Integrand size = 11, antiderivative size = 118 \[ \int \frac {x^7}{(a+b x)^7} \, dx=\frac {x}{b^7}+\frac {a^7}{6 b^8 (a+b x)^6}-\frac {7 a^6}{5 b^8 (a+b x)^5}+\frac {21 a^5}{4 b^8 (a+b x)^4}-\frac {35 a^4}{3 b^8 (a+b x)^3}+\frac {35 a^3}{2 b^8 (a+b x)^2}-\frac {21 a^2}{b^8 (a+b x)}-\frac {7 a \log (a+b x)}{b^8} \]
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Time = 0.05 (sec) , antiderivative size = 118, 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} \[ \int \frac {x^7}{(a+b x)^7} \, dx=\frac {a^7}{6 b^8 (a+b x)^6}-\frac {7 a^6}{5 b^8 (a+b x)^5}+\frac {21 a^5}{4 b^8 (a+b x)^4}-\frac {35 a^4}{3 b^8 (a+b x)^3}+\frac {35 a^3}{2 b^8 (a+b x)^2}-\frac {21 a^2}{b^8 (a+b x)}-\frac {7 a \log (a+b x)}{b^8}+\frac {x}{b^7} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{b^7}-\frac {a^7}{b^7 (a+b x)^7}+\frac {7 a^6}{b^7 (a+b x)^6}-\frac {21 a^5}{b^7 (a+b x)^5}+\frac {35 a^4}{b^7 (a+b x)^4}-\frac {35 a^3}{b^7 (a+b x)^3}+\frac {21 a^2}{b^7 (a+b x)^2}-\frac {7 a}{b^7 (a+b x)}\right ) \, dx \\ & = \frac {x}{b^7}+\frac {a^7}{6 b^8 (a+b x)^6}-\frac {7 a^6}{5 b^8 (a+b x)^5}+\frac {21 a^5}{4 b^8 (a+b x)^4}-\frac {35 a^4}{3 b^8 (a+b x)^3}+\frac {35 a^3}{2 b^8 (a+b x)^2}-\frac {21 a^2}{b^8 (a+b x)}-\frac {7 a \log (a+b x)}{b^8} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.88 \[ \int \frac {x^7}{(a+b x)^7} \, dx=-\frac {669 a^7+3594 a^6 b x+7725 a^5 b^2 x^2+8200 a^4 b^3 x^3+4050 a^3 b^4 x^4+360 a^2 b^5 x^5-360 a b^6 x^6-60 b^7 x^7+420 a (a+b x)^6 \log (a+b x)}{60 b^8 (a+b x)^6} \]
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Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {x}{b^{7}}+\frac {-21 a^{2} b^{4} x^{5}-\frac {175 a^{3} b^{3} x^{4}}{2}-\frac {455 a^{4} b^{2} x^{3}}{3}-\frac {539 a^{5} b \,x^{2}}{4}-\frac {609 a^{6} x}{10}-\frac {223 a^{7}}{20 b}}{b^{7} \left (b x +a \right )^{6}}-\frac {7 a \ln \left (b x +a \right )}{b^{8}}\) | \(87\) |
norman | \(\frac {\frac {x^{7}}{b}-\frac {343 a^{7}}{20 b^{8}}-\frac {42 a^{2} x^{5}}{b^{3}}-\frac {315 a^{3} x^{4}}{2 b^{4}}-\frac {770 a^{4} x^{3}}{3 b^{5}}-\frac {875 a^{5} x^{2}}{4 b^{6}}-\frac {959 a^{6} x}{10 b^{7}}}{\left (b x +a \right )^{6}}-\frac {7 a \ln \left (b x +a \right )}{b^{8}}\) | \(91\) |
default | \(\frac {x}{b^{7}}+\frac {a^{7}}{6 b^{8} \left (b x +a \right )^{6}}-\frac {7 a^{6}}{5 b^{8} \left (b x +a \right )^{5}}+\frac {21 a^{5}}{4 b^{8} \left (b x +a \right )^{4}}-\frac {35 a^{4}}{3 b^{8} \left (b x +a \right )^{3}}+\frac {35 a^{3}}{2 b^{8} \left (b x +a \right )^{2}}-\frac {21 a^{2}}{b^{8} \left (b x +a \right )}-\frac {7 a \ln \left (b x +a \right )}{b^{8}}\) | \(109\) |
parallelrisch | \(-\frac {420 \ln \left (b x +a \right ) x^{6} a \,b^{6}-60 b^{7} x^{7}+2520 \ln \left (b x +a \right ) x^{5} a^{2} b^{5}+6300 \ln \left (b x +a \right ) x^{4} a^{3} b^{4}+2520 a^{2} b^{5} x^{5}+8400 \ln \left (b x +a \right ) x^{3} a^{4} b^{3}+9450 a^{3} b^{4} x^{4}+6300 \ln \left (b x +a \right ) x^{2} a^{5} b^{2}+15400 a^{4} b^{3} x^{3}+2520 \ln \left (b x +a \right ) x \,a^{6} b +13125 a^{5} b^{2} x^{2}+420 \ln \left (b x +a \right ) a^{7}+5754 a^{6} b x +1029 a^{7}}{60 b^{8} \left (b x +a \right )^{6}}\) | \(185\) |
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Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.82 \[ \int \frac {x^7}{(a+b x)^7} \, dx=\frac {60 \, b^{7} x^{7} + 360 \, a b^{6} x^{6} - 360 \, a^{2} b^{5} x^{5} - 4050 \, a^{3} b^{4} x^{4} - 8200 \, a^{4} b^{3} x^{3} - 7725 \, a^{5} b^{2} x^{2} - 3594 \, a^{6} b x - 669 \, a^{7} - 420 \, {\left (a b^{6} x^{6} + 6 \, a^{2} b^{5} x^{5} + 15 \, a^{3} b^{4} x^{4} + 20 \, a^{4} b^{3} x^{3} + 15 \, a^{5} b^{2} x^{2} + 6 \, a^{6} b x + a^{7}\right )} \log \left (b x + a\right )}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} \]
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Time = 0.36 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.30 \[ \int \frac {x^7}{(a+b x)^7} \, dx=- \frac {7 a \log {\left (a + b x \right )}}{b^{8}} + \frac {- 669 a^{7} - 3654 a^{6} b x - 8085 a^{5} b^{2} x^{2} - 9100 a^{4} b^{3} x^{3} - 5250 a^{3} b^{4} x^{4} - 1260 a^{2} b^{5} x^{5}}{60 a^{6} b^{8} + 360 a^{5} b^{9} x + 900 a^{4} b^{10} x^{2} + 1200 a^{3} b^{11} x^{3} + 900 a^{2} b^{12} x^{4} + 360 a b^{13} x^{5} + 60 b^{14} x^{6}} + \frac {x}{b^{7}} \]
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Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.23 \[ \int \frac {x^7}{(a+b x)^7} \, dx=-\frac {1260 \, a^{2} b^{5} x^{5} + 5250 \, a^{3} b^{4} x^{4} + 9100 \, a^{4} b^{3} x^{3} + 8085 \, a^{5} b^{2} x^{2} + 3654 \, a^{6} b x + 669 \, a^{7}}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} + \frac {x}{b^{7}} - \frac {7 \, a \log \left (b x + a\right )}{b^{8}} \]
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75 \[ \int \frac {x^7}{(a+b x)^7} \, dx=\frac {x}{b^{7}} - \frac {7 \, a \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {1260 \, a^{2} b^{5} x^{5} + 5250 \, a^{3} b^{4} x^{4} + 9100 \, a^{4} b^{3} x^{3} + 8085 \, a^{5} b^{2} x^{2} + 3654 \, a^{6} b x + 669 \, a^{7}}{60 \, {\left (b x + a\right )}^{6} b^{8}} \]
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Time = 0.45 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int \frac {x^7}{(a+b x)^7} \, dx=-\frac {7\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {21\,a^2}{a+b\,x}-\frac {35\,a^3}{2\,{\left (a+b\,x\right )}^2}+\frac {35\,a^4}{3\,{\left (a+b\,x\right )}^3}-\frac {21\,a^5}{4\,{\left (a+b\,x\right )}^4}+\frac {7\,a^6}{5\,{\left (a+b\,x\right )}^5}-\frac {a^7}{6\,{\left (a+b\,x\right )}^6}}{b^8} \]
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