Optimal. Leaf size=177 \[ \frac {a^{11}}{9 b^{12} (a+b x)^9}-\frac {11 a^{10}}{8 b^{12} (a+b x)^8}+\frac {55 a^9}{7 b^{12} (a+b x)^7}-\frac {55 a^8}{2 b^{12} (a+b x)^6}+\frac {66 a^7}{b^{12} (a+b x)^5}-\frac {231 a^6}{2 b^{12} (a+b x)^4}+\frac {154 a^5}{b^{12} (a+b x)^3}-\frac {165 a^4}{b^{12} (a+b x)^2}+\frac {165 a^3}{b^{12} (a+b x)}+\frac {55 a^2 \log (a+b x)}{b^{12}}-\frac {10 a x}{b^{11}}+\frac {x^2}{2 b^{10}} \]
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Rubi [A] time = 0.14, antiderivative size = 177, 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 = {43} \begin {gather*} \frac {a^{11}}{9 b^{12} (a+b x)^9}-\frac {11 a^{10}}{8 b^{12} (a+b x)^8}+\frac {55 a^9}{7 b^{12} (a+b x)^7}-\frac {55 a^8}{2 b^{12} (a+b x)^6}+\frac {66 a^7}{b^{12} (a+b x)^5}-\frac {231 a^6}{2 b^{12} (a+b x)^4}+\frac {154 a^5}{b^{12} (a+b x)^3}-\frac {165 a^4}{b^{12} (a+b x)^2}+\frac {165 a^3}{b^{12} (a+b x)}+\frac {55 a^2 \log (a+b x)}{b^{12}}-\frac {10 a x}{b^{11}}+\frac {x^2}{2 b^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^{11}}{(a+b x)^{10}} \, dx &=\int \left (-\frac {10 a}{b^{11}}+\frac {x}{b^{10}}-\frac {a^{11}}{b^{11} (a+b x)^{10}}+\frac {11 a^{10}}{b^{11} (a+b x)^9}-\frac {55 a^9}{b^{11} (a+b x)^8}+\frac {165 a^8}{b^{11} (a+b x)^7}-\frac {330 a^7}{b^{11} (a+b x)^6}+\frac {462 a^6}{b^{11} (a+b x)^5}-\frac {462 a^5}{b^{11} (a+b x)^4}+\frac {330 a^4}{b^{11} (a+b x)^3}-\frac {165 a^3}{b^{11} (a+b x)^2}+\frac {55 a^2}{b^{11} (a+b x)}\right ) \, dx\\ &=-\frac {10 a x}{b^{11}}+\frac {x^2}{2 b^{10}}+\frac {a^{11}}{9 b^{12} (a+b x)^9}-\frac {11 a^{10}}{8 b^{12} (a+b x)^8}+\frac {55 a^9}{7 b^{12} (a+b x)^7}-\frac {55 a^8}{2 b^{12} (a+b x)^6}+\frac {66 a^7}{b^{12} (a+b x)^5}-\frac {231 a^6}{2 b^{12} (a+b x)^4}+\frac {154 a^5}{b^{12} (a+b x)^3}-\frac {165 a^4}{b^{12} (a+b x)^2}+\frac {165 a^3}{b^{12} (a+b x)}+\frac {55 a^2 \log (a+b x)}{b^{12}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 150, normalized size = 0.85 \begin {gather*} \frac {42131 a^{11}+351459 a^{10} b x+1281096 a^9 b^2 x^2+2656584 a^8 b^3 x^3+3402756 a^7 b^4 x^4+2704212 a^6 b^5 x^5+1220688 a^5 b^6 x^6+190512 a^4 b^7 x^7-77112 a^3 b^8 x^8-36288 a^2 b^9 x^9+27720 a^2 (a+b x)^9 \log (a+b x)-2772 a b^{10} x^{10}+252 b^{11} x^{11}}{504 b^{12} (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 {x^{11}}{(a+b x)^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.68, size = 327, normalized size = 1.85 \begin {gather*} \frac {252 \, b^{11} x^{11} - 2772 \, a b^{10} x^{10} - 36288 \, a^{2} b^{9} x^{9} - 77112 \, a^{3} b^{8} x^{8} + 190512 \, a^{4} b^{7} x^{7} + 1220688 \, a^{5} b^{6} x^{6} + 2704212 \, a^{6} b^{5} x^{5} + 3402756 \, a^{7} b^{4} x^{4} + 2656584 \, a^{8} b^{3} x^{3} + 1281096 \, a^{9} b^{2} x^{2} + 351459 \, a^{10} b x + 42131 \, a^{11} + 27720 \, {\left (a^{2} b^{9} x^{9} + 9 \, a^{3} b^{8} x^{8} + 36 \, a^{4} b^{7} x^{7} + 84 \, a^{5} b^{6} x^{6} + 126 \, a^{6} b^{5} x^{5} + 126 \, a^{7} b^{4} x^{4} + 84 \, a^{8} b^{3} x^{3} + 36 \, a^{9} b^{2} x^{2} + 9 \, a^{10} b x + a^{11}\right )} \log \left (b x + a\right )}{504 \, {\left (b^{21} x^{9} + 9 \, a b^{20} x^{8} + 36 \, a^{2} b^{19} x^{7} + 84 \, a^{3} b^{18} x^{6} + 126 \, a^{4} b^{17} x^{5} + 126 \, a^{5} b^{16} x^{4} + 84 \, a^{6} b^{15} x^{3} + 36 \, a^{7} b^{14} x^{2} + 9 \, a^{8} b^{13} x + a^{9} b^{12}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.70, size = 138, normalized size = 0.78 \begin {gather*} \frac {55 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{12}} + \frac {b^{10} x^{2} - 20 \, a b^{9} x}{2 \, b^{20}} + \frac {83160 \, a^{3} b^{8} x^{8} + 582120 \, a^{4} b^{7} x^{7} + 1823976 \, a^{5} b^{6} x^{6} + 3318084 \, a^{6} b^{5} x^{5} + 3817044 \, a^{7} b^{4} x^{4} + 2835756 \, a^{8} b^{3} x^{3} + 1326204 \, a^{9} b^{2} x^{2} + 356499 \, a^{10} b x + 42131 \, a^{11}}{504 \, {\left (b x + a\right )}^{9} b^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 166, normalized size = 0.94 \begin {gather*} \frac {a^{11}}{9 \left (b x +a \right )^{9} b^{12}}-\frac {11 a^{10}}{8 \left (b x +a \right )^{8} b^{12}}+\frac {55 a^{9}}{7 \left (b x +a \right )^{7} b^{12}}-\frac {55 a^{8}}{2 \left (b x +a \right )^{6} b^{12}}+\frac {66 a^{7}}{\left (b x +a \right )^{5} b^{12}}-\frac {231 a^{6}}{2 \left (b x +a \right )^{4} b^{12}}+\frac {154 a^{5}}{\left (b x +a \right )^{3} b^{12}}-\frac {165 a^{4}}{\left (b x +a \right )^{2} b^{12}}+\frac {x^{2}}{2 b^{10}}+\frac {165 a^{3}}{\left (b x +a \right ) b^{12}}+\frac {55 a^{2} \ln \left (b x +a \right )}{b^{12}}-\frac {10 a x}{b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.65, size = 223, normalized size = 1.26 \begin {gather*} \frac {83160 \, a^{3} b^{8} x^{8} + 582120 \, a^{4} b^{7} x^{7} + 1823976 \, a^{5} b^{6} x^{6} + 3318084 \, a^{6} b^{5} x^{5} + 3817044 \, a^{7} b^{4} x^{4} + 2835756 \, a^{8} b^{3} x^{3} + 1326204 \, a^{9} b^{2} x^{2} + 356499 \, a^{10} b x + 42131 \, a^{11}}{504 \, {\left (b^{21} x^{9} + 9 \, a b^{20} x^{8} + 36 \, a^{2} b^{19} x^{7} + 84 \, a^{3} b^{18} x^{6} + 126 \, a^{4} b^{17} x^{5} + 126 \, a^{5} b^{16} x^{4} + 84 \, a^{6} b^{15} x^{3} + 36 \, a^{7} b^{14} x^{2} + 9 \, a^{8} b^{13} x + a^{9} b^{12}\right )}} + \frac {55 \, a^{2} \log \left (b x + a\right )}{b^{12}} + \frac {b x^{2} - 20 \, a x}{2 \, b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 138, normalized size = 0.78 \begin {gather*} \frac {\frac {{\left (a+b\,x\right )}^2}{2}+\frac {165\,a^3}{a+b\,x}-\frac {165\,a^4}{{\left (a+b\,x\right )}^2}+\frac {154\,a^5}{{\left (a+b\,x\right )}^3}-\frac {231\,a^6}{2\,{\left (a+b\,x\right )}^4}+\frac {66\,a^7}{{\left (a+b\,x\right )}^5}-\frac {55\,a^8}{2\,{\left (a+b\,x\right )}^6}+\frac {55\,a^9}{7\,{\left (a+b\,x\right )}^7}-\frac {11\,a^{10}}{8\,{\left (a+b\,x\right )}^8}+\frac {a^{11}}{9\,{\left (a+b\,x\right )}^9}+55\,a^2\,\ln \left (a+b\,x\right )-11\,a\,b\,x}{b^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 236, normalized size = 1.33 \begin {gather*} \frac {55 a^{2} \log {\left (a + b x \right )}}{b^{12}} - \frac {10 a x}{b^{11}} + \frac {42131 a^{11} + 356499 a^{10} b x + 1326204 a^{9} b^{2} x^{2} + 2835756 a^{8} b^{3} x^{3} + 3817044 a^{7} b^{4} x^{4} + 3318084 a^{6} b^{5} x^{5} + 1823976 a^{5} b^{6} x^{6} + 582120 a^{4} b^{7} x^{7} + 83160 a^{3} b^{8} x^{8}}{504 a^{9} b^{12} + 4536 a^{8} b^{13} x + 18144 a^{7} b^{14} x^{2} + 42336 a^{6} b^{15} x^{3} + 63504 a^{5} b^{16} x^{4} + 63504 a^{4} b^{17} x^{5} + 42336 a^{3} b^{18} x^{6} + 18144 a^{2} b^{19} x^{7} + 4536 a b^{20} x^{8} + 504 b^{21} x^{9}} + \frac {x^{2}}{2 b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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