Optimal. Leaf size=27 \[ \frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b (1+12 m)} \]
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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {267}
\begin {gather*} \frac {\left (a+b x^{12 m+1}\right )^{13}}{13 b (12 m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rubi steps
\begin {align*} \int x^{12+12 (-1+m)} \left (a+b x^{1+12 m}\right )^{12} \, dx &=\frac {\left (a+b x^{1+12 m}\right )^{13}}{13 b (1+12 m)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(193\) vs. \(2(27)=54\).
time = 0.00, size = 193, normalized size = 7.15 \begin {gather*} \frac {x^{1+12 m} \left (13 a^{12}+78 a^{11} b x^{1+12 m}+286 a^{10} b^2 x^{2+24 m}+715 a^9 b^3 x^{3+36 m}+1287 a^8 b^4 x^{4+48 m}+1716 a^7 b^5 x^{5+60 m}+1716 a^6 b^6 x^{6+72 m}+1287 a^5 b^7 x^{7+84 m}+715 a^4 b^8 x^{8+96 m}+286 a^3 b^9 x^{9+108 m}+78 a^2 b^{10} x^{10+120 m}+13 a b^{11} x^{11+132 m}+b^{12} x^{12+144 m}\right )}{13+156 m} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(310\) vs.
\(2(25)=50\).
time = 0.00, size = 311, normalized size = 11.52
method | result | size |
risch | \(\frac {b^{12} x^{13} x^{156 m}}{13+156 m}+\frac {a \,b^{11} x^{12} x^{144 m}}{1+12 m}+\frac {6 a^{2} b^{10} x^{11} x^{132 m}}{1+12 m}+\frac {22 a^{3} b^{9} x^{10} x^{120 m}}{1+12 m}+\frac {55 a^{4} b^{8} x^{9} x^{108 m}}{1+12 m}+\frac {99 a^{5} b^{7} x^{8} x^{96 m}}{1+12 m}+\frac {132 a^{6} b^{6} x^{7} x^{84 m}}{1+12 m}+\frac {132 b^{5} a^{7} x^{6} x^{72 m}}{1+12 m}+\frac {99 a^{8} b^{4} x^{5} x^{60 m}}{1+12 m}+\frac {55 b^{3} a^{9} x^{4} x^{48 m}}{1+12 m}+\frac {22 a^{10} b^{2} x^{3} x^{36 m}}{1+12 m}+\frac {6 b \,a^{11} x^{2} x^{24 m}}{1+12 m}+\frac {a^{12} x \,x^{12 m}}{1+12 m}\) | \(311\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 25, normalized size = 0.93 \begin {gather*} \frac {{\left (b x^{12 \, m + 1} + a\right )}^{13}}{13 \, b {\left (12 \, m + 1\right )}} \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. 194 vs.
\(2 (25) = 50\).
time = 0.42, size = 194, normalized size = 7.19 \begin {gather*} \frac {b^{12} x^{156 \, m + 13} + 13 \, a b^{11} x^{144 \, m + 12} + 78 \, a^{2} b^{10} x^{132 \, m + 11} + 286 \, a^{3} b^{9} x^{120 \, m + 10} + 715 \, a^{4} b^{8} x^{108 \, m + 9} + 1287 \, a^{5} b^{7} x^{96 \, m + 8} + 1716 \, a^{6} b^{6} x^{84 \, m + 7} + 1716 \, a^{7} b^{5} x^{72 \, m + 6} + 1287 \, a^{8} b^{4} x^{60 \, m + 5} + 715 \, a^{9} b^{3} x^{48 \, m + 4} + 286 \, a^{10} b^{2} x^{36 \, m + 3} + 78 \, a^{11} b x^{24 \, m + 2} + 13 \, a^{12} x^{12 \, m + 1}}{13 \, {\left (12 \, m + 1\right )}} \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. 284 vs.
\(2 (19) = 38\).
time = 8.04, size = 284, normalized size = 10.52 \begin {gather*} \begin {cases} \frac {13 a^{12} x x^{12 m}}{156 m + 13} + \frac {78 a^{11} b x^{2} x^{24 m}}{156 m + 13} + \frac {286 a^{10} b^{2} x^{3} x^{36 m}}{156 m + 13} + \frac {715 a^{9} b^{3} x^{4} x^{48 m}}{156 m + 13} + \frac {1287 a^{8} b^{4} x^{5} x^{60 m}}{156 m + 13} + \frac {1716 a^{7} b^{5} x^{6} x^{72 m}}{156 m + 13} + \frac {1716 a^{6} b^{6} x^{7} x^{84 m}}{156 m + 13} + \frac {1287 a^{5} b^{7} x^{8} x^{96 m}}{156 m + 13} + \frac {715 a^{4} b^{8} x^{9} x^{108 m}}{156 m + 13} + \frac {286 a^{3} b^{9} x^{10} x^{120 m}}{156 m + 13} + \frac {78 a^{2} b^{10} x^{11} x^{132 m}}{156 m + 13} + \frac {13 a b^{11} x^{12} x^{144 m}}{156 m + 13} + \frac {b^{12} x^{13} x^{156 m}}{156 m + 13} & \text {for}\: m \neq - \frac {1}{12} \\\left (a + b\right )^{12} \log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.25, size = 25, normalized size = 0.93 \begin {gather*} \frac {{\left (b x^{12 \, m + 1} + a\right )}^{13}}{13 \, b {\left (12 \, m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 285, normalized size = 10.56 \begin {gather*} \frac {b^{12}\,x^{156\,m}\,x^{13}}{156\,m+13}+\frac {a^{12}\,x\,x^{12\,m}}{12\,m+1}+\frac {6\,a^{11}\,b\,x^{24\,m}\,x^2}{12\,m+1}+\frac {a\,b^{11}\,x^{144\,m}\,x^{12}}{12\,m+1}+\frac {22\,a^{10}\,b^2\,x^{36\,m}\,x^3}{12\,m+1}+\frac {55\,a^9\,b^3\,x^{48\,m}\,x^4}{12\,m+1}+\frac {99\,a^8\,b^4\,x^{60\,m}\,x^5}{12\,m+1}+\frac {132\,a^7\,b^5\,x^{72\,m}\,x^6}{12\,m+1}+\frac {132\,a^6\,b^6\,x^{84\,m}\,x^7}{12\,m+1}+\frac {99\,a^5\,b^7\,x^{96\,m}\,x^8}{12\,m+1}+\frac {55\,a^4\,b^8\,x^{108\,m}\,x^9}{12\,m+1}+\frac {22\,a^3\,b^9\,x^{120\,m}\,x^{10}}{12\,m+1}+\frac {6\,a^2\,b^{10}\,x^{132\,m}\,x^{11}}{12\,m+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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