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 = 0.05, 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} \[ \frac {1}{7} a^2 c x^7+\frac {1}{3} a c x^6+\frac {c x^5}{5} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.53, size = 23, normalized size = 0.79 \[ \frac {1}{7} \, a^{2} c x^{7} + \frac {1}{3} \, a c x^{6} + \frac {1}{5} \, c x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 23, normalized size = 0.79 \[ \frac {1}{7} \, a^{2} c x^{7} + \frac {1}{3} \, a c x^{6} + \frac {1}{5} \, c x^{5} \]
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 \[ c \left (\frac {1}{7} a^{2} x^{7}+\frac {1}{3} x^{6} a +\frac {1}{5} x^{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 23, normalized size = 0.79 \[ \frac {1}{7} \, a^{2} c x^{7} + \frac {1}{3} \, a c x^{6} + \frac {1}{5} \, c x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 20, normalized size = 0.69 \[ \frac {c\,x^5\,\left (15\,a^2\,x^2+35\,a\,x+21\right )}{105} \]
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 \[ \frac {a^{2} c x^{7}}{7} + \frac {a c x^{6}}{3} + \frac {c x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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