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 = 0.0597232, antiderivative size = 37, normalized size of antiderivative = 1., 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 6167
Rule 6129
Rule 43
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*}
Mathematica [A] time = 0.0191041, 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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Maple [A] time = 0.039, size = 47, normalized size = 1.3 \begin{align*}{c}^{5} \left ( -{\frac{{x}^{6}{a}^{5}}{6}}+{\frac{3\,{x}^{5}{a}^{4}}{5}}-{\frac{{x}^{4}{a}^{3}}{2}}-{\frac{2\,{x}^{3}{a}^{2}}{3}}+{\frac{3\,a{x}^{2}}{2}}-x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0412, size = 81, normalized size = 2.19 \begin{align*} -\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51856, size = 130, normalized size = 3.51 \begin{align*} -\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.096712, size = 66, normalized size = 1.78 \begin{align*} - \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14329, size = 81, normalized size = 2.19 \begin{align*} -\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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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