Optimal. Leaf size=54 \[ \frac {x^{1+m}}{1+m}+\frac {a x^{2+m}}{2+m}-\frac {a^2 x^{3+m}}{3+m}-\frac {a^3 x^{4+m}}{4+m} \]
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Rubi [A]
time = 0.07, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6285, 76}
\begin {gather*} -\frac {a^3 x^{m+4}}{m+4}-\frac {a^2 x^{m+3}}{m+3}+\frac {a x^{m+2}}{m+2}+\frac {x^{m+1}}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 6285
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^m \left (1-a^2 x^2\right )^{3/2} \, dx &=\int x^m (1-a x) (1+a x)^2 \, dx\\ &=\int \left (x^m+a x^{1+m}-a^2 x^{2+m}-a^3 x^{3+m}\right ) \, dx\\ &=\frac {x^{1+m}}{1+m}+\frac {a x^{2+m}}{2+m}-\frac {a^2 x^{3+m}}{3+m}-\frac {a^3 x^{4+m}}{4+m}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 54, normalized size = 1.00 \begin {gather*} \frac {x^{1+m} \left (-(1+a x)^3+(5+2 m) \left (\frac {1}{1+m}+\frac {2 a x}{2+m}+\frac {a^2 x^2}{3+m}\right )\right )}{4+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 65, normalized size = 1.20
method | result | size |
norman | \(\frac {x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {a \,x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}-\frac {a^{2} x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}-\frac {a^{3} x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}\) | \(65\) |
risch | \(-\frac {x \left (a^{3} m^{3} x^{3}+6 a^{3} m^{2} x^{3}+11 a^{3} m \,x^{3}+a^{2} m^{3} x^{2}+6 a^{3} x^{3}+7 a^{2} m^{2} x^{2}+14 a^{2} m \,x^{2}-a \,m^{3} x +8 a^{2} x^{2}-8 a \,m^{2} x -19 a m x -m^{3}-12 a x -9 m^{2}-26 m -24\right ) x^{m}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(141\) |
gosper | \(-\frac {x^{1+m} \left (a^{3} m^{3} x^{3}+6 a^{3} m^{2} x^{3}+11 a^{3} m \,x^{3}+a^{2} m^{3} x^{2}+6 a^{3} x^{3}+7 a^{2} m^{2} x^{2}+14 a^{2} m \,x^{2}-a \,m^{3} x +8 a^{2} x^{2}-8 a \,m^{2} x -19 a m x -m^{3}-12 a x -9 m^{2}-26 m -24\right )}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(142\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 54, normalized size = 1.00 \begin {gather*} -\frac {a^{3} x^{m + 4}}{m + 4} - \frac {a^{2} x^{m + 3}}{m + 3} + \frac {a x^{m + 2}}{m + 2} + \frac {x^{m + 1}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs.
\(2 (54) = 108\).
time = 0.34, size = 128, normalized size = 2.37 \begin {gather*} -\frac {{\left ({\left (a^{3} m^{3} + 6 \, a^{3} m^{2} + 11 \, a^{3} m + 6 \, a^{3}\right )} x^{4} + {\left (a^{2} m^{3} + 7 \, a^{2} m^{2} + 14 \, a^{2} m + 8 \, a^{2}\right )} x^{3} - {\left (a m^{3} + 8 \, a m^{2} + 19 \, a m + 12 \, a\right )} x^{2} - {\left (m^{3} + 9 \, m^{2} + 26 \, m + 24\right )} x\right )} x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 585 vs.
\(2 (41) = 82\).
time = 0.24, size = 585, normalized size = 10.83 \begin {gather*} \begin {cases} - a^{3} \log {\left (x \right )} + \frac {a^{2}}{x} - \frac {a}{2 x^{2}} - \frac {1}{3 x^{3}} & \text {for}\: m = -4 \\- a^{3} x - a^{2} \log {\left (x \right )} - \frac {a}{x} - \frac {1}{2 x^{2}} & \text {for}\: m = -3 \\- \frac {a^{3} x^{2}}{2} - a^{2} x + a \log {\left (x \right )} - \frac {1}{x} & \text {for}\: m = -2 \\- \frac {a^{3} x^{3}}{3} - \frac {a^{2} x^{2}}{2} + a x + \log {\left (x \right )} & \text {for}\: m = -1 \\- \frac {a^{3} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {6 a^{3} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {11 a^{3} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {6 a^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {a^{2} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {7 a^{2} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {14 a^{2} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {8 a^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {a m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 a m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {19 a m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {12 a x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {9 m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {26 m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs.
\(2 (54) = 108\).
time = 0.44, size = 197, normalized size = 3.65 \begin {gather*} -\frac {a^{3} m^{3} x^{4} x^{m} + 6 \, a^{3} m^{2} x^{4} x^{m} + a^{2} m^{3} x^{3} x^{m} + 11 \, a^{3} m x^{4} x^{m} + 7 \, a^{2} m^{2} x^{3} x^{m} + 6 \, a^{3} x^{4} x^{m} - a m^{3} x^{2} x^{m} + 14 \, a^{2} m x^{3} x^{m} - 8 \, a m^{2} x^{2} x^{m} + 8 \, a^{2} x^{3} x^{m} - m^{3} x x^{m} - 19 \, a m x^{2} x^{m} - 9 \, m^{2} x x^{m} - 12 \, a x^{2} x^{m} - 26 \, m x x^{m} - 24 \, x x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.00, size = 160, normalized size = 2.96 \begin {gather*} x^m\,\left (\frac {x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {a\,x^2\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {a^3\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {a^2\,x^3\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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