Integrand size = 11, antiderivative size = 159 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=\frac {x}{b^{10}}-\frac {a^{10}}{9 b^{11} (a+b x)^9}+\frac {5 a^9}{4 b^{11} (a+b x)^8}-\frac {45 a^8}{7 b^{11} (a+b x)^7}+\frac {20 a^7}{b^{11} (a+b x)^6}-\frac {42 a^6}{b^{11} (a+b x)^5}+\frac {63 a^5}{b^{11} (a+b x)^4}-\frac {70 a^4}{b^{11} (a+b x)^3}+\frac {60 a^3}{b^{11} (a+b x)^2}-\frac {45 a^2}{b^{11} (a+b x)}-\frac {10 a \log (a+b x)}{b^{11}} \]
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Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=-\frac {a^{10}}{9 b^{11} (a+b x)^9}+\frac {5 a^9}{4 b^{11} (a+b x)^8}-\frac {45 a^8}{7 b^{11} (a+b x)^7}+\frac {20 a^7}{b^{11} (a+b x)^6}-\frac {42 a^6}{b^{11} (a+b x)^5}+\frac {63 a^5}{b^{11} (a+b x)^4}-\frac {70 a^4}{b^{11} (a+b x)^3}+\frac {60 a^3}{b^{11} (a+b x)^2}-\frac {45 a^2}{b^{11} (a+b x)}-\frac {10 a \log (a+b x)}{b^{11}}+\frac {x}{b^{10}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.86 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=-\frac {4861 a^{10}+41229 a^9 b x+153576 a^8 b^2 x^2+328104 a^7 b^3 x^3+439236 a^6 b^4 x^4+375732 a^5 b^5 x^5+197568 a^4 b^6 x^6+54432 a^3 b^7 x^7+2268 a^2 b^8 x^8-2268 a b^9 x^9-252 b^{10} x^{10}+2520 a (a+b x)^9 \log (a+b x)}{252 b^{11} (a+b x)^9} \]
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Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {x}{b^{10}}+\frac {-45 a^{2} b^{7} x^{8}-300 a^{3} b^{6} x^{7}-910 a^{4} b^{5} x^{6}-1617 a^{5} b^{4} x^{5}-1827 a^{6} b^{3} x^{4}-1338 a^{7} b^{2} x^{3}-\frac {4329 a^{8} b \,x^{2}}{7}-\frac {4609 a^{9} x}{28}-\frac {4861 a^{10}}{252 b}}{b^{10} \left (b x +a \right )^{9}}-\frac {10 a \ln \left (b x +a \right )}{b^{11}}\) | \(120\) |
norman | \(\frac {\frac {x^{10}}{b}-\frac {7129 a^{10}}{252 b^{11}}-\frac {90 a^{2} x^{8}}{b^{3}}-\frac {540 a^{3} x^{7}}{b^{4}}-\frac {1540 a^{4} x^{6}}{b^{5}}-\frac {2625 a^{5} x^{5}}{b^{6}}-\frac {2877 a^{6} x^{4}}{b^{7}}-\frac {2058 a^{7} x^{3}}{b^{8}}-\frac {6534 a^{8} x^{2}}{7 b^{9}}-\frac {6849 a^{9} x}{28 b^{10}}}{\left (b x +a \right )^{9}}-\frac {10 a \ln \left (b x +a \right )}{b^{11}}\) | \(124\) |
default | \(\frac {x}{b^{10}}-\frac {a^{10}}{9 b^{11} \left (b x +a \right )^{9}}+\frac {5 a^{9}}{4 b^{11} \left (b x +a \right )^{8}}-\frac {45 a^{8}}{7 b^{11} \left (b x +a \right )^{7}}+\frac {20 a^{7}}{b^{11} \left (b x +a \right )^{6}}-\frac {42 a^{6}}{b^{11} \left (b x +a \right )^{5}}+\frac {63 a^{5}}{b^{11} \left (b x +a \right )^{4}}-\frac {70 a^{4}}{b^{11} \left (b x +a \right )^{3}}+\frac {60 a^{3}}{b^{11} \left (b x +a \right )^{2}}-\frac {45 a^{2}}{b^{11} \left (b x +a \right )}-\frac {10 a \ln \left (b x +a \right )}{b^{11}}\) | \(154\) |
parallelrisch | \(-\frac {7129 a^{10}-252 b^{10} x^{10}+2520 \ln \left (b x +a \right ) a^{10}+2520 \ln \left (b x +a \right ) x^{9} a \,b^{9}+22680 \ln \left (b x +a \right ) x^{8} a^{2} b^{8}+90720 \ln \left (b x +a \right ) x^{7} a^{3} b^{7}+211680 \ln \left (b x +a \right ) x^{3} a^{7} b^{3}+90720 \ln \left (b x +a \right ) x^{2} a^{8} b^{2}+22680 \ln \left (b x +a \right ) x \,a^{9} b +211680 \ln \left (b x +a \right ) x^{6} a^{4} b^{6}+317520 \ln \left (b x +a \right ) x^{5} a^{5} b^{5}+317520 \ln \left (b x +a \right ) x^{4} a^{6} b^{4}+61641 a^{9} b x +235224 a^{8} b^{2} x^{2}+518616 a^{7} b^{3} x^{3}+725004 a^{6} b^{4} x^{4}+661500 a^{5} b^{5} x^{5}+388080 a^{4} b^{6} x^{6}+136080 a^{3} b^{7} x^{7}+22680 a^{2} b^{8} x^{8}}{252 b^{11} \left (b x +a \right )^{9}}\) | \(269\) |
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Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (153) = 306\).
Time = 0.22 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.97 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=\frac {252 \, b^{10} x^{10} + 2268 \, a b^{9} x^{9} - 2268 \, a^{2} b^{8} x^{8} - 54432 \, a^{3} b^{7} x^{7} - 197568 \, a^{4} b^{6} x^{6} - 375732 \, a^{5} b^{5} x^{5} - 439236 \, a^{6} b^{4} x^{4} - 328104 \, a^{7} b^{3} x^{3} - 153576 \, a^{8} b^{2} x^{2} - 41229 \, a^{9} b x - 4861 \, a^{10} - 2520 \, {\left (a b^{9} x^{9} + 9 \, a^{2} b^{8} x^{8} + 36 \, a^{3} b^{7} x^{7} + 84 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 126 \, a^{6} b^{4} x^{4} + 84 \, a^{7} b^{3} x^{3} + 36 \, a^{8} b^{2} x^{2} + 9 \, a^{9} b x + a^{10}\right )} \log \left (b x + a\right )}{252 \, {\left (b^{20} x^{9} + 9 \, a b^{19} x^{8} + 36 \, a^{2} b^{18} x^{7} + 84 \, a^{3} b^{17} x^{6} + 126 \, a^{4} b^{16} x^{5} + 126 \, a^{5} b^{15} x^{4} + 84 \, a^{6} b^{14} x^{3} + 36 \, a^{7} b^{13} x^{2} + 9 \, a^{8} b^{12} x + a^{9} b^{11}\right )}} \]
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Time = 0.90 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.41 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=- \frac {10 a \log {\left (a + b x \right )}}{b^{11}} + \frac {- 4861 a^{10} - 41481 a^{9} b x - 155844 a^{8} b^{2} x^{2} - 337176 a^{7} b^{3} x^{3} - 460404 a^{6} b^{4} x^{4} - 407484 a^{5} b^{5} x^{5} - 229320 a^{4} b^{6} x^{6} - 75600 a^{3} b^{7} x^{7} - 11340 a^{2} b^{8} x^{8}}{252 a^{9} b^{11} + 2268 a^{8} b^{12} x + 9072 a^{7} b^{13} x^{2} + 21168 a^{6} b^{14} x^{3} + 31752 a^{5} b^{15} x^{4} + 31752 a^{4} b^{16} x^{5} + 21168 a^{3} b^{17} x^{6} + 9072 a^{2} b^{18} x^{7} + 2268 a b^{19} x^{8} + 252 b^{20} x^{9}} + \frac {x}{b^{10}} \]
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Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.33 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=-\frac {11340 \, a^{2} b^{8} x^{8} + 75600 \, a^{3} b^{7} x^{7} + 229320 \, a^{4} b^{6} x^{6} + 407484 \, a^{5} b^{5} x^{5} + 460404 \, a^{6} b^{4} x^{4} + 337176 \, a^{7} b^{3} x^{3} + 155844 \, a^{8} b^{2} x^{2} + 41481 \, a^{9} b x + 4861 \, a^{10}}{252 \, {\left (b^{20} x^{9} + 9 \, a b^{19} x^{8} + 36 \, a^{2} b^{18} x^{7} + 84 \, a^{3} b^{17} x^{6} + 126 \, a^{4} b^{16} x^{5} + 126 \, a^{5} b^{15} x^{4} + 84 \, a^{6} b^{14} x^{3} + 36 \, a^{7} b^{13} x^{2} + 9 \, a^{8} b^{12} x + a^{9} b^{11}\right )}} + \frac {x}{b^{10}} - \frac {10 \, a \log \left (b x + a\right )}{b^{11}} \]
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Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.76 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=\frac {x}{b^{10}} - \frac {10 \, a \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {11340 \, a^{2} b^{8} x^{8} + 75600 \, a^{3} b^{7} x^{7} + 229320 \, a^{4} b^{6} x^{6} + 407484 \, a^{5} b^{5} x^{5} + 460404 \, a^{6} b^{4} x^{4} + 337176 \, a^{7} b^{3} x^{3} + 155844 \, a^{8} b^{2} x^{2} + 41481 \, a^{9} b x + 4861 \, a^{10}}{252 \, {\left (b x + a\right )}^{9} b^{11}} \]
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Time = 1.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.80 \[ \int \frac {x^{10}}{(a+b x)^{10}} \, dx=-\frac {10\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {45\,a^2}{a+b\,x}-\frac {60\,a^3}{{\left (a+b\,x\right )}^2}+\frac {70\,a^4}{{\left (a+b\,x\right )}^3}-\frac {63\,a^5}{{\left (a+b\,x\right )}^4}+\frac {42\,a^6}{{\left (a+b\,x\right )}^5}-\frac {20\,a^7}{{\left (a+b\,x\right )}^6}+\frac {45\,a^8}{7\,{\left (a+b\,x\right )}^7}-\frac {5\,a^9}{4\,{\left (a+b\,x\right )}^8}+\frac {a^{10}}{9\,{\left (a+b\,x\right )}^9}}{b^{11}} \]
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