac {1}{3} \, c x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 23, normalized size = 0.79 \[ c \left (\frac {1}{5} x^{5} a^{2}+\frac {1}{2} x^{4} a +\frac {1}{3} x^{3}\right ) \]
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 \[ \frac {1}{5} \, a^{2} c x^{5} + \frac {1}{2} \, a c x^{4} + \frac {1}{3} \, c x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 20, normalized size = 0.69 \[ \frac {c\,x^3\,\left (6\,a^2\,x^2+15\,a\,x+10\right )}{30} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 24, normalized size = 0.83 \[ \frac {a^{2} c x^{5}}{5} + \frac {a c x^{4}}{2} + \frac {c x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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