Optimal. Leaf size=218 \[ -\frac {a^{10} A}{5 x^5}-\frac {a^9 (a B+10 A b)}{4 x^4}-\frac {5 a^8 b (2 a B+9 A b)}{3 x^3}-\frac {15 a^7 b^2 (3 a B+8 A b)}{2 x^2}-\frac {30 a^6 b^3 (4 a B+7 A b)}{x}+42 a^5 b^4 \log (x) (5 a B+6 A b)+42 a^4 b^5 x (6 a B+5 A b)+15 a^3 b^6 x^2 (7 a B+4 A b)+5 a^2 b^7 x^3 (8 a B+3 A b)+\frac {1}{5} b^9 x^5 (10 a B+A b)+\frac {5}{4} a b^8 x^4 (9 a B+2 A b)+\frac {1}{6} b^{10} B x^6 \]
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Rubi [A] time = 0.14, antiderivative size = 218, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {15 a^7 b^2 (3 a B+8 A b)}{2 x^2}+15 a^3 b^6 x^2 (7 a B+4 A b)+5 a^2 b^7 x^3 (8 a B+3 A b)-\frac {30 a^6 b^3 (4 a B+7 A b)}{x}+42 a^4 b^5 x (6 a B+5 A b)+42 a^5 b^4 \log (x) (5 a B+6 A b)-\frac {a^9 (a B+10 A b)}{4 x^4}-\frac {5 a^8 b (2 a B+9 A b)}{3 x^3}-\frac {a^{10} A}{5 x^5}+\frac {5}{4} a b^8 x^4 (9 a B+2 A b)+\frac {1}{5} b^9 x^5 (10 a B+A b)+\frac {1}{6} b^{10} B x^6 \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{x^6} \, dx &=\int \left (42 a^4 b^5 (5 A b+6 a B)+\frac {a^{10} A}{x^6}+\frac {a^9 (10 A b+a B)}{x^5}+\frac {5 a^8 b (9 A b+2 a B)}{x^4}+\frac {15 a^7 b^2 (8 A b+3 a B)}{x^3}+\frac {30 a^6 b^3 (7 A b+4 a B)}{x^2}+\frac {42 a^5 b^4 (6 A b+5 a B)}{x}+30 a^3 b^6 (4 A b+7 a B) x+15 a^2 b^7 (3 A b+8 a B) x^2+5 a b^8 (2 A b+9 a B) x^3+b^9 (A b+10 a B) x^4+b^{10} B x^5\right ) \, dx\\ &=-\frac {a^{10} A}{5 x^5}-\frac {a^9 (10 A b+a B)}{4 x^4}-\frac {5 a^8 b (9 A b+2 a B)}{3 x^3}-\frac {15 a^7 b^2 (8 A b+3 a B)}{2 x^2}-\frac {30 a^6 b^3 (7 A b+4 a B)}{x}+42 a^4 b^5 (5 A b+6 a B) x+15 a^3 b^6 (4 A b+7 a B) x^2+5 a^2 b^7 (3 A b+8 a B) x^3+\frac {5}{4} a b^8 (2 A b+9 a B) x^4+\frac {1}{5} b^9 (A b+10 a B) x^5+\frac {1}{6} b^{10} B x^6+42 a^5 b^4 (6 A b+5 a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.11, size = 210, normalized size = 0.96 \begin {gather*} -\frac {a^{10} (4 A+5 B x)}{20 x^5}-\frac {5 a^9 b (3 A+4 B x)}{6 x^4}-\frac {15 a^8 b^2 (2 A+3 B x)}{2 x^3}-\frac {60 a^7 b^3 (A+2 B x)}{x^2}-\frac {210 a^6 A b^4}{x}+42 a^5 b^4 \log (x) (5 a B+6 A b)+252 a^5 b^5 B x+105 a^4 b^6 x (2 A+B x)+20 a^3 b^7 x^2 (3 A+2 B x)+\frac {15}{4} a^2 b^8 x^3 (4 A+3 B x)+\frac {1}{2} a b^9 x^4 (5 A+4 B x)+\frac {1}{30} b^{10} x^5 (6 A+5 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^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.40, size = 245, normalized size = 1.12 \begin {gather*} \frac {10 \, B b^{10} x^{11} - 12 \, A a^{10} + 12 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 75 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 300 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 900 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 2520 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 2520 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} \log \relax (x) - 1800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 450 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 100 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 15 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 241, normalized size = 1.11 \begin {gather*} \frac {1}{6} \, B b^{10} x^{6} + 2 \, B a b^{9} x^{5} + \frac {1}{5} \, A b^{10} x^{5} + \frac {45}{4} \, B a^{2} b^{8} x^{4} + \frac {5}{2} \, A a b^{9} x^{4} + 40 \, B a^{3} b^{7} x^{3} + 15 \, A a^{2} b^{8} x^{3} + 105 \, B a^{4} b^{6} x^{2} + 60 \, A a^{3} b^{7} x^{2} + 252 \, B a^{5} b^{5} x + 210 \, A a^{4} b^{6} x + 42 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} \log \left ({\left | x \right |}\right ) - \frac {12 \, A a^{10} + 1800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 450 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 100 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 15 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 240, normalized size = 1.10 \begin {gather*} \frac {B \,b^{10} x^{6}}{6}+\frac {A \,b^{10} x^{5}}{5}+2 B a \,b^{9} x^{5}+\frac {5 A a \,b^{9} x^{4}}{2}+\frac {45 B \,a^{2} b^{8} x^{4}}{4}+15 A \,a^{2} b^{8} x^{3}+40 B \,a^{3} b^{7} x^{3}+60 A \,a^{3} b^{7} x^{2}+105 B \,a^{4} b^{6} x^{2}+252 A \,a^{5} b^{5} \ln \relax (x )+210 A \,a^{4} b^{6} x +210 B \,a^{6} b^{4} \ln \relax (x )+252 B \,a^{5} b^{5} x -\frac {210 A \,a^{6} b^{4}}{x}-\frac {120 B \,a^{7} b^{3}}{x}-\frac {60 A \,a^{7} b^{3}}{x^{2}}-\frac {45 B \,a^{8} b^{2}}{2 x^{2}}-\frac {15 A \,a^{8} b^{2}}{x^{3}}-\frac {10 B \,a^{9} b}{3 x^{3}}-\frac {5 A \,a^{9} b}{2 x^{4}}-\frac {B \,a^{10}}{4 x^{4}}-\frac {A \,a^{10}}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 241, normalized size = 1.11 \begin {gather*} \frac {1}{6} \, B b^{10} x^{6} + \frac {1}{5} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{5} + \frac {5}{4} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{4} + 5 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{3} + 15 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{2} + 42 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x + 42 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} \log \relax (x) - \frac {12 \, A a^{10} + 1800 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 450 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 100 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 15 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 221, normalized size = 1.01 \begin {gather*} x^5\,\left (\frac {A\,b^{10}}{5}+2\,B\,a\,b^9\right )-\frac {x\,\left (\frac {B\,a^{10}}{4}+\frac {5\,A\,b\,a^9}{2}\right )+\frac {A\,a^{10}}{5}+x^2\,\left (\frac {10\,B\,a^9\,b}{3}+15\,A\,a^8\,b^2\right )+x^3\,\left (\frac {45\,B\,a^8\,b^2}{2}+60\,A\,a^7\,b^3\right )+x^4\,\left (120\,B\,a^7\,b^3+210\,A\,a^6\,b^4\right )}{x^5}+\ln \relax (x)\,\left (210\,B\,a^6\,b^4+252\,A\,a^5\,b^5\right )+\frac {B\,b^{10}\,x^6}{6}+15\,a^3\,b^6\,x^2\,\left (4\,A\,b+7\,B\,a\right )+5\,a^2\,b^7\,x^3\,\left (3\,A\,b+8\,B\,a\right )+42\,a^4\,b^5\,x\,\left (5\,A\,b+6\,B\,a\right )+\frac {5\,a\,b^8\,x^4\,\left (2\,A\,b+9\,B\,a\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.34, size = 250, normalized size = 1.15 \begin {gather*} \frac {B b^{10} x^{6}}{6} + 42 a^{5} b^{4} \left (6 A b + 5 B a\right ) \log {\relax (x )} + x^{5} \left (\frac {A b^{10}}{5} + 2 B a b^{9}\right ) + x^{4} \left (\frac {5 A a b^{9}}{2} + \frac {45 B a^{2} b^{8}}{4}\right ) + x^{3} \left (15 A a^{2} b^{8} + 40 B a^{3} b^{7}\right ) + x^{2} \left (60 A a^{3} b^{7} + 105 B a^{4} b^{6}\right ) + x \left (210 A a^{4} b^{6} + 252 B a^{5} b^{5}\right ) + \frac {- 12 A a^{10} + x^{4} \left (- 12600 A a^{6} b^{4} - 7200 B a^{7} b^{3}\right ) + x^{3} \left (- 3600 A a^{7} b^{3} - 1350 B a^{8} b^{2}\right ) + x^{2} \left (- 900 A a^{8} b^{2} - 200 B a^{9} b\right ) + x \left (- 150 A a^{9} b - 15 B a^{10}\right )}{60 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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