Optimal. Leaf size=119 \[ -\frac {a^{10}}{8 x^8}-\frac {10 a^9 b}{7 x^7}-\frac {15 a^8 b^2}{2 x^6}-\frac {24 a^7 b^3}{x^5}-\frac {105 a^6 b^4}{2 x^4}-\frac {84 a^5 b^5}{x^3}-\frac {105 a^4 b^6}{x^2}-\frac {120 a^3 b^7}{x}+10 a b^9 x+\frac {b^{10} x^2}{2}+45 a^2 b^8 \log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} -\frac {a^{10}}{8 x^8}-\frac {10 a^9 b}{7 x^7}-\frac {15 a^8 b^2}{2 x^6}-\frac {24 a^7 b^3}{x^5}-\frac {105 a^6 b^4}{2 x^4}-\frac {84 a^5 b^5}{x^3}-\frac {105 a^4 b^6}{x^2}-\frac {120 a^3 b^7}{x}+45 a^2 b^8 \log (x)+10 a b^9 x+\frac {b^{10} x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^{10}}{x^9} \, dx &=\int \left (10 a b^9+\frac {a^{10}}{x^9}+\frac {10 a^9 b}{x^8}+\frac {45 a^8 b^2}{x^7}+\frac {120 a^7 b^3}{x^6}+\frac {210 a^6 b^4}{x^5}+\frac {252 a^5 b^5}{x^4}+\frac {210 a^4 b^6}{x^3}+\frac {120 a^3 b^7}{x^2}+\frac {45 a^2 b^8}{x}+b^{10} x\right ) \, dx\\ &=-\frac {a^{10}}{8 x^8}-\frac {10 a^9 b}{7 x^7}-\frac {15 a^8 b^2}{2 x^6}-\frac {24 a^7 b^3}{x^5}-\frac {105 a^6 b^4}{2 x^4}-\frac {84 a^5 b^5}{x^3}-\frac {105 a^4 b^6}{x^2}-\frac {120 a^3 b^7}{x}+10 a b^9 x+\frac {b^{10} x^2}{2}+45 a^2 b^8 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 119, normalized size = 1.00 \begin {gather*} -\frac {a^{10}}{8 x^8}-\frac {10 a^9 b}{7 x^7}-\frac {15 a^8 b^2}{2 x^6}-\frac {24 a^7 b^3}{x^5}-\frac {105 a^6 b^4}{2 x^4}-\frac {84 a^5 b^5}{x^3}-\frac {105 a^4 b^6}{x^2}-\frac {120 a^3 b^7}{x}+10 a b^9 x+\frac {b^{10} x^2}{2}+45 a^2 b^8 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 110, normalized size = 0.92
method | result | size |
default | \(-\frac {a^{10}}{8 x^{8}}-\frac {10 a^{9} b}{7 x^{7}}-\frac {15 a^{8} b^{2}}{2 x^{6}}-\frac {24 a^{7} b^{3}}{x^{5}}-\frac {105 a^{6} b^{4}}{2 x^{4}}-\frac {84 a^{5} b^{5}}{x^{3}}-\frac {105 a^{4} b^{6}}{x^{2}}-\frac {120 a^{3} b^{7}}{x}+10 a \,b^{9} x +\frac {b^{10} x^{2}}{2}+45 a^{2} b^{8} \ln \left (x \right )\) | \(110\) |
risch | \(\frac {b^{10} x^{2}}{2}+10 a \,b^{9} x +\frac {-120 a^{3} b^{7} x^{7}-105 a^{4} b^{6} x^{6}-84 a^{5} b^{5} x^{5}-\frac {105}{2} a^{6} b^{4} x^{4}-24 a^{7} b^{3} x^{3}-\frac {15}{2} a^{8} b^{2} x^{2}-\frac {10}{7} a^{9} b x -\frac {1}{8} a^{10}}{x^{8}}+45 a^{2} b^{8} \ln \left (x \right )\) | \(110\) |
norman | \(\frac {-\frac {1}{8} a^{10}+\frac {1}{2} b^{10} x^{10}+10 a \,b^{9} x^{9}-120 a^{3} b^{7} x^{7}-105 a^{4} b^{6} x^{6}-84 a^{5} b^{5} x^{5}-\frac {105}{2} a^{6} b^{4} x^{4}-24 a^{7} b^{3} x^{3}-\frac {15}{2} a^{8} b^{2} x^{2}-\frac {10}{7} a^{9} b x}{x^{8}}+45 a^{2} b^{8} \ln \left (x \right )\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 110, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, b^{10} x^{2} + 10 \, a b^{9} x + 45 \, a^{2} b^{8} \log \left (x\right ) - \frac {6720 \, a^{3} b^{7} x^{7} + 5880 \, a^{4} b^{6} x^{6} + 4704 \, a^{5} b^{5} x^{5} + 2940 \, a^{6} b^{4} x^{4} + 1344 \, a^{7} b^{3} x^{3} + 420 \, a^{8} b^{2} x^{2} + 80 \, a^{9} b x + 7 \, a^{10}}{56 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 114, normalized size = 0.96 \begin {gather*} \frac {28 \, b^{10} x^{10} + 560 \, a b^{9} x^{9} + 2520 \, a^{2} b^{8} x^{8} \log \left (x\right ) - 6720 \, a^{3} b^{7} x^{7} - 5880 \, a^{4} b^{6} x^{6} - 4704 \, a^{5} b^{5} x^{5} - 2940 \, a^{6} b^{4} x^{4} - 1344 \, a^{7} b^{3} x^{3} - 420 \, a^{8} b^{2} x^{2} - 80 \, a^{9} b x - 7 \, a^{10}}{56 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 119, normalized size = 1.00 \begin {gather*} 45 a^{2} b^{8} \log {\left (x \right )} + 10 a b^{9} x + \frac {b^{10} x^{2}}{2} + \frac {- 7 a^{10} - 80 a^{9} b x - 420 a^{8} b^{2} x^{2} - 1344 a^{7} b^{3} x^{3} - 2940 a^{6} b^{4} x^{4} - 4704 a^{5} b^{5} x^{5} - 5880 a^{4} b^{6} x^{6} - 6720 a^{3} b^{7} x^{7}}{56 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.39, size = 111, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, b^{10} x^{2} + 10 \, a b^{9} x + 45 \, a^{2} b^{8} \log \left ({\left | x \right |}\right ) - \frac {6720 \, a^{3} b^{7} x^{7} + 5880 \, a^{4} b^{6} x^{6} + 4704 \, a^{5} b^{5} x^{5} + 2940 \, a^{6} b^{4} x^{4} + 1344 \, a^{7} b^{3} x^{3} + 420 \, a^{8} b^{2} x^{2} + 80 \, a^{9} b x + 7 \, a^{10}}{56 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 110, normalized size = 0.92 \begin {gather*} \frac {b^{10}\,x^2}{2}-\frac {\frac {a^{10}}{8}+\frac {10\,a^9\,b\,x}{7}+\frac {15\,a^8\,b^2\,x^2}{2}+24\,a^7\,b^3\,x^3+\frac {105\,a^6\,b^4\,x^4}{2}+84\,a^5\,b^5\,x^5+105\,a^4\,b^6\,x^6+120\,a^3\,b^7\,x^7}{x^8}+45\,a^2\,b^8\,\ln \left (x\right )+10\,a\,b^9\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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