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 = 0.05, 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} \[ \frac {1}{6} a^2 c x^6+\frac {2}{5} a c x^5+\frac {c x^4}{4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.63, size = 23, normalized size = 0.79 \[ \frac {1}{6} \, a^{2} c x^{6} + \frac {2}{5} \, a c x^{5} + \frac {1}{4} \, c x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 23, normalized size = 0.79 \[ \frac {1}{6} \, a^{2} c x^{6} + \frac {2}{5} \, a c x^{5} + \frac {1}{4} \, c x^{4} \]
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 \[ c \left (\frac {1}{6} a^{2} x^{6}+\frac {2}{5} a \,x^{5}+\frac {1}{4} x^{4}\right ) \]
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 \[ \frac {1}{6} \, a^{2} c x^{6} + \frac {2}{5} \, a c x^{5} + \frac {1}{4} \, c x^{4} \]
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 \[ \frac {c\,x^4\,\left (10\,a^2\,x^2+24\,a\,x+15\right )}{60} \]
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 \[ \frac {a^{2} c x^{6}}{6} + \frac {2 a c x^{5}}{5} + \frac {c x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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