Optimal. Leaf size=37 \[ \frac {2 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \]
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Rubi [A] time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ \frac {2 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^5 \, dx\\ &=-\left (c^5 \int (1-a x)^4 (1+a x) \, dx\right )\\ &=-\left (c^5 \int \left (2 (1-a x)^4-(1-a x)^5\right ) \, dx\right )\\ &=\frac {2 c^5 (1-a x)^5}{5 a}-\frac {c^5 (1-a x)^6}{6 a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.62 \[ -\frac {c^5 (a x-1)^5 (5 a x+7)}{30 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 60, normalized size = 1.62 \[ -\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {3}{5} \, a^{4} c^{5} x^{5} - \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} + \frac {3}{2} \, a c^{5} x^{2} - c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 60, normalized size = 1.62 \[ -\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {3}{5} \, a^{4} c^{5} x^{5} - \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} + \frac {3}{2} \, a c^{5} x^{2} - c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 47, normalized size = 1.27 \[ c^{5} \left (-\frac {1}{6} x^{6} a^{5}+\frac {3}{5} a^{4} x^{5}-\frac {1}{2} x^{4} a^{3}-\frac {2}{3} x^{3} a^{2}+\frac {3}{2} a \,x^{2}-x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 60, normalized size = 1.62 \[ -\frac {1}{6} \, a^{5} c^{5} x^{6} + \frac {3}{5} \, a^{4} c^{5} x^{5} - \frac {1}{2} \, a^{3} c^{5} x^{4} - \frac {2}{3} \, a^{2} c^{5} x^{3} + \frac {3}{2} \, a c^{5} x^{2} - c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 60, normalized size = 1.62 \[ -\frac {a^5\,c^5\,x^6}{6}+\frac {3\,a^4\,c^5\,x^5}{5}-\frac {a^3\,c^5\,x^4}{2}-\frac {2\,a^2\,c^5\,x^3}{3}+\frac {3\,a\,c^5\,x^2}{2}-c^5\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.08, size = 66, normalized size = 1.78 \[ - \frac {a^{5} c^{5} x^{6}}{6} + \frac {3 a^{4} c^{5} x^{5}}{5} - \frac {a^{3} c^{5} x^{4}}{2} - \frac {2 a^{2} c^{5} x^{3}}{3} + \frac {3 a c^{5} x^{2}}{2} - c^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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