Optimal. Leaf size=240 \[ \frac{28 a^2 (a+b x)^{17} (A b-3 a B)}{17 b^{10}}-\frac{7 a^3 (a+b x)^{16} (4 A b-9 a B)}{8 b^{10}}+\frac{14 a^4 (a+b x)^{15} (5 A b-9 a B)}{15 b^{10}}-\frac{2 a^5 (a+b x)^{14} (2 A b-3 a B)}{b^{10}}+\frac{4 a^6 (a+b x)^{13} (7 A b-9 a B)}{13 b^{10}}-\frac{a^7 (a+b x)^{12} (8 A b-9 a B)}{12 b^{10}}+\frac{a^8 (a+b x)^{11} (A b-a B)}{11 b^{10}}+\frac{(a+b x)^{19} (A b-9 a B)}{19 b^{10}}-\frac{2 a (a+b x)^{18} (2 A b-9 a B)}{9 b^{10}}+\frac{B (a+b x)^{20}}{20 b^{10}} \]
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Rubi [A] time = 0.193844, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ \frac{28 a^2 (a+b x)^{17} (A b-3 a B)}{17 b^{10}}-\frac{7 a^3 (a+b x)^{16} (4 A b-9 a B)}{8 b^{10}}+\frac{14 a^4 (a+b x)^{15} (5 A b-9 a B)}{15 b^{10}}-\frac{2 a^5 (a+b x)^{14} (2 A b-3 a B)}{b^{10}}+\frac{4 a^6 (a+b x)^{13} (7 A b-9 a B)}{13 b^{10}}-\frac{a^7 (a+b x)^{12} (8 A b-9 a B)}{12 b^{10}}+\frac{a^8 (a+b x)^{11} (A b-a B)}{11 b^{10}}+\frac{(a+b x)^{19} (A b-9 a B)}{19 b^{10}}-\frac{2 a (a+b x)^{18} (2 A b-9 a B)}{9 b^{10}}+\frac{B (a+b x)^{20}}{20 b^{10}} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int x^8 (a+b x)^{10} (A+B x) \, dx &=\int \left (-\frac{a^8 (-A b+a B) (a+b x)^{10}}{b^9}+\frac{a^7 (-8 A b+9 a B) (a+b x)^{11}}{b^9}-\frac{4 a^6 (-7 A b+9 a B) (a+b x)^{12}}{b^9}+\frac{28 a^5 (-2 A b+3 a B) (a+b x)^{13}}{b^9}-\frac{14 a^4 (-5 A b+9 a B) (a+b x)^{14}}{b^9}+\frac{14 a^3 (-4 A b+9 a B) (a+b x)^{15}}{b^9}-\frac{28 a^2 (-A b+3 a B) (a+b x)^{16}}{b^9}+\frac{4 a (-2 A b+9 a B) (a+b x)^{17}}{b^9}+\frac{(A b-9 a B) (a+b x)^{18}}{b^9}+\frac{B (a+b x)^{19}}{b^9}\right ) \, dx\\ &=\frac{a^8 (A b-a B) (a+b x)^{11}}{11 b^{10}}-\frac{a^7 (8 A b-9 a B) (a+b x)^{12}}{12 b^{10}}+\frac{4 a^6 (7 A b-9 a B) (a+b x)^{13}}{13 b^{10}}-\frac{2 a^5 (2 A b-3 a B) (a+b x)^{14}}{b^{10}}+\frac{14 a^4 (5 A b-9 a B) (a+b x)^{15}}{15 b^{10}}-\frac{7 a^3 (4 A b-9 a B) (a+b x)^{16}}{8 b^{10}}+\frac{28 a^2 (A b-3 a B) (a+b x)^{17}}{17 b^{10}}-\frac{2 a (2 A b-9 a B) (a+b x)^{18}}{9 b^{10}}+\frac{(A b-9 a B) (a+b x)^{19}}{19 b^{10}}+\frac{B (a+b x)^{20}}{20 b^{10}}\\ \end{align*}
Mathematica [A] time = 0.0290651, size = 229, normalized size = 0.95 \[ \frac{15}{17} a^2 b^7 x^{17} (8 a B+3 A b)+\frac{15}{8} a^3 b^6 x^{16} (7 a B+4 A b)+\frac{14}{5} a^4 b^5 x^{15} (6 a B+5 A b)+3 a^5 b^4 x^{14} (5 a B+6 A b)+\frac{30}{13} a^6 b^3 x^{13} (4 a B+7 A b)+\frac{5}{4} a^7 b^2 x^{12} (3 a B+8 A b)+\frac{5}{11} a^8 b x^{11} (2 a B+9 A b)+\frac{1}{10} a^9 x^{10} (a B+10 A b)+\frac{1}{9} a^{10} A x^9+\frac{1}{19} b^9 x^{19} (10 a B+A b)+\frac{5}{18} a b^8 x^{18} (9 a B+2 A b)+\frac{1}{20} b^{10} B x^{20} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 244, normalized size = 1. \begin{align*}{\frac{{b}^{10}B{x}^{20}}{20}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{19}}{19}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{18}}{18}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{17}}{17}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{16}}{16}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{15}}{15}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{14}}{14}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{13}}{13}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{12}}{12}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{10}}{10}}+{\frac{{a}^{10}A{x}^{9}}{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990628, size = 328, normalized size = 1.37 \begin{align*} \frac{1}{20} \, B b^{10} x^{20} + \frac{1}{9} \, A a^{10} x^{9} + \frac{1}{19} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{19} + \frac{5}{18} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{18} + \frac{15}{17} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{17} + \frac{15}{8} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{16} + \frac{14}{5} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{15} + 3 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{14} + \frac{30}{13} \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{13} + \frac{5}{4} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{12} + \frac{5}{11} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{11} + \frac{1}{10} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22433, size = 617, normalized size = 2.57 \begin{align*} \frac{1}{20} x^{20} b^{10} B + \frac{10}{19} x^{19} b^{9} a B + \frac{1}{19} x^{19} b^{10} A + \frac{5}{2} x^{18} b^{8} a^{2} B + \frac{5}{9} x^{18} b^{9} a A + \frac{120}{17} x^{17} b^{7} a^{3} B + \frac{45}{17} x^{17} b^{8} a^{2} A + \frac{105}{8} x^{16} b^{6} a^{4} B + \frac{15}{2} x^{16} b^{7} a^{3} A + \frac{84}{5} x^{15} b^{5} a^{5} B + 14 x^{15} b^{6} a^{4} A + 15 x^{14} b^{4} a^{6} B + 18 x^{14} b^{5} a^{5} A + \frac{120}{13} x^{13} b^{3} a^{7} B + \frac{210}{13} x^{13} b^{4} a^{6} A + \frac{15}{4} x^{12} b^{2} a^{8} B + 10 x^{12} b^{3} a^{7} A + \frac{10}{11} x^{11} b a^{9} B + \frac{45}{11} x^{11} b^{2} a^{8} A + \frac{1}{10} x^{10} a^{10} B + x^{10} b a^{9} A + \frac{1}{9} x^{9} a^{10} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.12603, size = 264, normalized size = 1.1 \begin{align*} \frac{A a^{10} x^{9}}{9} + \frac{B b^{10} x^{20}}{20} + x^{19} \left (\frac{A b^{10}}{19} + \frac{10 B a b^{9}}{19}\right ) + x^{18} \left (\frac{5 A a b^{9}}{9} + \frac{5 B a^{2} b^{8}}{2}\right ) + x^{17} \left (\frac{45 A a^{2} b^{8}}{17} + \frac{120 B a^{3} b^{7}}{17}\right ) + x^{16} \left (\frac{15 A a^{3} b^{7}}{2} + \frac{105 B a^{4} b^{6}}{8}\right ) + x^{15} \left (14 A a^{4} b^{6} + \frac{84 B a^{5} b^{5}}{5}\right ) + x^{14} \left (18 A a^{5} b^{5} + 15 B a^{6} b^{4}\right ) + x^{13} \left (\frac{210 A a^{6} b^{4}}{13} + \frac{120 B a^{7} b^{3}}{13}\right ) + x^{12} \left (10 A a^{7} b^{3} + \frac{15 B a^{8} b^{2}}{4}\right ) + x^{11} \left (\frac{45 A a^{8} b^{2}}{11} + \frac{10 B a^{9} b}{11}\right ) + x^{10} \left (A a^{9} b + \frac{B a^{10}}{10}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17321, size = 329, normalized size = 1.37 \begin{align*} \frac{1}{20} \, B b^{10} x^{20} + \frac{10}{19} \, B a b^{9} x^{19} + \frac{1}{19} \, A b^{10} x^{19} + \frac{5}{2} \, B a^{2} b^{8} x^{18} + \frac{5}{9} \, A a b^{9} x^{18} + \frac{120}{17} \, B a^{3} b^{7} x^{17} + \frac{45}{17} \, A a^{2} b^{8} x^{17} + \frac{105}{8} \, B a^{4} b^{6} x^{16} + \frac{15}{2} \, A a^{3} b^{7} x^{16} + \frac{84}{5} \, B a^{5} b^{5} x^{15} + 14 \, A a^{4} b^{6} x^{15} + 15 \, B a^{6} b^{4} x^{14} + 18 \, A a^{5} b^{5} x^{14} + \frac{120}{13} \, B a^{7} b^{3} x^{13} + \frac{210}{13} \, A a^{6} b^{4} x^{13} + \frac{15}{4} \, B a^{8} b^{2} x^{12} + 10 \, A a^{7} b^{3} x^{12} + \frac{10}{11} \, B a^{9} b x^{11} + \frac{45}{11} \, A a^{8} b^{2} x^{11} + \frac{1}{10} \, B a^{10} x^{10} + A a^{9} b x^{10} + \frac{1}{9} \, A a^{10} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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