Optimal. Leaf size=29 \[ \frac {\left (a+b x^{12 n+p+1}\right )^{13}}{13 b (12 n+p+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1584, 261} \begin {gather*} \frac {\left (a+b x^{12 n+p+1}\right )^{13}}{13 b (12 n+p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 261
Rule 1584
Rubi steps
\begin {align*} \int x^p \left (a x^n+b x^{1+13 n+p}\right )^{12} \, dx &=\int x^{12 n+p} \left (a+b x^{1+12 n+p}\right )^{12} \, dx\\ &=\frac {\left (a+b x^{1+12 n+p}\right )^{13}}{13 b (1+12 n+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^{12 n+p+1}\right )^{13}}{13 b (12 n+p+1)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.09, size = 232, normalized size = 8.00 \begin {gather*} \frac {x^{12 n+p+1} \left (13 a^{12}+78 a^{11} b x^{12 n+p+1}+286 a^{10} b^2 x^{24 n+2 p+2}+715 a^9 b^3 x^{36 n+3 p+3}+1287 a^8 b^4 x^{48 n+4 p+4}+1716 a^7 b^5 x^{60 n+5 p+5}+1716 a^6 b^6 x^{72 n+6 p+6}+1287 a^5 b^7 x^{84 n+7 p+7}+715 a^4 b^8 x^{96 n+8 p+8}+286 a^3 b^9 x^{108 n+9 p+9}+78 a^2 b^{10} x^{120 n+10 p+10}+13 a b^{11} x^{132 n+11 p+11}+b^{12} x^{144 n+12 p+12}\right )}{13 (12 n+p+1)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 297, normalized size = 10.24 \begin {gather*} \frac {78 \, a^{2} b^{10} x^{2 \, n} x^{143 \, n + 11 \, p + 11} + 286 \, a^{3} b^{9} x^{3 \, n} x^{130 \, n + 10 \, p + 10} + 715 \, a^{4} b^{8} x^{4 \, n} x^{117 \, n + 9 \, p + 9} + 1287 \, a^{5} b^{7} x^{5 \, n} x^{104 \, n + 8 \, p + 8} + 1716 \, a^{6} b^{6} x^{6 \, n} x^{91 \, n + 7 \, p + 7} + 1716 \, a^{7} b^{5} x^{7 \, n} x^{78 \, n + 6 \, p + 6} + 1287 \, a^{8} b^{4} x^{8 \, n} x^{65 \, n + 5 \, p + 5} + 715 \, a^{9} b^{3} x^{9 \, n} x^{52 \, n + 4 \, p + 4} + 286 \, a^{10} b^{2} x^{10 \, n} x^{39 \, n + 3 \, p + 3} + 78 \, a^{11} b x^{11 \, n} x^{26 \, n + 2 \, p + 2} + 13 \, a^{12} x^{12 \, n} x^{13 \, n + p + 1} + 13 \, a b^{11} x^{156 \, n + 12 \, p + 12} x^{n} + b^{12} x^{169 \, n + 13 \, p + 13}}{13 \, {\left (12 \, n + p + 1\right )} x^{13 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.47, size = 269, normalized size = 9.28 \begin {gather*} \frac {b^{12} x^{13} x^{156 \, n} x^{13 \, p} + 13 \, a b^{11} x^{12} x^{144 \, n} x^{12 \, p} + 78 \, a^{2} b^{10} x^{11} x^{132 \, n} x^{11 \, p} + 286 \, a^{3} b^{9} x^{10} x^{120 \, n} x^{10 \, p} + 715 \, a^{4} b^{8} x^{9} x^{108 \, n} x^{9 \, p} + 1287 \, a^{5} b^{7} x^{8} x^{96 \, n} x^{8 \, p} + 1716 \, a^{6} b^{6} x^{7} x^{84 \, n} x^{7 \, p} + 1716 \, a^{7} b^{5} x^{6} x^{72 \, n} x^{6 \, p} + 1287 \, a^{8} b^{4} x^{5} x^{60 \, n} x^{5 \, p} + 715 \, a^{9} b^{3} x^{4} x^{48 \, n} x^{4 \, p} + 286 \, a^{10} b^{2} x^{3} x^{36 \, n} x^{3 \, p} + 78 \, a^{11} b x^{2} x^{24 \, n} x^{2 \, p} + 13 \, a^{12} x x^{12 \, n} x^{p}}{13 \, {\left (12 \, n + p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 363, normalized size = 12.52 \begin {gather*} \frac {b^{12} x^{13} x^{156 n} x^{13 p}}{13+156 n +13 p}+\frac {a \,b^{11} x^{12} x^{144 n} x^{12 p}}{12 n +p +1}+\frac {6 a^{2} b^{10} x^{11} x^{132 n} x^{11 p}}{12 n +p +1}+\frac {22 a^{3} b^{9} x^{10} x^{120 n} x^{10 p}}{12 n +p +1}+\frac {55 a^{4} b^{8} x^{9} x^{108 n} x^{9 p}}{12 n +p +1}+\frac {99 a^{5} b^{7} x^{8} x^{96 n} x^{8 p}}{12 n +p +1}+\frac {132 a^{6} b^{6} x^{7} x^{84 n} x^{7 p}}{12 n +p +1}+\frac {132 a^{7} b^{5} x^{6} x^{72 n} x^{6 p}}{12 n +p +1}+\frac {99 a^{8} b^{4} x^{5} x^{60 n} x^{5 p}}{12 n +p +1}+\frac {55 a^{9} b^{3} x^{4} x^{48 n} x^{4 p}}{12 n +p +1}+\frac {22 a^{10} b^{2} x^{3} x^{36 n} x^{3 p}}{12 n +p +1}+\frac {6 a^{11} b \,x^{2} x^{24 n} x^{2 p}}{12 n +p +1}+\frac {a^{12} x \,x^{p} x^{12 n}}{12 n +p +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.48, size = 325, normalized size = 11.21 \begin {gather*} \frac {b^{12} x^{156 \, n + 13 \, p + 13}}{13 \, {\left (12 \, n + p + 1\right )}} + \frac {a b^{11} x^{144 \, n + 12 \, p + 12}}{12 \, n + p + 1} + \frac {6 \, a^{2} b^{10} x^{132 \, n + 11 \, p + 11}}{12 \, n + p + 1} + \frac {22 \, a^{3} b^{9} x^{120 \, n + 10 \, p + 10}}{12 \, n + p + 1} + \frac {55 \, a^{4} b^{8} x^{108 \, n + 9 \, p + 9}}{12 \, n + p + 1} + \frac {99 \, a^{5} b^{7} x^{96 \, n + 8 \, p + 8}}{12 \, n + p + 1} + \frac {132 \, a^{6} b^{6} x^{84 \, n + 7 \, p + 7}}{12 \, n + p + 1} + \frac {132 \, a^{7} b^{5} x^{72 \, n + 6 \, p + 6}}{12 \, n + p + 1} + \frac {99 \, a^{8} b^{4} x^{60 \, n + 5 \, p + 5}}{12 \, n + p + 1} + \frac {55 \, a^{9} b^{3} x^{48 \, n + 4 \, p + 4}}{12 \, n + p + 1} + \frac {22 \, a^{10} b^{2} x^{36 \, n + 3 \, p + 3}}{12 \, n + p + 1} + \frac {6 \, a^{11} b x^{24 \, n + 2 \, p + 2}}{12 \, n + p + 1} + \frac {a^{12} x^{12 \, n + p + 1}}{12 \, n + p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.78, size = 363, normalized size = 12.52 \begin {gather*} \frac {a^{12}\,x\,x^p\,x^{12\,n}}{12\,n+p+1}+\frac {b^{12}\,x^{156\,n}\,x^{13\,p}\,x^{13}}{156\,n+13\,p+13}+\frac {22\,a^{10}\,b^2\,x^{36\,n}\,x^{3\,p}\,x^3}{12\,n+p+1}+\frac {55\,a^9\,b^3\,x^{48\,n}\,x^{4\,p}\,x^4}{12\,n+p+1}+\frac {99\,a^8\,b^4\,x^{60\,n}\,x^{5\,p}\,x^5}{12\,n+p+1}+\frac {132\,a^7\,b^5\,x^{72\,n}\,x^{6\,p}\,x^6}{12\,n+p+1}+\frac {132\,a^6\,b^6\,x^{84\,n}\,x^{7\,p}\,x^7}{12\,n+p+1}+\frac {99\,a^5\,b^7\,x^{96\,n}\,x^{8\,p}\,x^8}{12\,n+p+1}+\frac {55\,a^4\,b^8\,x^{108\,n}\,x^{9\,p}\,x^9}{12\,n+p+1}+\frac {22\,a^3\,b^9\,x^{120\,n}\,x^{10\,p}\,x^{10}}{12\,n+p+1}+\frac {6\,a^2\,b^{10}\,x^{132\,n}\,x^{11\,p}\,x^{11}}{12\,n+p+1}+\frac {6\,a^{11}\,b\,x^{24\,n}\,x^{2\,p}\,x^2}{12\,n+p+1}+\frac {a\,b^{11}\,x^{144\,n}\,x^{12\,p}\,x^{12}}{12\,n+p+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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