Optimal. Leaf size=84 \[ \frac {c^3 (a x+1)^9}{9 a^3}-\frac {3 c^3 (a x+1)^8}{4 a^3}+\frac {13 c^3 (a x+1)^7}{7 a^3}-\frac {2 c^3 (a x+1)^6}{a^3}+\frac {4 c^3 (a x+1)^5}{5 a^3} \]
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Rubi [A] time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 88} \[ \frac {c^3 (a x+1)^9}{9 a^3}-\frac {3 c^3 (a x+1)^8}{4 a^3}+\frac {13 c^3 (a x+1)^7}{7 a^3}-\frac {2 c^3 (a x+1)^6}{a^3}+\frac {4 c^3 (a x+1)^5}{5 a^3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^2 \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^2 (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (\frac {4 (1+a x)^4}{a^2}-\frac {12 (1+a x)^5}{a^2}+\frac {13 (1+a x)^6}{a^2}-\frac {6 (1+a x)^7}{a^2}+\frac {(1+a x)^8}{a^2}\right ) \, dx\\ &=\frac {4 c^3 (1+a x)^5}{5 a^3}-\frac {2 c^3 (1+a x)^6}{a^3}+\frac {13 c^3 (1+a x)^7}{7 a^3}-\frac {3 c^3 (1+a x)^8}{4 a^3}+\frac {c^3 (1+a x)^9}{9 a^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 70, normalized size = 0.83 \[ c^3 \left (\frac {a^6 x^9}{9}+\frac {a^5 x^8}{4}-\frac {a^4 x^7}{7}-\frac {2 a^3 x^6}{3}-\frac {a^2 x^5}{5}+\frac {a x^4}{2}+\frac {x^3}{3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 73, normalized size = 0.87 \[ \frac {1}{9} \, a^{6} c^{3} x^{9} + \frac {1}{4} \, a^{5} c^{3} x^{8} - \frac {1}{7} \, a^{4} c^{3} x^{7} - \frac {2}{3} \, a^{3} c^{3} x^{6} - \frac {1}{5} \, a^{2} c^{3} x^{5} + \frac {1}{2} \, a c^{3} x^{4} + \frac {1}{3} \, c^{3} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.73, size = 73, normalized size = 0.87 \[ \frac {1}{9} \, a^{6} c^{3} x^{9} + \frac {1}{4} \, a^{5} c^{3} x^{8} - \frac {1}{7} \, a^{4} c^{3} x^{7} - \frac {2}{3} \, a^{3} c^{3} x^{6} - \frac {1}{5} \, a^{2} c^{3} x^{5} + \frac {1}{2} \, a c^{3} x^{4} + \frac {1}{3} \, c^{3} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 57, normalized size = 0.68 \[ c^{3} \left (\frac {1}{9} x^{9} a^{6}+\frac {1}{4} a^{5} x^{8}-\frac {1}{7} x^{7} a^{4}-\frac {2}{3} x^{6} a^{3}-\frac {1}{5} x^{5} a^{2}+\frac {1}{2} x^{4} a +\frac {1}{3} x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 73, normalized size = 0.87 \[ \frac {1}{9} \, a^{6} c^{3} x^{9} + \frac {1}{4} \, a^{5} c^{3} x^{8} - \frac {1}{7} \, a^{4} c^{3} x^{7} - \frac {2}{3} \, a^{3} c^{3} x^{6} - \frac {1}{5} \, a^{2} c^{3} x^{5} + \frac {1}{2} \, a c^{3} x^{4} + \frac {1}{3} \, c^{3} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 73, normalized size = 0.87 \[ \frac {a^6\,c^3\,x^9}{9}+\frac {a^5\,c^3\,x^8}{4}-\frac {a^4\,c^3\,x^7}{7}-\frac {2\,a^3\,c^3\,x^6}{3}-\frac {a^2\,c^3\,x^5}{5}+\frac {a\,c^3\,x^4}{2}+\frac {c^3\,x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 78, normalized size = 0.93 \[ \frac {a^{6} c^{3} x^{9}}{9} + \frac {a^{5} c^{3} x^{8}}{4} - \frac {a^{4} c^{3} x^{7}}{7} - \frac {2 a^{3} c^{3} x^{6}}{3} - \frac {a^{2} c^{3} x^{5}}{5} + \frac {a c^{3} x^{4}}{2} + \frac {c^{3} x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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