Optimal. Leaf size=69 \[ \frac{c^3 (a x+1)^8}{8 a^2}-\frac{5 c^3 (a x+1)^7}{7 a^2}+\frac{4 c^3 (a x+1)^6}{3 a^2}-\frac{4 c^3 (a x+1)^5}{5 a^2} \]
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Rubi [A] time = 0.0799643, antiderivative size = 69, normalized size of antiderivative = 1., 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, 77} \[ \frac{c^3 (a x+1)^8}{8 a^2}-\frac{5 c^3 (a x+1)^7}{7 a^2}+\frac{4 c^3 (a x+1)^6}{3 a^2}-\frac{4 c^3 (a x+1)^5}{5 a^2} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 77
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (-\frac{4 (1+a x)^4}{a}+\frac{8 (1+a x)^5}{a}-\frac{5 (1+a x)^6}{a}+\frac{(1+a x)^7}{a}\right ) \, dx\\ &=-\frac{4 c^3 (1+a x)^5}{5 a^2}+\frac{4 c^3 (1+a x)^6}{3 a^2}-\frac{5 c^3 (1+a x)^7}{7 a^2}+\frac{c^3 (1+a x)^8}{8 a^2}\\ \end{align*}
Mathematica [A] time = 0.0297239, size = 70, normalized size = 1.01 \[ c^3 \left (\frac{a^6 x^8}{8}+\frac{2 a^5 x^7}{7}-\frac{a^4 x^6}{6}-\frac{4 a^3 x^5}{5}-\frac{a^2 x^4}{4}+\frac{2 a x^3}{3}+\frac{x^2}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 57, normalized size = 0.8 \begin{align*}{c}^{3} \left ({\frac{{x}^{8}{a}^{6}}{8}}+{\frac{2\,{a}^{5}{x}^{7}}{7}}-{\frac{{x}^{6}{a}^{4}}{6}}-{\frac{4\,{x}^{5}{a}^{3}}{5}}-{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{2\,{x}^{3}a}{3}}+{\frac{{x}^{2}}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961356, size = 99, normalized size = 1.43 \begin{align*} \frac{1}{8} \, a^{6} c^{3} x^{8} + \frac{2}{7} \, a^{5} c^{3} x^{7} - \frac{1}{6} \, a^{4} c^{3} x^{6} - \frac{4}{5} \, a^{3} c^{3} x^{5} - \frac{1}{4} \, a^{2} c^{3} x^{4} + \frac{2}{3} \, a c^{3} x^{3} + \frac{1}{2} \, c^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71556, size = 161, normalized size = 2.33 \begin{align*} \frac{1}{8} \, a^{6} c^{3} x^{8} + \frac{2}{7} \, a^{5} c^{3} x^{7} - \frac{1}{6} \, a^{4} c^{3} x^{6} - \frac{4}{5} \, a^{3} c^{3} x^{5} - \frac{1}{4} \, a^{2} c^{3} x^{4} + \frac{2}{3} \, a c^{3} x^{3} + \frac{1}{2} \, c^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.096283, size = 82, normalized size = 1.19 \begin{align*} \frac{a^{6} c^{3} x^{8}}{8} + \frac{2 a^{5} c^{3} x^{7}}{7} - \frac{a^{4} c^{3} x^{6}}{6} - \frac{4 a^{3} c^{3} x^{5}}{5} - \frac{a^{2} c^{3} x^{4}}{4} + \frac{2 a c^{3} x^{3}}{3} + \frac{c^{3} x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13904, size = 99, normalized size = 1.43 \begin{align*} \frac{1}{8} \, a^{6} c^{3} x^{8} + \frac{2}{7} \, a^{5} c^{3} x^{7} - \frac{1}{6} \, a^{4} c^{3} x^{6} - \frac{4}{5} \, a^{3} c^{3} x^{5} - \frac{1}{4} \, a^{2} c^{3} x^{4} + \frac{2}{3} \, a c^{3} x^{3} + \frac{1}{2} \, c^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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