Optimal. Leaf size=216 \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{7 x^7}-\frac{5 a^6 b^3 (4 a B+7 A b)}{x^6}-\frac{42 a^5 b^4 (5 a B+6 A b)}{5 x^5}-\frac{21 a^4 b^5 (6 a B+5 A b)}{2 x^4}-\frac{10 a^3 b^6 (7 a B+4 A b)}{x^3}-\frac{15 a^2 b^7 (8 a B+3 A b)}{2 x^2}-\frac{a^9 (a B+10 A b)}{9 x^9}-\frac{5 a^8 b (2 a B+9 A b)}{8 x^8}-\frac{a^{10} A}{10 x^{10}}-\frac{5 a b^8 (9 a B+2 A b)}{x}+b^9 \log (x) (10 a B+A b)+b^{10} B x \]
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Rubi [A] time = 0.144374, antiderivative size = 216, 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} \[ -\frac{15 a^7 b^2 (3 a B+8 A b)}{7 x^7}-\frac{5 a^6 b^3 (4 a B+7 A b)}{x^6}-\frac{42 a^5 b^4 (5 a B+6 A b)}{5 x^5}-\frac{21 a^4 b^5 (6 a B+5 A b)}{2 x^4}-\frac{10 a^3 b^6 (7 a B+4 A b)}{x^3}-\frac{15 a^2 b^7 (8 a B+3 A b)}{2 x^2}-\frac{a^9 (a B+10 A b)}{9 x^9}-\frac{5 a^8 b (2 a B+9 A b)}{8 x^8}-\frac{a^{10} A}{10 x^{10}}-\frac{5 a b^8 (9 a B+2 A b)}{x}+b^9 \log (x) (10 a B+A b)+b^{10} B x \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int \frac{(a+b x)^{10} (A+B x)}{x^{11}} \, dx &=\int \left (b^{10} B+\frac{a^{10} A}{x^{11}}+\frac{a^9 (10 A b+a B)}{x^{10}}+\frac{5 a^8 b (9 A b+2 a B)}{x^9}+\frac{15 a^7 b^2 (8 A b+3 a B)}{x^8}+\frac{30 a^6 b^3 (7 A b+4 a B)}{x^7}+\frac{42 a^5 b^4 (6 A b+5 a B)}{x^6}+\frac{42 a^4 b^5 (5 A b+6 a B)}{x^5}+\frac{30 a^3 b^6 (4 A b+7 a B)}{x^4}+\frac{15 a^2 b^7 (3 A b+8 a B)}{x^3}+\frac{5 a b^8 (2 A b+9 a B)}{x^2}+\frac{b^9 (A b+10 a B)}{x}\right ) \, dx\\ &=-\frac{a^{10} A}{10 x^{10}}-\frac{a^9 (10 A b+a B)}{9 x^9}-\frac{5 a^8 b (9 A b+2 a B)}{8 x^8}-\frac{15 a^7 b^2 (8 A b+3 a B)}{7 x^7}-\frac{5 a^6 b^3 (7 A b+4 a B)}{x^6}-\frac{42 a^5 b^4 (6 A b+5 a B)}{5 x^5}-\frac{21 a^4 b^5 (5 A b+6 a B)}{2 x^4}-\frac{10 a^3 b^6 (4 A b+7 a B)}{x^3}-\frac{15 a^2 b^7 (3 A b+8 a B)}{2 x^2}-\frac{5 a b^8 (2 A b+9 a B)}{x}+b^{10} B x+b^9 (A b+10 a B) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0926875, size = 209, normalized size = 0.97 \[ -\frac{45 a^8 b^2 (7 A+8 B x)}{56 x^8}-\frac{20 a^7 b^3 (6 A+7 B x)}{7 x^7}-\frac{7 a^6 b^4 (5 A+6 B x)}{x^6}-\frac{63 a^5 b^5 (4 A+5 B x)}{5 x^5}-\frac{35 a^4 b^6 (3 A+4 B x)}{2 x^4}-\frac{20 a^3 b^7 (2 A+3 B x)}{x^3}-\frac{45 a^2 b^8 (A+2 B x)}{2 x^2}-\frac{5 a^9 b (8 A+9 B x)}{36 x^9}-\frac{a^{10} (9 A+10 B x)}{90 x^{10}}+b^9 \log (x) (10 a B+A b)-\frac{10 a A b^9}{x}+b^{10} B x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 240, normalized size = 1.1 \begin{align*}{b}^{10}Bx+A\ln \left ( x \right ){b}^{10}+10\,B\ln \left ( x \right ) a{b}^{9}-40\,{\frac{{a}^{3}{b}^{7}A}{{x}^{3}}}-70\,{\frac{{a}^{4}{b}^{6}B}{{x}^{3}}}-{\frac{252\,{a}^{5}{b}^{5}A}{5\,{x}^{5}}}-42\,{\frac{{a}^{6}{b}^{4}B}{{x}^{5}}}-{\frac{105\,{a}^{4}{b}^{6}A}{2\,{x}^{4}}}-63\,{\frac{{a}^{5}{b}^{5}B}{{x}^{4}}}-{\frac{45\,{a}^{8}{b}^{2}A}{8\,{x}^{8}}}-{\frac{5\,{a}^{9}bB}{4\,{x}^{8}}}-{\frac{45\,{a}^{2}{b}^{8}A}{2\,{x}^{2}}}-60\,{\frac{{a}^{3}{b}^{7}B}{{x}^{2}}}-35\,{\frac{{a}^{6}{b}^{4}A}{{x}^{6}}}-20\,{\frac{{a}^{7}{b}^{3}B}{{x}^{6}}}-{\frac{120\,{a}^{7}{b}^{3}A}{7\,{x}^{7}}}-{\frac{45\,{a}^{8}{b}^{2}B}{7\,{x}^{7}}}-10\,{\frac{a{b}^{9}A}{x}}-45\,{\frac{{a}^{2}{b}^{8}B}{x}}-{\frac{10\,{a}^{9}bA}{9\,{x}^{9}}}-{\frac{{a}^{10}B}{9\,{x}^{9}}}-{\frac{A{a}^{10}}{10\,{x}^{10}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04763, size = 323, normalized size = 1.5 \begin{align*} B b^{10} x +{\left (10 \, B a b^{9} + A b^{10}\right )} \log \left (x\right ) - \frac{252 \, A a^{10} + 12600 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 18900 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 25200 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 26460 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 21168 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 12600 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5400 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1575 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 280 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27044, size = 581, normalized size = 2.69 \begin{align*} \frac{2520 \, B b^{10} x^{11} + 2520 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} \log \left (x\right ) - 252 \, A a^{10} - 12600 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} - 18900 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} - 25200 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} - 26460 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} - 21168 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 12600 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 5400 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 1575 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 280 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.6507, size = 236, normalized size = 1.09 \begin{align*} B b^{10} x + b^{9} \left (A b + 10 B a\right ) \log{\left (x \right )} - \frac{252 A a^{10} + x^{9} \left (25200 A a b^{9} + 113400 B a^{2} b^{8}\right ) + x^{8} \left (56700 A a^{2} b^{8} + 151200 B a^{3} b^{7}\right ) + x^{7} \left (100800 A a^{3} b^{7} + 176400 B a^{4} b^{6}\right ) + x^{6} \left (132300 A a^{4} b^{6} + 158760 B a^{5} b^{5}\right ) + x^{5} \left (127008 A a^{5} b^{5} + 105840 B a^{6} b^{4}\right ) + x^{4} \left (88200 A a^{6} b^{4} + 50400 B a^{7} b^{3}\right ) + x^{3} \left (43200 A a^{7} b^{3} + 16200 B a^{8} b^{2}\right ) + x^{2} \left (14175 A a^{8} b^{2} + 3150 B a^{9} b\right ) + x \left (2800 A a^{9} b + 280 B a^{10}\right )}{2520 x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29908, size = 324, normalized size = 1.5 \begin{align*} B b^{10} x +{\left (10 \, B a b^{9} + A b^{10}\right )} \log \left ({\left | x \right |}\right ) - \frac{252 \, A a^{10} + 12600 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 18900 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 25200 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 26460 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 21168 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 12600 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5400 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 1575 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 280 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2520 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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