Optimal. Leaf size=216 \[ -\frac {a^{10} A}{2 x^2}-\frac {a^9 (a B+10 A b)}{x}+5 a^8 b \log (x) (2 a B+9 A b)+15 a^7 b^2 x (3 a B+8 A b)+15 a^6 b^3 x^2 (4 a B+7 A b)+14 a^5 b^4 x^3 (5 a B+6 A b)+\frac {21}{2} a^4 b^5 x^4 (6 a B+5 A b)+6 a^3 b^6 x^5 (7 a B+4 A b)+\frac {5}{2} a^2 b^7 x^6 (8 a B+3 A b)+\frac {1}{8} b^9 x^8 (10 a B+A b)+\frac {5}{7} a b^8 x^7 (9 a B+2 A b)+\frac {1}{9} b^{10} B x^9 \]
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Rubi [A] time = 0.14, antiderivative size = 216, 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*} 15 a^6 b^3 x^2 (4 a B+7 A b)+14 a^5 b^4 x^3 (5 a B+6 A b)+\frac {21}{2} a^4 b^5 x^4 (6 a B+5 A b)+6 a^3 b^6 x^5 (7 a B+4 A b)+\frac {5}{2} a^2 b^7 x^6 (8 a B+3 A b)+15 a^7 b^2 x (3 a B+8 A b)-\frac {a^9 (a B+10 A b)}{x}+5 a^8 b \log (x) (2 a B+9 A b)-\frac {a^{10} A}{2 x^2}+\frac {5}{7} a b^8 x^7 (9 a B+2 A b)+\frac {1}{8} b^9 x^8 (10 a B+A b)+\frac {1}{9} b^{10} B x^9 \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^3} \, dx &=\int \left (15 a^7 b^2 (8 A b+3 a B)+\frac {a^{10} A}{x^3}+\frac {a^9 (10 A b+a B)}{x^2}+\frac {5 a^8 b (9 A b+2 a B)}{x}+30 a^6 b^3 (7 A b+4 a B) x+42 a^5 b^4 (6 A b+5 a B) x^2+42 a^4 b^5 (5 A b+6 a B) x^3+30 a^3 b^6 (4 A b+7 a B) x^4+15 a^2 b^7 (3 A b+8 a B) x^5+5 a b^8 (2 A b+9 a B) x^6+b^9 (A b+10 a B) x^7+b^{10} B x^8\right ) \, dx\\ &=-\frac {a^{10} A}{2 x^2}-\frac {a^9 (10 A b+a B)}{x}+15 a^7 b^2 (8 A b+3 a B) x+15 a^6 b^3 (7 A b+4 a B) x^2+14 a^5 b^4 (6 A b+5 a B) x^3+\frac {21}{2} a^4 b^5 (5 A b+6 a B) x^4+6 a^3 b^6 (4 A b+7 a B) x^5+\frac {5}{2} a^2 b^7 (3 A b+8 a B) x^6+\frac {5}{7} a b^8 (2 A b+9 a B) x^7+\frac {1}{8} b^9 (A b+10 a B) x^8+\frac {1}{9} b^{10} B x^9+5 a^8 b (9 A b+2 a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.11, size = 206, normalized size = 0.95 \begin {gather*} -\frac {a^{10} (A+2 B x)}{2 x^2}-\frac {10 a^9 A b}{x}+5 a^8 b \log (x) (2 a B+9 A b)+45 a^8 b^2 B x+60 a^7 b^3 x (2 A+B x)+35 a^6 b^4 x^2 (3 A+2 B x)+21 a^5 b^5 x^3 (4 A+3 B x)+\frac {21}{2} a^4 b^6 x^4 (5 A+4 B x)+4 a^3 b^7 x^5 (6 A+5 B x)+\frac {15}{14} a^2 b^8 x^6 (7 A+6 B x)+\frac {5}{28} a b^9 x^7 (8 A+7 B x)+\frac {1}{72} b^{10} x^8 (9 A+8 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^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.18, size = 245, normalized size = 1.13 \begin {gather*} \frac {56 \, B b^{10} x^{11} - 252 \, A a^{10} + 63 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 360 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 1260 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 3024 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 5292 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 7056 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 7560 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 7560 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 2520 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} \log \relax (x) - 504 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{504 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.77, size = 240, normalized size = 1.11 \begin {gather*} \frac {1}{9} \, B b^{10} x^{9} + \frac {5}{4} \, B a b^{9} x^{8} + \frac {1}{8} \, A b^{10} x^{8} + \frac {45}{7} \, B a^{2} b^{8} x^{7} + \frac {10}{7} \, A a b^{9} x^{7} + 20 \, B a^{3} b^{7} x^{6} + \frac {15}{2} \, A a^{2} b^{8} x^{6} + 42 \, B a^{4} b^{6} x^{5} + 24 \, A a^{3} b^{7} x^{5} + 63 \, B a^{5} b^{5} x^{4} + \frac {105}{2} \, A a^{4} b^{6} x^{4} + 70 \, B a^{6} b^{4} x^{3} + 84 \, A a^{5} b^{5} x^{3} + 60 \, B a^{7} b^{3} x^{2} + 105 \, A a^{6} b^{4} x^{2} + 45 \, B a^{8} b^{2} x + 120 \, A a^{7} b^{3} x + 5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{10} + 2 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2 \, x^{2}} \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 = 1.11 \begin {gather*} \frac {B \,b^{10} x^{9}}{9}+\frac {A \,b^{10} x^{8}}{8}+\frac {5 B a \,b^{9} x^{8}}{4}+\frac {10 A a \,b^{9} x^{7}}{7}+\frac {45 B \,a^{2} b^{8} x^{7}}{7}+\frac {15 A \,a^{2} b^{8} x^{6}}{2}+20 B \,a^{3} b^{7} x^{6}+24 A \,a^{3} b^{7} x^{5}+42 B \,a^{4} b^{6} x^{5}+\frac {105 A \,a^{4} b^{6} x^{4}}{2}+63 B \,a^{5} b^{5} x^{4}+84 A \,a^{5} b^{5} x^{3}+70 B \,a^{6} b^{4} x^{3}+105 A \,a^{6} b^{4} x^{2}+60 B \,a^{7} b^{3} x^{2}+45 A \,a^{8} b^{2} \ln \relax (x )+120 A \,a^{7} b^{3} x +10 B \,a^{9} b \ln \relax (x )+45 B \,a^{8} b^{2} x -\frac {10 A \,a^{9} b}{x}-\frac {B \,a^{10}}{x}-\frac {A \,a^{10}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 240, normalized size = 1.11 \begin {gather*} \frac {1}{9} \, B b^{10} x^{9} + \frac {1}{8} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{8} + \frac {5}{7} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{7} + \frac {5}{2} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{6} + 6 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{5} + \frac {21}{2} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{4} + 14 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{3} + 15 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{2} + 15 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x + 5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} \log \relax (x) - \frac {A a^{10} + 2 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 211, normalized size = 0.98 \begin {gather*} \ln \relax (x)\,\left (10\,B\,a^9\,b+45\,A\,a^8\,b^2\right )-\frac {x\,\left (B\,a^{10}+10\,A\,b\,a^9\right )+\frac {A\,a^{10}}{2}}{x^2}+x^8\,\left (\frac {A\,b^{10}}{8}+\frac {5\,B\,a\,b^9}{4}\right )+\frac {B\,b^{10}\,x^9}{9}+15\,a^6\,b^3\,x^2\,\left (7\,A\,b+4\,B\,a\right )+14\,a^5\,b^4\,x^3\,\left (6\,A\,b+5\,B\,a\right )+\frac {21\,a^4\,b^5\,x^4\,\left (5\,A\,b+6\,B\,a\right )}{2}+6\,a^3\,b^6\,x^5\,\left (4\,A\,b+7\,B\,a\right )+\frac {5\,a^2\,b^7\,x^6\,\left (3\,A\,b+8\,B\,a\right )}{2}+15\,a^7\,b^2\,x\,\left (8\,A\,b+3\,B\,a\right )+\frac {5\,a\,b^8\,x^7\,\left (2\,A\,b+9\,B\,a\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.10, size = 248, normalized size = 1.15 \begin {gather*} \frac {B b^{10} x^{9}}{9} + 5 a^{8} b \left (9 A b + 2 B a\right ) \log {\relax (x )} + x^{8} \left (\frac {A b^{10}}{8} + \frac {5 B a b^{9}}{4}\right ) + x^{7} \left (\frac {10 A a b^{9}}{7} + \frac {45 B a^{2} b^{8}}{7}\right ) + x^{6} \left (\frac {15 A a^{2} b^{8}}{2} + 20 B a^{3} b^{7}\right ) + x^{5} \left (24 A a^{3} b^{7} + 42 B a^{4} b^{6}\right ) + x^{4} \left (\frac {105 A a^{4} b^{6}}{2} + 63 B a^{5} b^{5}\right ) + x^{3} \left (84 A a^{5} b^{5} + 70 B a^{6} b^{4}\right ) + x^{2} \left (105 A a^{6} b^{4} + 60 B a^{7} b^{3}\right ) + x \left (120 A a^{7} b^{3} + 45 B a^{8} b^{2}\right ) + \frac {- A a^{10} + x \left (- 20 A a^{9} b - 2 B a^{10}\right )}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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