Optimal. Leaf size=132 \[ \frac{9}{4} a^2 b^8 x^{20}+\frac{120}{19} a^3 b^7 x^{19}+\frac{35}{3} a^4 b^6 x^{18}+\frac{252}{17} a^5 b^5 x^{17}+\frac{105}{8} a^6 b^4 x^{16}+8 a^7 b^3 x^{15}+\frac{45}{14} a^8 b^2 x^{14}+\frac{10}{13} a^9 b x^{13}+\frac{a^{10} x^{12}}{12}+\frac{10}{21} a b^9 x^{21}+\frac{b^{10} x^{22}}{22} \]
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Rubi [A] time = 0.0759259, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{9}{4} a^2 b^8 x^{20}+\frac{120}{19} a^3 b^7 x^{19}+\frac{35}{3} a^4 b^6 x^{18}+\frac{252}{17} a^5 b^5 x^{17}+\frac{105}{8} a^6 b^4 x^{16}+8 a^7 b^3 x^{15}+\frac{45}{14} a^8 b^2 x^{14}+\frac{10}{13} a^9 b x^{13}+\frac{a^{10} x^{12}}{12}+\frac{10}{21} a b^9 x^{21}+\frac{b^{10} x^{22}}{22} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^{11} (a+b x)^{10} \, dx &=\int \left (a^{10} x^{11}+10 a^9 b x^{12}+45 a^8 b^2 x^{13}+120 a^7 b^3 x^{14}+210 a^6 b^4 x^{15}+252 a^5 b^5 x^{16}+210 a^4 b^6 x^{17}+120 a^3 b^7 x^{18}+45 a^2 b^8 x^{19}+10 a b^9 x^{20}+b^{10} x^{21}\right ) \, dx\\ &=\frac{a^{10} x^{12}}{12}+\frac{10}{13} a^9 b x^{13}+\frac{45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac{105}{8} a^6 b^4 x^{16}+\frac{252}{17} a^5 b^5 x^{17}+\frac{35}{3} a^4 b^6 x^{18}+\frac{120}{19} a^3 b^7 x^{19}+\frac{9}{4} a^2 b^8 x^{20}+\frac{10}{21} a b^9 x^{21}+\frac{b^{10} x^{22}}{22}\\ \end{align*}
Mathematica [A] time = 0.0064768, size = 132, normalized size = 1. \[ \frac{9}{4} a^2 b^8 x^{20}+\frac{120}{19} a^3 b^7 x^{19}+\frac{35}{3} a^4 b^6 x^{18}+\frac{252}{17} a^5 b^5 x^{17}+\frac{105}{8} a^6 b^4 x^{16}+8 a^7 b^3 x^{15}+\frac{45}{14} a^8 b^2 x^{14}+\frac{10}{13} a^9 b x^{13}+\frac{a^{10} x^{12}}{12}+\frac{10}{21} a b^9 x^{21}+\frac{b^{10} x^{22}}{22} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 113, normalized size = 0.9 \begin{align*}{\frac{{a}^{10}{x}^{12}}{12}}+{\frac{10\,{a}^{9}b{x}^{13}}{13}}+{\frac{45\,{a}^{8}{b}^{2}{x}^{14}}{14}}+8\,{a}^{7}{b}^{3}{x}^{15}+{\frac{105\,{a}^{6}{b}^{4}{x}^{16}}{8}}+{\frac{252\,{a}^{5}{b}^{5}{x}^{17}}{17}}+{\frac{35\,{a}^{4}{b}^{6}{x}^{18}}{3}}+{\frac{120\,{a}^{3}{b}^{7}{x}^{19}}{19}}+{\frac{9\,{a}^{2}{b}^{8}{x}^{20}}{4}}+{\frac{10\,a{b}^{9}{x}^{21}}{21}}+{\frac{{b}^{10}{x}^{22}}{22}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06356, size = 151, normalized size = 1.14 \begin{align*} \frac{1}{22} \, b^{10} x^{22} + \frac{10}{21} \, a b^{9} x^{21} + \frac{9}{4} \, a^{2} b^{8} x^{20} + \frac{120}{19} \, a^{3} b^{7} x^{19} + \frac{35}{3} \, a^{4} b^{6} x^{18} + \frac{252}{17} \, a^{5} b^{5} x^{17} + \frac{105}{8} \, a^{6} b^{4} x^{16} + 8 \, a^{7} b^{3} x^{15} + \frac{45}{14} \, a^{8} b^{2} x^{14} + \frac{10}{13} \, a^{9} b x^{13} + \frac{1}{12} \, a^{10} x^{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50884, size = 288, normalized size = 2.18 \begin{align*} \frac{1}{22} x^{22} b^{10} + \frac{10}{21} x^{21} b^{9} a + \frac{9}{4} x^{20} b^{8} a^{2} + \frac{120}{19} x^{19} b^{7} a^{3} + \frac{35}{3} x^{18} b^{6} a^{4} + \frac{252}{17} x^{17} b^{5} a^{5} + \frac{105}{8} x^{16} b^{4} a^{6} + 8 x^{15} b^{3} a^{7} + \frac{45}{14} x^{14} b^{2} a^{8} + \frac{10}{13} x^{13} b a^{9} + \frac{1}{12} x^{12} a^{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.099574, size = 133, normalized size = 1.01 \begin{align*} \frac{a^{10} x^{12}}{12} + \frac{10 a^{9} b x^{13}}{13} + \frac{45 a^{8} b^{2} x^{14}}{14} + 8 a^{7} b^{3} x^{15} + \frac{105 a^{6} b^{4} x^{16}}{8} + \frac{252 a^{5} b^{5} x^{17}}{17} + \frac{35 a^{4} b^{6} x^{18}}{3} + \frac{120 a^{3} b^{7} x^{19}}{19} + \frac{9 a^{2} b^{8} x^{20}}{4} + \frac{10 a b^{9} x^{21}}{21} + \frac{b^{10} x^{22}}{22} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14462, size = 151, normalized size = 1.14 \begin{align*} \frac{1}{22} \, b^{10} x^{22} + \frac{10}{21} \, a b^{9} x^{21} + \frac{9}{4} \, a^{2} b^{8} x^{20} + \frac{120}{19} \, a^{3} b^{7} x^{19} + \frac{35}{3} \, a^{4} b^{6} x^{18} + \frac{252}{17} \, a^{5} b^{5} x^{17} + \frac{105}{8} \, a^{6} b^{4} x^{16} + 8 \, a^{7} b^{3} x^{15} + \frac{45}{14} \, a^{8} b^{2} x^{14} + \frac{10}{13} \, a^{9} b x^{13} + \frac{1}{12} \, a^{10} x^{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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