Optimal. Leaf size=148 \[ a^{10} A \log (x)+10 a^9 A b x+\frac {45}{2} a^8 A b^2 x^2+40 a^7 A b^3 x^3+\frac {105}{2} a^6 A b^4 x^4+\frac {252}{5} a^5 A b^5 x^5+35 a^4 A b^6 x^6+\frac {120}{7} a^3 A b^7 x^7+\frac {45}{8} a^2 A b^8 x^8+\frac {10}{9} a A b^9 x^9+\frac {B (a+b x)^{11}}{11 b}+\frac {1}{10} A b^{10} x^{10} \]
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Rubi [A] time = 0.06, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 43} \begin {gather*} \frac {45}{8} a^2 A b^8 x^8+\frac {120}{7} a^3 A b^7 x^7+35 a^4 A b^6 x^6+\frac {252}{5} a^5 A b^5 x^5+\frac {105}{2} a^6 A b^4 x^4+40 a^7 A b^3 x^3+\frac {45}{2} a^8 A b^2 x^2+10 a^9 A b x+a^{10} A \log (x)+\frac {10}{9} a A b^9 x^9+\frac {B (a+b x)^{11}}{11 b}+\frac {1}{10} A b^{10} x^{10} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 80
Rubi steps
\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{x} \, dx &=\frac {B (a+b x)^{11}}{11 b}+A \int \frac {(a+b x)^{10}}{x} \, dx\\ &=\frac {B (a+b x)^{11}}{11 b}+A \int \left (10 a^9 b+\frac {a^{10}}{x}+45 a^8 b^2 x+120 a^7 b^3 x^2+210 a^6 b^4 x^3+252 a^5 b^5 x^4+210 a^4 b^6 x^5+120 a^3 b^7 x^6+45 a^2 b^8 x^7+10 a b^9 x^8+b^{10} x^9\right ) \, dx\\ &=10 a^9 A b x+\frac {45}{2} a^8 A b^2 x^2+40 a^7 A b^3 x^3+\frac {105}{2} a^6 A b^4 x^4+\frac {252}{5} a^5 A b^5 x^5+35 a^4 A b^6 x^6+\frac {120}{7} a^3 A b^7 x^7+\frac {45}{8} a^2 A b^8 x^8+\frac {10}{9} a A b^9 x^9+\frac {1}{10} A b^{10} x^{10}+\frac {B (a+b x)^{11}}{11 b}+a^{10} A \log (x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 208, normalized size = 1.41 \begin {gather*} a^{10} A \log (x)+a^{10} B x+5 a^9 b x (2 A+B x)+\frac {15}{2} a^8 b^2 x^2 (3 A+2 B x)+10 a^7 b^3 x^3 (4 A+3 B x)+\frac {21}{2} a^6 b^4 x^4 (5 A+4 B x)+\frac {42}{5} a^5 b^5 x^5 (6 A+5 B x)+5 a^4 b^6 x^6 (7 A+6 B x)+\frac {15}{7} a^3 b^7 x^7 (8 A+7 B x)+\frac {5}{8} a^2 b^8 x^8 (9 A+8 B x)+\frac {1}{9} a b^9 x^9 (10 A+9 B x)+\frac {1}{110} b^{10} x^{10} (11 A+10 B x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^{10} (A+B x)}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.21, size = 238, normalized size = 1.61 \begin {gather*} \frac {1}{11} \, B b^{10} x^{11} + A a^{10} \log \relax (x) + \frac {1}{10} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + \frac {5}{9} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + \frac {15}{8} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + \frac {30}{7} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + \frac {42}{5} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + \frac {15}{2} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + \frac {5}{2} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + {\left (B a^{10} + 10 \, A a^{9} b\right )} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.36, size = 238, normalized size = 1.61 \begin {gather*} \frac {1}{11} \, B b^{10} x^{11} + B a b^{9} x^{10} + \frac {1}{10} \, A b^{10} x^{10} + 5 \, B a^{2} b^{8} x^{9} + \frac {10}{9} \, A a b^{9} x^{9} + 15 \, B a^{3} b^{7} x^{8} + \frac {45}{8} \, A a^{2} b^{8} x^{8} + 30 \, B a^{4} b^{6} x^{7} + \frac {120}{7} \, A a^{3} b^{7} x^{7} + 42 \, B a^{5} b^{5} x^{6} + 35 \, A a^{4} b^{6} x^{6} + 42 \, B a^{6} b^{4} x^{5} + \frac {252}{5} \, A a^{5} b^{5} x^{5} + 30 \, B a^{7} b^{3} x^{4} + \frac {105}{2} \, A a^{6} b^{4} x^{4} + 15 \, B a^{8} b^{2} x^{3} + 40 \, A a^{7} b^{3} x^{3} + 5 \, B a^{9} b x^{2} + \frac {45}{2} \, A a^{8} b^{2} x^{2} + B a^{10} x + 10 \, A a^{9} b x + A a^{10} \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 238, normalized size = 1.61 \begin {gather*} \frac {B \,b^{10} x^{11}}{11}+\frac {A \,b^{10} x^{10}}{10}+B a \,b^{9} x^{10}+\frac {10 A a \,b^{9} x^{9}}{9}+5 B \,a^{2} b^{8} x^{9}+\frac {45 A \,a^{2} b^{8} x^{8}}{8}+15 B \,a^{3} b^{7} x^{8}+\frac {120 A \,a^{3} b^{7} x^{7}}{7}+30 B \,a^{4} b^{6} x^{7}+35 A \,a^{4} b^{6} x^{6}+42 B \,a^{5} b^{5} x^{6}+\frac {252 A \,a^{5} b^{5} x^{5}}{5}+42 B \,a^{6} b^{4} x^{5}+\frac {105 A \,a^{6} b^{4} x^{4}}{2}+30 B \,a^{7} b^{3} x^{4}+40 A \,a^{7} b^{3} x^{3}+15 B \,a^{8} b^{2} x^{3}+\frac {45 A \,a^{8} b^{2} x^{2}}{2}+5 B \,a^{9} b \,x^{2}+A \,a^{10} \ln \relax (x )+10 A \,a^{9} b x +B \,a^{10} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 238, normalized size = 1.61 \begin {gather*} \frac {1}{11} \, B b^{10} x^{11} + A a^{10} \log \relax (x) + \frac {1}{10} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + \frac {5}{9} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + \frac {15}{8} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + \frac {30}{7} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + \frac {42}{5} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + \frac {15}{2} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 5 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + \frac {5}{2} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + {\left (B a^{10} + 10 \, A a^{9} b\right )} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 205, normalized size = 1.39 \begin {gather*} x\,\left (B\,a^{10}+10\,A\,b\,a^9\right )+x^{10}\,\left (\frac {A\,b^{10}}{10}+B\,a\,b^9\right )+\frac {B\,b^{10}\,x^{11}}{11}+A\,a^{10}\,\ln \relax (x)+5\,a^7\,b^2\,x^3\,\left (8\,A\,b+3\,B\,a\right )+\frac {15\,a^6\,b^3\,x^4\,\left (7\,A\,b+4\,B\,a\right )}{2}+\frac {42\,a^5\,b^4\,x^5\,\left (6\,A\,b+5\,B\,a\right )}{5}+7\,a^4\,b^5\,x^6\,\left (5\,A\,b+6\,B\,a\right )+\frac {30\,a^3\,b^6\,x^7\,\left (4\,A\,b+7\,B\,a\right )}{7}+\frac {15\,a^2\,b^7\,x^8\,\left (3\,A\,b+8\,B\,a\right )}{8}+\frac {5\,a^8\,b\,x^2\,\left (9\,A\,b+2\,B\,a\right )}{2}+\frac {5\,a\,b^8\,x^9\,\left (2\,A\,b+9\,B\,a\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 246, normalized size = 1.66 \begin {gather*} A a^{10} \log {\relax (x )} + \frac {B b^{10} x^{11}}{11} + x^{10} \left (\frac {A b^{10}}{10} + B a b^{9}\right ) + x^{9} \left (\frac {10 A a b^{9}}{9} + 5 B a^{2} b^{8}\right ) + x^{8} \left (\frac {45 A a^{2} b^{8}}{8} + 15 B a^{3} b^{7}\right ) + x^{7} \left (\frac {120 A a^{3} b^{7}}{7} + 30 B a^{4} b^{6}\right ) + x^{6} \left (35 A a^{4} b^{6} + 42 B a^{5} b^{5}\right ) + x^{5} \left (\frac {252 A a^{5} b^{5}}{5} + 42 B a^{6} b^{4}\right ) + x^{4} \left (\frac {105 A a^{6} b^{4}}{2} + 30 B a^{7} b^{3}\right ) + x^{3} \left (40 A a^{7} b^{3} + 15 B a^{8} b^{2}\right ) + x^{2} \left (\frac {45 A a^{8} b^{2}}{2} + 5 B a^{9} b\right ) + x \left (10 A a^{9} b + B a^{10}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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