Optimal. Leaf size=42 \[ \frac {a^2 c x^{m+3}}{m+3}+\frac {2 a c x^{m+2}}{m+2}+\frac {c x^{m+1}}{m+1} \]
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Rubi [A] time = 0.06, antiderivative size = 42, 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, 43} \[ \frac {a^2 c x^{m+3}}{m+3}+\frac {2 a c x^{m+2}}{m+2}+\frac {c x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right ) \, dx &=c \int x^m (1+a x)^2 \, dx\\ &=c \int \left (x^m+2 a x^{1+m}+a^2 x^{2+m}\right ) \, dx\\ &=\frac {c x^{1+m}}{1+m}+\frac {2 a c x^{2+m}}{2+m}+\frac {a^2 c x^{3+m}}{3+m}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 34, normalized size = 0.81 \[ c x^{m+1} \left (\frac {a^2 x^2}{m+3}+\frac {2 a x}{m+2}+\frac {1}{m+1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 82, normalized size = 1.95 \[ \frac {{\left ({\left (a^{2} c m^{2} + 3 \, a^{2} c m + 2 \, a^{2} c\right )} x^{3} + 2 \, {\left (a c m^{2} + 4 \, a c m + 3 \, a c\right )} x^{2} + {\left (c m^{2} + 5 \, c m + 6 \, c\right )} x\right )} x^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} c x^{2} - c\right )} {\left (a x + 1\right )}^{2} x^{m}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 74, normalized size = 1.76 \[ \frac {c \,x^{1+m} \left (a^{2} m^{2} x^{2}+3 a^{2} m \,x^{2}+2 a^{2} x^{2}+2 a \,m^{2} x +8 a m x +6 a x +m^{2}+5 m +6\right )}{\left (3+m \right ) \left (2+m \right ) \left (1+m \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 62, normalized size = 1.48 \[ \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} a^{2} c x^{3} + 2 \, {\left (m^{2} + 4 \, m + 3\right )} a c x^{2} + {\left (m^{2} + 5 \, m + 6\right )} c x\right )} x^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 92, normalized size = 2.19 \[ x^m\,\left (\frac {c\,x\,\left (m^2+5\,m+6\right )}{m^3+6\,m^2+11\,m+6}+\frac {2\,a\,c\,x^2\,\left (m^2+4\,m+3\right )}{m^3+6\,m^2+11\,m+6}+\frac {a^2\,c\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.10, size = 299, normalized size = 7.12 \[ \begin {cases} a^{2} c \log {\relax (x )} - \frac {2 a c}{x} - \frac {c}{2 x^{2}} & \text {for}\: m = -3 \\a^{2} c x + 2 a c \log {\relax (x )} - \frac {c}{x} & \text {for}\: m = -2 \\\frac {a^{2} c x^{2}}{2} + 2 a c x + c \log {\relax (x )} & \text {for}\: m = -1 \\\frac {a^{2} c m^{2} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {3 a^{2} c m x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {2 a^{2} c x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {2 a c m^{2} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {8 a c m x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {6 a c x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {c m^{2} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {5 c m x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac {6 c x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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