Optimal. Leaf size=38 \[ \frac{(a+b x)^{11} (A b-a B)}{11 b^2}+\frac{B (a+b x)^{12}}{12 b^2} \]
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Rubi [A] time = 0.0157818, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{(a+b x)^{11} (A b-a B)}{11 b^2}+\frac{B (a+b x)^{12}}{12 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^{10} (A+B x) \, dx &=\int \left (\frac{(A b-a B) (a+b x)^{10}}{b}+\frac{B (a+b x)^{11}}{b}\right ) \, dx\\ &=\frac{(A b-a B) (a+b x)^{11}}{11 b^2}+\frac{B (a+b x)^{12}}{12 b^2}\\ \end{align*}
Mathematica [B] time = 0.0601074, size = 198, normalized size = 5.21 \[ \frac{1}{132} x \left (495 a^8 b^2 x^2 (4 A+3 B x)+792 a^7 b^3 x^3 (5 A+4 B x)+924 a^6 b^4 x^4 (6 A+5 B x)+792 a^5 b^5 x^5 (7 A+6 B x)+495 a^4 b^6 x^6 (8 A+7 B x)+220 a^3 b^7 x^7 (9 A+8 B x)+66 a^2 b^8 x^8 (10 A+9 B x)+220 a^9 b x (3 A+2 B x)+66 a^{10} (2 A+B x)+12 a b^9 x^9 (11 A+10 B x)+b^{10} x^{10} (12 A+11 B x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 241, normalized size = 6.3 \begin{align*}{\frac{{b}^{10}B{x}^{12}}{12}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{10}}{10}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{9}}{9}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{8}}{8}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{7}}{7}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{6}}{6}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{5}}{5}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{4}}{4}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{3}}{3}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{2}}{2}}+{a}^{10}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.81584, size = 324, normalized size = 8.53 \begin{align*} \frac{1}{12} \, B b^{10} x^{12} + A a^{10} x + \frac{1}{11} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{11} + \frac{1}{2} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{10} + \frac{5}{3} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{9} + \frac{15}{4} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{8} + 6 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{7} + 7 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{6} + 6 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{5} + \frac{15}{4} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{4} + \frac{5}{3} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52664, size = 554, normalized size = 14.58 \begin{align*} \frac{1}{12} x^{12} b^{10} B + \frac{10}{11} x^{11} b^{9} a B + \frac{1}{11} x^{11} b^{10} A + \frac{9}{2} x^{10} b^{8} a^{2} B + x^{10} b^{9} a A + \frac{40}{3} x^{9} b^{7} a^{3} B + 5 x^{9} b^{8} a^{2} A + \frac{105}{4} x^{8} b^{6} a^{4} B + 15 x^{8} b^{7} a^{3} A + 36 x^{7} b^{5} a^{5} B + 30 x^{7} b^{6} a^{4} A + 35 x^{6} b^{4} a^{6} B + 42 x^{6} b^{5} a^{5} A + 24 x^{5} b^{3} a^{7} B + 42 x^{5} b^{4} a^{6} A + \frac{45}{4} x^{4} b^{2} a^{8} B + 30 x^{4} b^{3} a^{7} A + \frac{10}{3} x^{3} b a^{9} B + 15 x^{3} b^{2} a^{8} A + \frac{1}{2} x^{2} a^{10} B + 5 x^{2} b a^{9} A + x a^{10} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.105304, size = 248, normalized size = 6.53 \begin{align*} A a^{10} x + \frac{B b^{10} x^{12}}{12} + x^{11} \left (\frac{A b^{10}}{11} + \frac{10 B a b^{9}}{11}\right ) + x^{10} \left (A a b^{9} + \frac{9 B a^{2} b^{8}}{2}\right ) + x^{9} \left (5 A a^{2} b^{8} + \frac{40 B a^{3} b^{7}}{3}\right ) + x^{8} \left (15 A a^{3} b^{7} + \frac{105 B a^{4} b^{6}}{4}\right ) + x^{7} \left (30 A a^{4} b^{6} + 36 B a^{5} b^{5}\right ) + x^{6} \left (42 A a^{5} b^{5} + 35 B a^{6} b^{4}\right ) + x^{5} \left (42 A a^{6} b^{4} + 24 B a^{7} b^{3}\right ) + x^{4} \left (30 A a^{7} b^{3} + \frac{45 B a^{8} b^{2}}{4}\right ) + x^{3} \left (15 A a^{8} b^{2} + \frac{10 B a^{9} b}{3}\right ) + x^{2} \left (5 A a^{9} b + \frac{B a^{10}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.9355, size = 325, normalized size = 8.55 \begin{align*} \frac{1}{12} \, B b^{10} x^{12} + \frac{10}{11} \, B a b^{9} x^{11} + \frac{1}{11} \, A b^{10} x^{11} + \frac{9}{2} \, B a^{2} b^{8} x^{10} + A a b^{9} x^{10} + \frac{40}{3} \, B a^{3} b^{7} x^{9} + 5 \, A a^{2} b^{8} x^{9} + \frac{105}{4} \, B a^{4} b^{6} x^{8} + 15 \, A a^{3} b^{7} x^{8} + 36 \, B a^{5} b^{5} x^{7} + 30 \, A a^{4} b^{6} x^{7} + 35 \, B a^{6} b^{4} x^{6} + 42 \, A a^{5} b^{5} x^{6} + 24 \, B a^{7} b^{3} x^{5} + 42 \, A a^{6} b^{4} x^{5} + \frac{45}{4} \, B a^{8} b^{2} x^{4} + 30 \, A a^{7} b^{3} x^{4} + \frac{10}{3} \, B a^{9} b x^{3} + 15 \, A a^{8} b^{2} x^{3} + \frac{1}{2} \, B a^{10} x^{2} + 5 \, A a^{9} b x^{2} + A a^{10} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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