Optimal. Leaf size=141 \[ -\frac {\log (a+b x)}{a^{10}}+\frac {\log (x)}{a^{10}}+\frac {1}{a^9 (a+b x)}+\frac {1}{2 a^8 (a+b x)^2}+\frac {1}{3 a^7 (a+b x)^3}+\frac {1}{4 a^6 (a+b x)^4}+\frac {1}{5 a^5 (a+b x)^5}+\frac {1}{6 a^4 (a+b x)^6}+\frac {1}{7 a^3 (a+b x)^7}+\frac {1}{8 a^2 (a+b x)^8}+\frac {1}{9 a (a+b x)^9} \]
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Rubi [A] time = 0.07, antiderivative size = 141, 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} \begin {gather*} \frac {1}{a^9 (a+b x)}+\frac {1}{2 a^8 (a+b x)^2}+\frac {1}{3 a^7 (a+b x)^3}+\frac {1}{4 a^6 (a+b x)^4}+\frac {1}{5 a^5 (a+b x)^5}+\frac {1}{6 a^4 (a+b x)^6}+\frac {1}{7 a^3 (a+b x)^7}+\frac {1}{8 a^2 (a+b x)^8}-\frac {\log (a+b x)}{a^{10}}+\frac {\log (x)}{a^{10}}+\frac {1}{9 a (a+b x)^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)^{10}} \, dx &=\int \left (\frac {1}{a^{10} x}-\frac {b}{a (a+b x)^{10}}-\frac {b}{a^2 (a+b x)^9}-\frac {b}{a^3 (a+b x)^8}-\frac {b}{a^4 (a+b x)^7}-\frac {b}{a^5 (a+b x)^6}-\frac {b}{a^6 (a+b x)^5}-\frac {b}{a^7 (a+b x)^4}-\frac {b}{a^8 (a+b x)^3}-\frac {b}{a^9 (a+b x)^2}-\frac {b}{a^{10} (a+b x)}\right ) \, dx\\ &=\frac {1}{9 a (a+b x)^9}+\frac {1}{8 a^2 (a+b x)^8}+\frac {1}{7 a^3 (a+b x)^7}+\frac {1}{6 a^4 (a+b x)^6}+\frac {1}{5 a^5 (a+b x)^5}+\frac {1}{4 a^6 (a+b x)^4}+\frac {1}{3 a^7 (a+b x)^3}+\frac {1}{2 a^8 (a+b x)^2}+\frac {1}{a^9 (a+b x)}+\frac {\log (x)}{a^{10}}-\frac {\log (a+b x)}{a^{10}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 127, normalized size = 0.90 \begin {gather*} -\frac {\log (a+b x)}{a^{10}}+\frac {\log (x)}{a^{10}}+\frac {280 a^8+315 a^7 (a+b x)+360 a^6 (a+b x)^2+420 a^5 (a+b x)^3+504 a^4 (a+b x)^4+630 a^3 (a+b x)^5+840 a^2 (a+b x)^6+1260 a (a+b x)^7+2520 (a+b x)^8}{2520 a^9 (a+b x)^9} \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 {1}{x (a+b x)^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.08, size = 388, normalized size = 2.75 \begin {gather*} \frac {2520 \, a b^{8} x^{8} + 21420 \, a^{2} b^{7} x^{7} + 80220 \, a^{3} b^{6} x^{6} + 173250 \, a^{4} b^{5} x^{5} + 236754 \, a^{5} b^{4} x^{4} + 210756 \, a^{6} b^{3} x^{3} + 120564 \, a^{7} b^{2} x^{2} + 41481 \, a^{8} b x + 7129 \, a^{9} - 2520 \, {\left (b^{9} x^{9} + 9 \, a b^{8} x^{8} + 36 \, a^{2} b^{7} x^{7} + 84 \, a^{3} b^{6} x^{6} + 126 \, a^{4} b^{5} x^{5} + 126 \, a^{5} b^{4} x^{4} + 84 \, a^{6} b^{3} x^{3} + 36 \, a^{7} b^{2} x^{2} + 9 \, a^{8} b x + a^{9}\right )} \log \left (b x + a\right ) + 2520 \, {\left (b^{9} x^{9} + 9 \, a b^{8} x^{8} + 36 \, a^{2} b^{7} x^{7} + 84 \, a^{3} b^{6} x^{6} + 126 \, a^{4} b^{5} x^{5} + 126 \, a^{5} b^{4} x^{4} + 84 \, a^{6} b^{3} x^{3} + 36 \, a^{7} b^{2} x^{2} + 9 \, a^{8} b x + a^{9}\right )} \log \relax (x)}{2520 \, {\left (a^{10} b^{9} x^{9} + 9 \, a^{11} b^{8} x^{8} + 36 \, a^{12} b^{7} x^{7} + 84 \, a^{13} b^{6} x^{6} + 126 \, a^{14} b^{5} x^{5} + 126 \, a^{15} b^{4} x^{4} + 84 \, a^{16} b^{3} x^{3} + 36 \, a^{17} b^{2} x^{2} + 9 \, a^{18} b x + a^{19}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 120, normalized size = 0.85 \begin {gather*} -\frac {\log \left ({\left | b x + a \right |}\right )}{a^{10}} + \frac {\log \left ({\left | x \right |}\right )}{a^{10}} + \frac {2520 \, a b^{8} x^{8} + 21420 \, a^{2} b^{7} x^{7} + 80220 \, a^{3} b^{6} x^{6} + 173250 \, a^{4} b^{5} x^{5} + 236754 \, a^{5} b^{4} x^{4} + 210756 \, a^{6} b^{3} x^{3} + 120564 \, a^{7} b^{2} x^{2} + 41481 \, a^{8} b x + 7129 \, a^{9}}{2520 \, {\left (b x + a\right )}^{9} a^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 126, normalized size = 0.89 \begin {gather*} \frac {1}{9 \left (b x +a \right )^{9} a}+\frac {1}{8 \left (b x +a \right )^{8} a^{2}}+\frac {1}{7 \left (b x +a \right )^{7} a^{3}}+\frac {1}{6 \left (b x +a \right )^{6} a^{4}}+\frac {1}{5 \left (b x +a \right )^{5} a^{5}}+\frac {1}{4 \left (b x +a \right )^{4} a^{6}}+\frac {1}{3 \left (b x +a \right )^{3} a^{7}}+\frac {1}{2 \left (b x +a \right )^{2} a^{8}}+\frac {1}{\left (b x +a \right ) a^{9}}+\frac {\ln \relax (x )}{a^{10}}-\frac {\ln \left (b x +a \right )}{a^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.75, size = 205, normalized size = 1.45 \begin {gather*} \frac {2520 \, b^{8} x^{8} + 21420 \, a b^{7} x^{7} + 80220 \, a^{2} b^{6} x^{6} + 173250 \, a^{3} b^{5} x^{5} + 236754 \, a^{4} b^{4} x^{4} + 210756 \, a^{5} b^{3} x^{3} + 120564 \, a^{6} b^{2} x^{2} + 41481 \, a^{7} b x + 7129 \, a^{8}}{2520 \, {\left (a^{9} b^{9} x^{9} + 9 \, a^{10} b^{8} x^{8} + 36 \, a^{11} b^{7} x^{7} + 84 \, a^{12} b^{6} x^{6} + 126 \, a^{13} b^{5} x^{5} + 126 \, a^{14} b^{4} x^{4} + 84 \, a^{15} b^{3} x^{3} + 36 \, a^{16} b^{2} x^{2} + 9 \, a^{17} b x + a^{18}\right )}} - \frac {\log \left (b x + a\right )}{a^{10}} + \frac {\log \relax (x)}{a^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.76, size = 145, normalized size = 1.03 \begin {gather*} \frac {1}{9\,a\,{\left (a+b\,x\right )}^9}-\frac {\ln \left (\frac {a+b\,x}{x}\right )-\frac {14\,b^2\,x^2}{{\left (a+b\,x\right )}^2}+\frac {56\,b^3\,x^3}{3\,{\left (a+b\,x\right )}^3}-\frac {35\,b^4\,x^4}{2\,{\left (a+b\,x\right )}^4}+\frac {56\,b^5\,x^5}{5\,{\left (a+b\,x\right )}^5}-\frac {14\,b^6\,x^6}{3\,{\left (a+b\,x\right )}^6}+\frac {8\,b^7\,x^7}{7\,{\left (a+b\,x\right )}^7}-\frac {b^8\,x^8}{8\,{\left (a+b\,x\right )}^8}+\frac {8\,b\,x}{a+b\,x}}{a^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.02, size = 212, normalized size = 1.50 \begin {gather*} \frac {7129 a^{8} + 41481 a^{7} b x + 120564 a^{6} b^{2} x^{2} + 210756 a^{5} b^{3} x^{3} + 236754 a^{4} b^{4} x^{4} + 173250 a^{3} b^{5} x^{5} + 80220 a^{2} b^{6} x^{6} + 21420 a b^{7} x^{7} + 2520 b^{8} x^{8}}{2520 a^{18} + 22680 a^{17} b x + 90720 a^{16} b^{2} x^{2} + 211680 a^{15} b^{3} x^{3} + 317520 a^{14} b^{4} x^{4} + 317520 a^{13} b^{5} x^{5} + 211680 a^{12} b^{6} x^{6} + 90720 a^{11} b^{7} x^{7} + 22680 a^{10} b^{8} x^{8} + 2520 a^{9} b^{9} x^{9}} + \frac {\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}}{a^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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