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 = 0.08, antiderivative size = 69, 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, 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 77
Rule 6150
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.49, size = 73, normalized size = 1.06 \[ \frac {1}{8} \, a^{6} c^{3} x^{8} + \frac {2}{7} \, a^{5} c^{3} x^{7} - \frac {1}{6} \, a^{4} c^{3} x^{6} - \frac {4}{5} \, a^{3} c^{3} x^{5} - \frac {1}{4} \, a^{2} c^{3} x^{4} + \frac {2}{3} \, a c^{3} x^{3} + \frac {1}{2} \, c^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 73, normalized size = 1.06 \[ \frac {1}{8} \, a^{6} c^{3} x^{8} + \frac {2}{7} \, a^{5} c^{3} x^{7} - \frac {1}{6} \, a^{4} c^{3} x^{6} - \frac {4}{5} \, a^{3} c^{3} x^{5} - \frac {1}{4} \, a^{2} c^{3} x^{4} + \frac {2}{3} \, a c^{3} x^{3} + \frac {1}{2} \, c^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 57, normalized size = 0.83 \[ c^{3} \left (\frac {1}{8} x^{8} a^{6}+\frac {2}{7} a^{5} x^{7}-\frac {1}{6} x^{6} a^{4}-\frac {4}{5} a^{3} x^{5}-\frac {1}{4} a^{2} x^{4}+\frac {2}{3} x^{3} a +\frac {1}{2} x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 73, normalized size = 1.06 \[ \frac {1}{8} \, a^{6} c^{3} x^{8} + \frac {2}{7} \, a^{5} c^{3} x^{7} - \frac {1}{6} \, a^{4} c^{3} x^{6} - \frac {4}{5} \, a^{3} c^{3} x^{5} - \frac {1}{4} \, a^{2} c^{3} x^{4} + \frac {2}{3} \, a c^{3} x^{3} + \frac {1}{2} \, c^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 73, normalized size = 1.06 \[ \frac {a^6\,c^3\,x^8}{8}+\frac {2\,a^5\,c^3\,x^7}{7}-\frac {a^4\,c^3\,x^6}{6}-\frac {4\,a^3\,c^3\,x^5}{5}-\frac {a^2\,c^3\,x^4}{4}+\frac {2\,a\,c^3\,x^3}{3}+\frac {c^3\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 82, normalized size = 1.19 \[ \frac {a^{6} c^{3} x^{8}}{8} + \frac {2 a^{5} c^{3} x^{7}}{7} - \frac {a^{4} c^{3} x^{6}}{6} - \frac {4 a^{3} c^{3} x^{5}}{5} - \frac {a^{2} c^{3} x^{4}}{4} + \frac {2 a c^{3} x^{3}}{3} + \frac {c^{3} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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