Optimal. Leaf size=132 \[ \frac{45}{19} a^2 b^8 x^{19}+\frac{20}{3} a^3 b^7 x^{18}+\frac{210}{17} a^4 b^6 x^{17}+\frac{63}{4} a^5 b^5 x^{16}+14 a^6 b^4 x^{15}+\frac{60}{7} a^7 b^3 x^{14}+\frac{45}{13} a^8 b^2 x^{13}+\frac{5}{6} a^9 b x^{12}+\frac{a^{10} x^{11}}{11}+\frac{1}{2} a b^9 x^{20}+\frac{b^{10} x^{21}}{21} \]
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Rubi [A] time = 0.0592057, 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} \[ \frac{45}{19} a^2 b^8 x^{19}+\frac{20}{3} a^3 b^7 x^{18}+\frac{210}{17} a^4 b^6 x^{17}+\frac{63}{4} a^5 b^5 x^{16}+14 a^6 b^4 x^{15}+\frac{60}{7} a^7 b^3 x^{14}+\frac{45}{13} a^8 b^2 x^{13}+\frac{5}{6} a^9 b x^{12}+\frac{a^{10} x^{11}}{11}+\frac{1}{2} a b^9 x^{20}+\frac{b^{10} x^{21}}{21} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^{10} (a+b x)^{10} \, dx &=\int \left (a^{10} x^{10}+10 a^9 b x^{11}+45 a^8 b^2 x^{12}+120 a^7 b^3 x^{13}+210 a^6 b^4 x^{14}+252 a^5 b^5 x^{15}+210 a^4 b^6 x^{16}+120 a^3 b^7 x^{17}+45 a^2 b^8 x^{18}+10 a b^9 x^{19}+b^{10} x^{20}\right ) \, dx\\ &=\frac{a^{10} x^{11}}{11}+\frac{5}{6} a^9 b x^{12}+\frac{45}{13} a^8 b^2 x^{13}+\frac{60}{7} a^7 b^3 x^{14}+14 a^6 b^4 x^{15}+\frac{63}{4} a^5 b^5 x^{16}+\frac{210}{17} a^4 b^6 x^{17}+\frac{20}{3} a^3 b^7 x^{18}+\frac{45}{19} a^2 b^8 x^{19}+\frac{1}{2} a b^9 x^{20}+\frac{b^{10} x^{21}}{21}\\ \end{align*}
Mathematica [A] time = 0.0045776, size = 132, normalized size = 1. \[ \frac{45}{19} a^2 b^8 x^{19}+\frac{20}{3} a^3 b^7 x^{18}+\frac{210}{17} a^4 b^6 x^{17}+\frac{63}{4} a^5 b^5 x^{16}+14 a^6 b^4 x^{15}+\frac{60}{7} a^7 b^3 x^{14}+\frac{45}{13} a^8 b^2 x^{13}+\frac{5}{6} a^9 b x^{12}+\frac{a^{10} x^{11}}{11}+\frac{1}{2} a b^9 x^{20}+\frac{b^{10} x^{21}}{21} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 113, normalized size = 0.9 \begin{align*}{\frac{{a}^{10}{x}^{11}}{11}}+{\frac{5\,{a}^{9}b{x}^{12}}{6}}+{\frac{45\,{a}^{8}{b}^{2}{x}^{13}}{13}}+{\frac{60\,{a}^{7}{b}^{3}{x}^{14}}{7}}+14\,{a}^{6}{b}^{4}{x}^{15}+{\frac{63\,{a}^{5}{b}^{5}{x}^{16}}{4}}+{\frac{210\,{a}^{4}{b}^{6}{x}^{17}}{17}}+{\frac{20\,{a}^{3}{b}^{7}{x}^{18}}{3}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{19}}{19}}+{\frac{a{b}^{9}{x}^{20}}{2}}+{\frac{{b}^{10}{x}^{21}}{21}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05995, size = 151, normalized size = 1.14 \begin{align*} \frac{1}{21} \, b^{10} x^{21} + \frac{1}{2} \, a b^{9} x^{20} + \frac{45}{19} \, a^{2} b^{8} x^{19} + \frac{20}{3} \, a^{3} b^{7} x^{18} + \frac{210}{17} \, a^{4} b^{6} x^{17} + \frac{63}{4} \, a^{5} b^{5} x^{16} + 14 \, a^{6} b^{4} x^{15} + \frac{60}{7} \, a^{7} b^{3} x^{14} + \frac{45}{13} \, a^{8} b^{2} x^{13} + \frac{5}{6} \, a^{9} b x^{12} + \frac{1}{11} \, a^{10} x^{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50767, size = 282, normalized size = 2.14 \begin{align*} \frac{1}{21} x^{21} b^{10} + \frac{1}{2} x^{20} b^{9} a + \frac{45}{19} x^{19} b^{8} a^{2} + \frac{20}{3} x^{18} b^{7} a^{3} + \frac{210}{17} x^{17} b^{6} a^{4} + \frac{63}{4} x^{16} b^{5} a^{5} + 14 x^{15} b^{4} a^{6} + \frac{60}{7} x^{14} b^{3} a^{7} + \frac{45}{13} x^{13} b^{2} a^{8} + \frac{5}{6} x^{12} b a^{9} + \frac{1}{11} x^{11} a^{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.12072, size = 131, normalized size = 0.99 \begin{align*} \frac{a^{10} x^{11}}{11} + \frac{5 a^{9} b x^{12}}{6} + \frac{45 a^{8} b^{2} x^{13}}{13} + \frac{60 a^{7} b^{3} x^{14}}{7} + 14 a^{6} b^{4} x^{15} + \frac{63 a^{5} b^{5} x^{16}}{4} + \frac{210 a^{4} b^{6} x^{17}}{17} + \frac{20 a^{3} b^{7} x^{18}}{3} + \frac{45 a^{2} b^{8} x^{19}}{19} + \frac{a b^{9} x^{20}}{2} + \frac{b^{10} x^{21}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17783, size = 151, normalized size = 1.14 \begin{align*} \frac{1}{21} \, b^{10} x^{21} + \frac{1}{2} \, a b^{9} x^{20} + \frac{45}{19} \, a^{2} b^{8} x^{19} + \frac{20}{3} \, a^{3} b^{7} x^{18} + \frac{210}{17} \, a^{4} b^{6} x^{17} + \frac{63}{4} \, a^{5} b^{5} x^{16} + 14 \, a^{6} b^{4} x^{15} + \frac{60}{7} \, a^{7} b^{3} x^{14} + \frac{45}{13} \, a^{8} b^{2} x^{13} + \frac{5}{6} \, a^{9} b x^{12} + \frac{1}{11} \, a^{10} x^{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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