Optimal. Leaf size=82 \[ \frac {x^{1+m}}{1+m}+\frac {a x^{2+m}}{2+m}-\frac {2 a^2 x^{3+m}}{3+m}-\frac {2 a^3 x^{4+m}}{4+m}+\frac {a^4 x^{5+m}}{5+m}+\frac {a^5 x^{6+m}}{6+m} \]
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Rubi [A]
time = 0.08, antiderivative size = 82, 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, 90}
\begin {gather*} \frac {a^5 x^{m+6}}{m+6}+\frac {a^4 x^{m+5}}{m+5}-\frac {2 a^3 x^{m+4}}{m+4}-\frac {2 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 90
Rule 6285
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^m \left (1-a^2 x^2\right )^{5/2} \, dx &=\int x^m (1-a x)^2 (1+a x)^3 \, dx\\ &=\int \left (x^m+a x^{1+m}-2 a^2 x^{2+m}-2 a^3 x^{3+m}+a^4 x^{4+m}+a^5 x^{5+m}\right ) \, dx\\ &=\frac {x^{1+m}}{1+m}+\frac {a x^{2+m}}{2+m}-\frac {2 a^2 x^{3+m}}{3+m}-\frac {2 a^3 x^{4+m}}{4+m}+\frac {a^4 x^{5+m}}{5+m}+\frac {a^5 x^{6+m}}{6+m}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 70, normalized size = 0.85 \begin {gather*} x^{1+m} \left (\frac {1}{1+m}+\frac {a x}{2+m}-\frac {2 a^2 x^2}{3+m}-\frac {2 a^3 x^3}{4+m}+\frac {a^4 x^4}{5+m}+\frac {a^5 x^5}{6+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 99, normalized size = 1.21
method | result | size |
norman | \(\frac {x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {a \,x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {a^{4} x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {a^{5} x^{6} {\mathrm e}^{m \ln \left (x \right )}}{6+m}-\frac {2 a^{2} x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}-\frac {2 a^{3} x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}\) | \(99\) |
risch | \(\frac {x \left (a^{5} m^{5} x^{5}+15 a^{5} m^{4} x^{5}+85 a^{5} m^{3} x^{5}+a^{4} m^{5} x^{4}+225 a^{5} m^{2} x^{5}+16 a^{4} m^{4} x^{4}+274 m \,x^{5} a^{5}+95 a^{4} m^{3} x^{4}-2 a^{3} m^{5} x^{3}+120 x^{5} a^{5}+260 a^{4} m^{2} x^{4}-34 a^{3} m^{4} x^{3}+324 m \,x^{4} a^{4}-214 a^{3} m^{3} x^{3}-2 a^{2} m^{5} x^{2}+144 a^{4} x^{4}-614 a^{3} m^{2} x^{3}-36 a^{2} m^{4} x^{2}-792 a^{3} m \,x^{3}-242 a^{2} m^{3} x^{2}+a \,m^{5} x -360 a^{3} x^{3}-744 a^{2} m^{2} x^{2}+19 a \,m^{4} x -1016 a^{2} m \,x^{2}+137 a \,m^{3} x +m^{5}-480 a^{2} x^{2}+461 a \,m^{2} x +20 m^{4}+702 a m x +155 m^{3}+360 a x +580 m^{2}+1044 m +720\right ) x^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(337\) |
gosper | \(\frac {x^{1+m} \left (a^{5} m^{5} x^{5}+15 a^{5} m^{4} x^{5}+85 a^{5} m^{3} x^{5}+a^{4} m^{5} x^{4}+225 a^{5} m^{2} x^{5}+16 a^{4} m^{4} x^{4}+274 m \,x^{5} a^{5}+95 a^{4} m^{3} x^{4}-2 a^{3} m^{5} x^{3}+120 x^{5} a^{5}+260 a^{4} m^{2} x^{4}-34 a^{3} m^{4} x^{3}+324 m \,x^{4} a^{4}-214 a^{3} m^{3} x^{3}-2 a^{2} m^{5} x^{2}+144 a^{4} x^{4}-614 a^{3} m^{2} x^{3}-36 a^{2} m^{4} x^{2}-792 a^{3} m \,x^{3}-242 a^{2} m^{3} x^{2}+a \,m^{5} x -360 a^{3} x^{3}-744 a^{2} m^{2} x^{2}+19 a \,m^{4} x -1016 a^{2} m \,x^{2}+137 a \,m^{3} x +m^{5}-480 a^{2} x^{2}+461 a \,m^{2} x +20 m^{4}+702 a m x +155 m^{3}+360 a x +580 m^{2}+1044 m +720\right )}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(338\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 82, normalized size = 1.00 \begin {gather*} \frac {a^{5} x^{m + 6}}{m + 6} + \frac {a^{4} x^{m + 5}}{m + 5} - \frac {2 \, a^{3} x^{m + 4}}{m + 4} - \frac {2 \, 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. 285 vs.
\(2 (82) = 164\).
time = 0.36, size = 285, normalized size = 3.48 \begin {gather*} \frac {{\left ({\left (a^{5} m^{5} + 15 \, a^{5} m^{4} + 85 \, a^{5} m^{3} + 225 \, a^{5} m^{2} + 274 \, a^{5} m + 120 \, a^{5}\right )} x^{6} + {\left (a^{4} m^{5} + 16 \, a^{4} m^{4} + 95 \, a^{4} m^{3} + 260 \, a^{4} m^{2} + 324 \, a^{4} m + 144 \, a^{4}\right )} x^{5} - 2 \, {\left (a^{3} m^{5} + 17 \, a^{3} m^{4} + 107 \, a^{3} m^{3} + 307 \, a^{3} m^{2} + 396 \, a^{3} m + 180 \, a^{3}\right )} x^{4} - 2 \, {\left (a^{2} m^{5} + 18 \, a^{2} m^{4} + 121 \, a^{2} m^{3} + 372 \, a^{2} m^{2} + 508 \, a^{2} m + 240 \, a^{2}\right )} x^{3} + {\left (a m^{5} + 19 \, a m^{4} + 137 \, a m^{3} + 461 \, a m^{2} + 702 \, a m + 360 \, a\right )} x^{2} + {\left (m^{5} + 20 \, m^{4} + 155 \, m^{3} + 580 \, m^{2} + 1044 \, m + 720\right )} x\right )} x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \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. 1760 vs.
\(2 (68) = 136\).
time = 0.46, size = 1760, normalized size = 21.46 \begin {gather*} \begin {cases} a^{5} \log {\left (x \right )} - \frac {a^{4}}{x} + \frac {a^{3}}{x^{2}} + \frac {2 a^{2}}{3 x^{3}} - \frac {a}{4 x^{4}} - \frac {1}{5 x^{5}} & \text {for}\: m = -6 \\a^{5} x + a^{4} \log {\left (x \right )} + \frac {2 a^{3}}{x} + \frac {a^{2}}{x^{2}} - \frac {a}{3 x^{3}} - \frac {1}{4 x^{4}} & \text {for}\: m = -5 \\\frac {a^{5} x^{2}}{2} + a^{4} x - 2 a^{3} \log {\left (x \right )} + \frac {2 a^{2}}{x} - \frac {a}{2 x^{2}} - \frac {1}{3 x^{3}} & \text {for}\: m = -4 \\\frac {a^{5} x^{3}}{3} + \frac {a^{4} x^{2}}{2} - 2 a^{3} x - 2 a^{2} \log {\left (x \right )} - \frac {a}{x} - \frac {1}{2 x^{2}} & \text {for}\: m = -3 \\\frac {a^{5} x^{4}}{4} + \frac {a^{4} x^{3}}{3} - a^{3} x^{2} - 2 a^{2} x + a \log {\left (x \right )} - \frac {1}{x} & \text {for}\: m = -2 \\\frac {a^{5} x^{5}}{5} + \frac {a^{4} x^{4}}{4} - \frac {2 a^{3} x^{3}}{3} - a^{2} x^{2} + a x + \log {\left (x \right )} & \text {for}\: m = -1 \\\frac {a^{5} m^{5} x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {15 a^{5} m^{4} x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {85 a^{5} m^{3} x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {225 a^{5} m^{2} x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {274 a^{5} m x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {120 a^{5} x^{6} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {a^{4} m^{5} x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {16 a^{4} m^{4} x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {95 a^{4} m^{3} x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {260 a^{4} m^{2} x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {324 a^{4} m x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {144 a^{4} x^{5} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {2 a^{3} m^{5} x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {34 a^{3} m^{4} x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {214 a^{3} m^{3} x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {614 a^{3} m^{2} x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {792 a^{3} m x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {360 a^{3} x^{4} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {2 a^{2} m^{5} x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {36 a^{2} m^{4} x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {242 a^{2} m^{3} x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {744 a^{2} m^{2} x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {1016 a^{2} m x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} - \frac {480 a^{2} x^{3} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {a m^{5} x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {19 a m^{4} x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {137 a m^{3} x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {461 a m^{2} x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {702 a m x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {360 a x^{2} x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {m^{5} x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {20 m^{4} x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {155 m^{3} x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {580 m^{2} x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {1044 m x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} + \frac {720 x x^{m}}{m^{6} + 21 m^{5} + 175 m^{4} + 735 m^{3} + 1624 m^{2} + 1764 m + 720} & \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. 460 vs.
\(2 (82) = 164\).
time = 0.43, size = 460, normalized size = 5.61 \begin {gather*} \frac {a^{5} m^{5} x^{6} x^{m} + 15 \, a^{5} m^{4} x^{6} x^{m} + a^{4} m^{5} x^{5} x^{m} + 85 \, a^{5} m^{3} x^{6} x^{m} + 16 \, a^{4} m^{4} x^{5} x^{m} + 225 \, a^{5} m^{2} x^{6} x^{m} - 2 \, a^{3} m^{5} x^{4} x^{m} + 95 \, a^{4} m^{3} x^{5} x^{m} + 274 \, a^{5} m x^{6} x^{m} - 34 \, a^{3} m^{4} x^{4} x^{m} + 260 \, a^{4} m^{2} x^{5} x^{m} + 120 \, a^{5} x^{6} x^{m} - 2 \, a^{2} m^{5} x^{3} x^{m} - 214 \, a^{3} m^{3} x^{4} x^{m} + 324 \, a^{4} m x^{5} x^{m} - 36 \, a^{2} m^{4} x^{3} x^{m} - 614 \, a^{3} m^{2} x^{4} x^{m} + 144 \, a^{4} x^{5} x^{m} + a m^{5} x^{2} x^{m} - 242 \, a^{2} m^{3} x^{3} x^{m} - 792 \, a^{3} m x^{4} x^{m} + 19 \, a m^{4} x^{2} x^{m} - 744 \, a^{2} m^{2} x^{3} x^{m} - 360 \, a^{3} x^{4} x^{m} + m^{5} x x^{m} + 137 \, a m^{3} x^{2} x^{m} - 1016 \, a^{2} m x^{3} x^{m} + 20 \, m^{4} x x^{m} + 461 \, a m^{2} x^{2} x^{m} - 480 \, a^{2} x^{3} x^{m} + 155 \, m^{3} x x^{m} + 702 \, a m x^{2} x^{m} + 580 \, m^{2} x x^{m} + 360 \, a x^{2} x^{m} + 1044 \, m x x^{m} + 720 \, x x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.16, size = 374, normalized size = 4.56 \begin {gather*} \frac {x\,x^m\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a\,x^m\,x^2\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^5\,x^m\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^4\,x^m\,x^5\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^3\,x^m\,x^4\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {2\,a^2\,x^m\,x^3\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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