Optimal. Leaf size=159 \[ -\frac {a^{10}}{9 b^{11} (a+b x)^9}+\frac {5 a^9}{4 b^{11} (a+b x)^8}-\frac {45 a^8}{7 b^{11} (a+b x)^7}+\frac {20 a^7}{b^{11} (a+b x)^6}-\frac {42 a^6}{b^{11} (a+b x)^5}+\frac {63 a^5}{b^{11} (a+b x)^4}-\frac {70 a^4}{b^{11} (a+b x)^3}+\frac {60 a^3}{b^{11} (a+b x)^2}-\frac {45 a^2}{b^{11} (a+b x)}-\frac {10 a \log (a+b x)}{b^{11}}+\frac {x}{b^{10}} \]
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Rubi [A] time = 0.12, antiderivative size = 159, 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^{10}}{9 b^{11} (a+b x)^9}+\frac {5 a^9}{4 b^{11} (a+b x)^8}-\frac {45 a^8}{7 b^{11} (a+b x)^7}+\frac {20 a^7}{b^{11} (a+b x)^6}-\frac {42 a^6}{b^{11} (a+b x)^5}+\frac {63 a^5}{b^{11} (a+b x)^4}-\frac {70 a^4}{b^{11} (a+b x)^3}+\frac {60 a^3}{b^{11} (a+b x)^2}-\frac {45 a^2}{b^{11} (a+b x)}-\frac {10 a \log (a+b x)}{b^{11}}+\frac {x}{b^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^{10}}{(a+b x)^{10}} \, dx &=\int \left (\frac {1}{b^{10}}+\frac {a^{10}}{b^{10} (a+b x)^{10}}-\frac {10 a^9}{b^{10} (a+b x)^9}+\frac {45 a^8}{b^{10} (a+b x)^8}-\frac {120 a^7}{b^{10} (a+b x)^7}+\frac {210 a^6}{b^{10} (a+b x)^6}-\frac {252 a^5}{b^{10} (a+b x)^5}+\frac {210 a^4}{b^{10} (a+b x)^4}-\frac {120 a^3}{b^{10} (a+b x)^3}+\frac {45 a^2}{b^{10} (a+b x)^2}-\frac {10 a}{b^{10} (a+b x)}\right ) \, dx\\ &=\frac {x}{b^{10}}-\frac {a^{10}}{9 b^{11} (a+b x)^9}+\frac {5 a^9}{4 b^{11} (a+b x)^8}-\frac {45 a^8}{7 b^{11} (a+b x)^7}+\frac {20 a^7}{b^{11} (a+b x)^6}-\frac {42 a^6}{b^{11} (a+b x)^5}+\frac {63 a^5}{b^{11} (a+b x)^4}-\frac {70 a^4}{b^{11} (a+b x)^3}+\frac {60 a^3}{b^{11} (a+b x)^2}-\frac {45 a^2}{b^{11} (a+b x)}-\frac {10 a \log (a+b x)}{b^{11}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 137, normalized size = 0.86 \begin {gather*} -\frac {4861 a^{10}+41229 a^9 b x+153576 a^8 b^2 x^2+328104 a^7 b^3 x^3+439236 a^6 b^4 x^4+375732 a^5 b^5 x^5+197568 a^4 b^6 x^6+54432 a^3 b^7 x^7+2268 a^2 b^8 x^8-2268 a b^9 x^9+2520 a (a+b x)^9 \log (a+b x)-252 b^{10} x^{10}}{252 b^{11} (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^{10}}{(a+b x)^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.42, size = 314, normalized size = 1.97 \begin {gather*} \frac {252 \, b^{10} x^{10} + 2268 \, a b^{9} x^{9} - 2268 \, a^{2} b^{8} x^{8} - 54432 \, a^{3} b^{7} x^{7} - 197568 \, a^{4} b^{6} x^{6} - 375732 \, a^{5} b^{5} x^{5} - 439236 \, a^{6} b^{4} x^{4} - 328104 \, a^{7} b^{3} x^{3} - 153576 \, a^{8} b^{2} x^{2} - 41229 \, a^{9} b x - 4861 \, a^{10} - 2520 \, {\left (a b^{9} x^{9} + 9 \, a^{2} b^{8} x^{8} + 36 \, a^{3} b^{7} x^{7} + 84 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 126 \, a^{6} b^{4} x^{4} + 84 \, a^{7} b^{3} x^{3} + 36 \, a^{8} b^{2} x^{2} + 9 \, a^{9} b x + a^{10}\right )} \log \left (b x + a\right )}{252 \, {\left (b^{20} x^{9} + 9 \, a b^{19} x^{8} + 36 \, a^{2} b^{18} x^{7} + 84 \, a^{3} b^{17} x^{6} + 126 \, a^{4} b^{16} x^{5} + 126 \, a^{5} b^{15} x^{4} + 84 \, a^{6} b^{14} x^{3} + 36 \, a^{7} b^{13} x^{2} + 9 \, a^{8} b^{12} x + a^{9} b^{11}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 121, normalized size = 0.76 \begin {gather*} \frac {x}{b^{10}} - \frac {10 \, a \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {11340 \, a^{2} b^{8} x^{8} + 75600 \, a^{3} b^{7} x^{7} + 229320 \, a^{4} b^{6} x^{6} + 407484 \, a^{5} b^{5} x^{5} + 460404 \, a^{6} b^{4} x^{4} + 337176 \, a^{7} b^{3} x^{3} + 155844 \, a^{8} b^{2} x^{2} + 41481 \, a^{9} b x + 4861 \, a^{10}}{252 \, {\left (b x + a\right )}^{9} b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 154, normalized size = 0.97 \begin {gather*} -\frac {a^{10}}{9 \left (b x +a \right )^{9} b^{11}}+\frac {5 a^{9}}{4 \left (b x +a \right )^{8} b^{11}}-\frac {45 a^{8}}{7 \left (b x +a \right )^{7} b^{11}}+\frac {20 a^{7}}{\left (b x +a \right )^{6} b^{11}}-\frac {42 a^{6}}{\left (b x +a \right )^{5} b^{11}}+\frac {63 a^{5}}{\left (b x +a \right )^{4} b^{11}}-\frac {70 a^{4}}{\left (b x +a \right )^{3} b^{11}}+\frac {60 a^{3}}{\left (b x +a \right )^{2} b^{11}}-\frac {45 a^{2}}{\left (b x +a \right ) b^{11}}-\frac {10 a \ln \left (b x +a \right )}{b^{11}}+\frac {x}{b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.66, size = 211, normalized size = 1.33 \begin {gather*} -\frac {11340 \, a^{2} b^{8} x^{8} + 75600 \, a^{3} b^{7} x^{7} + 229320 \, a^{4} b^{6} x^{6} + 407484 \, a^{5} b^{5} x^{5} + 460404 \, a^{6} b^{4} x^{4} + 337176 \, a^{7} b^{3} x^{3} + 155844 \, a^{8} b^{2} x^{2} + 41481 \, a^{9} b x + 4861 \, a^{10}}{252 \, {\left (b^{20} x^{9} + 9 \, a b^{19} x^{8} + 36 \, a^{2} b^{18} x^{7} + 84 \, a^{3} b^{17} x^{6} + 126 \, a^{4} b^{16} x^{5} + 126 \, a^{5} b^{15} x^{4} + 84 \, a^{6} b^{14} x^{3} + 36 \, a^{7} b^{13} x^{2} + 9 \, a^{8} b^{12} x + a^{9} b^{11}\right )}} + \frac {x}{b^{10}} - \frac {10 \, a \log \left (b x + a\right )}{b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 127, normalized size = 0.80 \begin {gather*} -\frac {10\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {45\,a^2}{a+b\,x}-\frac {60\,a^3}{{\left (a+b\,x\right )}^2}+\frac {70\,a^4}{{\left (a+b\,x\right )}^3}-\frac {63\,a^5}{{\left (a+b\,x\right )}^4}+\frac {42\,a^6}{{\left (a+b\,x\right )}^5}-\frac {20\,a^7}{{\left (a+b\,x\right )}^6}+\frac {45\,a^8}{7\,{\left (a+b\,x\right )}^7}-\frac {5\,a^9}{4\,{\left (a+b\,x\right )}^8}+\frac {a^{10}}{9\,{\left (a+b\,x\right )}^9}}{b^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.33, size = 224, normalized size = 1.41 \begin {gather*} - \frac {10 a \log {\left (a + b x \right )}}{b^{11}} + \frac {- 4861 a^{10} - 41481 a^{9} b x - 155844 a^{8} b^{2} x^{2} - 337176 a^{7} b^{3} x^{3} - 460404 a^{6} b^{4} x^{4} - 407484 a^{5} b^{5} x^{5} - 229320 a^{4} b^{6} x^{6} - 75600 a^{3} b^{7} x^{7} - 11340 a^{2} b^{8} x^{8}}{252 a^{9} b^{11} + 2268 a^{8} b^{12} x + 9072 a^{7} b^{13} x^{2} + 21168 a^{6} b^{14} x^{3} + 31752 a^{5} b^{15} x^{4} + 31752 a^{4} b^{16} x^{5} + 21168 a^{3} b^{17} x^{6} + 9072 a^{2} b^{18} x^{7} + 2268 a b^{19} x^{8} + 252 b^{20} x^{9}} + \frac {x}{b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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