Optimal. Leaf size=134 \[ -\frac {b^9 \log (x)}{a^{10}}+\frac {b^9 \log (a+b x)}{a^{10}}-\frac {b^8}{a^9 x}+\frac {b^7}{2 a^8 x^2}-\frac {b^6}{3 a^7 x^3}+\frac {b^5}{4 a^6 x^4}-\frac {b^4}{5 a^5 x^5}+\frac {b^3}{6 a^4 x^6}-\frac {b^2}{7 a^3 x^7}+\frac {b}{8 a^2 x^8}-\frac {1}{9 a x^9} \]
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Rubi [A] time = 0.06, antiderivative size = 134, 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 = {44} \[ \frac {b^7}{2 a^8 x^2}-\frac {b^6}{3 a^7 x^3}+\frac {b^5}{4 a^6 x^4}-\frac {b^4}{5 a^5 x^5}+\frac {b^3}{6 a^4 x^6}-\frac {b^2}{7 a^3 x^7}-\frac {b^8}{a^9 x}-\frac {b^9 \log (x)}{a^{10}}+\frac {b^9 \log (a+b x)}{a^{10}}+\frac {b}{8 a^2 x^8}-\frac {1}{9 a x^9} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x^{10} (a+b x)} \, dx &=\int \left (\frac {1}{a x^{10}}-\frac {b}{a^2 x^9}+\frac {b^2}{a^3 x^8}-\frac {b^3}{a^4 x^7}+\frac {b^4}{a^5 x^6}-\frac {b^5}{a^6 x^5}+\frac {b^6}{a^7 x^4}-\frac {b^7}{a^8 x^3}+\frac {b^8}{a^9 x^2}-\frac {b^9}{a^{10} x}+\frac {b^{10}}{a^{10} (a+b x)}\right ) \, dx\\ &=-\frac {1}{9 a x^9}+\frac {b}{8 a^2 x^8}-\frac {b^2}{7 a^3 x^7}+\frac {b^3}{6 a^4 x^6}-\frac {b^4}{5 a^5 x^5}+\frac {b^5}{4 a^6 x^4}-\frac {b^6}{3 a^7 x^3}+\frac {b^7}{2 a^8 x^2}-\frac {b^8}{a^9 x}-\frac {b^9 \log (x)}{a^{10}}+\frac {b^9 \log (a+b x)}{a^{10}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 134, normalized size = 1.00 \[ -\frac {b^9 \log (x)}{a^{10}}+\frac {b^9 \log (a+b x)}{a^{10}}-\frac {b^8}{a^9 x}+\frac {b^7}{2 a^8 x^2}-\frac {b^6}{3 a^7 x^3}+\frac {b^5}{4 a^6 x^4}-\frac {b^4}{5 a^5 x^5}+\frac {b^3}{6 a^4 x^6}-\frac {b^2}{7 a^3 x^7}+\frac {b}{8 a^2 x^8}-\frac {1}{9 a x^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 120, normalized size = 0.90 \[ \frac {2520 \, b^{9} x^{9} \log \left (b x + a\right ) - 2520 \, b^{9} x^{9} \log \relax (x) - 2520 \, a b^{8} x^{8} + 1260 \, a^{2} b^{7} x^{7} - 840 \, a^{3} b^{6} x^{6} + 630 \, a^{4} b^{5} x^{5} - 504 \, a^{5} b^{4} x^{4} + 420 \, a^{6} b^{3} x^{3} - 360 \, a^{7} b^{2} x^{2} + 315 \, a^{8} b x - 280 \, a^{9}}{2520 \, a^{10} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 122, normalized size = 0.91 \[ \frac {b^{9} \log \left ({\left | b x + a \right |}\right )}{a^{10}} - \frac {b^{9} \log \left ({\left | x \right |}\right )}{a^{10}} - \frac {2520 \, a b^{8} x^{8} - 1260 \, a^{2} b^{7} x^{7} + 840 \, a^{3} b^{6} x^{6} - 630 \, a^{4} b^{5} x^{5} + 504 \, a^{5} b^{4} x^{4} - 420 \, a^{6} b^{3} x^{3} + 360 \, a^{7} b^{2} x^{2} - 315 \, a^{8} b x + 280 \, a^{9}}{2520 \, a^{10} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 119, normalized size = 0.89 \[ -\frac {b^{9} \ln \relax (x )}{a^{10}}+\frac {b^{9} \ln \left (b x +a \right )}{a^{10}}-\frac {b^{8}}{a^{9} x}+\frac {b^{7}}{2 a^{8} x^{2}}-\frac {b^{6}}{3 a^{7} x^{3}}+\frac {b^{5}}{4 a^{6} x^{4}}-\frac {b^{4}}{5 a^{5} x^{5}}+\frac {b^{3}}{6 a^{4} x^{6}}-\frac {b^{2}}{7 a^{3} x^{7}}+\frac {b}{8 a^{2} x^{8}}-\frac {1}{9 a \,x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 117, normalized size = 0.87 \[ \frac {b^{9} \log \left (b x + a\right )}{a^{10}} - \frac {b^{9} \log \relax (x)}{a^{10}} - \frac {2520 \, b^{8} x^{8} - 1260 \, a b^{7} x^{7} + 840 \, a^{2} b^{6} x^{6} - 630 \, a^{3} b^{5} x^{5} + 504 \, a^{4} b^{4} x^{4} - 420 \, a^{5} b^{3} x^{3} + 360 \, a^{6} b^{2} x^{2} - 315 \, a^{7} b x + 280 \, a^{8}}{2520 \, a^{9} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 114, normalized size = 0.85 \[ -\frac {280\,a^9+2520\,a\,b^8\,x^8-5040\,b^9\,x^9\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )+360\,a^7\,b^2\,x^2-420\,a^6\,b^3\,x^3+504\,a^5\,b^4\,x^4-630\,a^4\,b^5\,x^5+840\,a^3\,b^6\,x^6-1260\,a^2\,b^7\,x^7-315\,a^8\,b\,x}{2520\,a^{10}\,x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 116, normalized size = 0.87 \[ \frac {- 280 a^{8} + 315 a^{7} b x - 360 a^{6} b^{2} x^{2} + 420 a^{5} b^{3} x^{3} - 504 a^{4} b^{4} x^{4} + 630 a^{3} b^{5} x^{5} - 840 a^{2} b^{6} x^{6} + 1260 a b^{7} x^{7} - 2520 b^{8} x^{8}}{2520 a^{9} x^{9}} + \frac {b^{9} \left (- \log {\relax (x )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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