Integrand size = 11, antiderivative size = 128 \[ \int \frac {x^8}{(a+b x)^7} \, dx=-\frac {7 a x}{b^8}+\frac {x^2}{2 b^7}-\frac {a^8}{6 b^9 (a+b x)^6}+\frac {8 a^7}{5 b^9 (a+b x)^5}-\frac {7 a^6}{b^9 (a+b x)^4}+\frac {56 a^5}{3 b^9 (a+b x)^3}-\frac {35 a^4}{b^9 (a+b x)^2}+\frac {56 a^3}{b^9 (a+b x)}+\frac {28 a^2 \log (a+b x)}{b^9} \]
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Time = 0.06 (sec) , antiderivative size = 128, 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} \[ \int \frac {x^8}{(a+b x)^7} \, dx=-\frac {a^8}{6 b^9 (a+b x)^6}+\frac {8 a^7}{5 b^9 (a+b x)^5}-\frac {7 a^6}{b^9 (a+b x)^4}+\frac {56 a^5}{3 b^9 (a+b x)^3}-\frac {35 a^4}{b^9 (a+b x)^2}+\frac {56 a^3}{b^9 (a+b x)}+\frac {28 a^2 \log (a+b x)}{b^9}-\frac {7 a x}{b^8}+\frac {x^2}{2 b^7} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {7 a}{b^8}+\frac {x}{b^7}+\frac {a^8}{b^8 (a+b x)^7}-\frac {8 a^7}{b^8 (a+b x)^6}+\frac {28 a^6}{b^8 (a+b x)^5}-\frac {56 a^5}{b^8 (a+b x)^4}+\frac {70 a^4}{b^8 (a+b x)^3}-\frac {56 a^3}{b^8 (a+b x)^2}+\frac {28 a^2}{b^8 (a+b x)}\right ) \, dx \\ & = -\frac {7 a x}{b^8}+\frac {x^2}{2 b^7}-\frac {a^8}{6 b^9 (a+b x)^6}+\frac {8 a^7}{5 b^9 (a+b x)^5}-\frac {7 a^6}{b^9 (a+b x)^4}+\frac {56 a^5}{3 b^9 (a+b x)^3}-\frac {35 a^4}{b^9 (a+b x)^2}+\frac {56 a^3}{b^9 (a+b x)}+\frac {28 a^2 \log (a+b x)}{b^9} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.81 \[ \int \frac {x^8}{(a+b x)^7} \, dx=\frac {-210 a b x+15 b^2 x^2-\frac {5 a^8}{(a+b x)^6}+\frac {48 a^7}{(a+b x)^5}-\frac {210 a^6}{(a+b x)^4}+\frac {560 a^5}{(a+b x)^3}-\frac {1050 a^4}{(a+b x)^2}+\frac {1680 a^3}{a+b x}+840 a^2 \log (a+b x)}{30 b^9} \]
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Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {x^{2}}{2 b^{7}}-\frac {7 a x}{b^{8}}+\frac {56 a^{3} b^{4} x^{5}+245 a^{4} b^{3} x^{4}+\frac {1316 a^{5} b^{2} x^{3}}{3}+399 a^{6} b \,x^{2}+\frac {918 a^{7} x}{5}+\frac {341 a^{8}}{10 b}}{b^{8} \left (b x +a \right )^{6}}+\frac {28 a^{2} \ln \left (b x +a \right )}{b^{9}}\) | \(99\) |
norman | \(\frac {\frac {x^{8}}{2 b}-\frac {4 a \,x^{7}}{b^{2}}+\frac {343 a^{8}}{5 b^{9}}+\frac {168 a^{3} x^{5}}{b^{4}}+\frac {630 a^{4} x^{4}}{b^{5}}+\frac {3080 a^{5} x^{3}}{3 b^{6}}+\frac {875 a^{6} x^{2}}{b^{7}}+\frac {1918 a^{7} x}{5 b^{8}}}{\left (b x +a \right )^{6}}+\frac {28 a^{2} \ln \left (b x +a \right )}{b^{9}}\) | \(103\) |
default | \(-\frac {-\frac {1}{2} b \,x^{2}+7 a x}{b^{8}}+\frac {28 a^{2} \ln \left (b x +a \right )}{b^{9}}-\frac {a^{8}}{6 b^{9} \left (b x +a \right )^{6}}-\frac {7 a^{6}}{b^{9} \left (b x +a \right )^{4}}+\frac {56 a^{5}}{3 b^{9} \left (b x +a \right )^{3}}+\frac {8 a^{7}}{5 b^{9} \left (b x +a \right )^{5}}-\frac {35 a^{4}}{b^{9} \left (b x +a \right )^{2}}+\frac {56 a^{3}}{b^{9} \left (b x +a \right )}\) | \(122\) |
parallelrisch | \(\frac {15 b^{8} x^{8}+840 \ln \left (b x +a \right ) x^{6} a^{2} b^{6}-120 a \,x^{7} b^{7}+5040 \ln \left (b x +a \right ) x^{5} a^{3} b^{5}+12600 \ln \left (b x +a \right ) x^{4} a^{4} b^{4}+5040 a^{3} x^{5} b^{5}+16800 \ln \left (b x +a \right ) x^{3} a^{5} b^{3}+18900 a^{4} x^{4} b^{4}+12600 \ln \left (b x +a \right ) x^{2} a^{6} b^{2}+30800 a^{5} b^{3} x^{3}+5040 \ln \left (b x +a \right ) x \,a^{7} b +26250 a^{6} x^{2} b^{2}+840 \ln \left (b x +a \right ) a^{8}+11508 a^{7} x b +2058 a^{8}}{30 b^{9} \left (b x +a \right )^{6}}\) | \(196\) |
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Time = 0.22 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.78 \[ \int \frac {x^8}{(a+b x)^7} \, dx=\frac {15 \, b^{8} x^{8} - 120 \, a b^{7} x^{7} - 1035 \, a^{2} b^{6} x^{6} - 1170 \, a^{3} b^{5} x^{5} + 3375 \, a^{4} b^{4} x^{4} + 10100 \, a^{5} b^{3} x^{3} + 10725 \, a^{6} b^{2} x^{2} + 5298 \, a^{7} b x + 1023 \, a^{8} + 840 \, {\left (a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{5} + 15 \, a^{4} b^{4} x^{4} + 20 \, a^{5} b^{3} x^{3} + 15 \, a^{6} b^{2} x^{2} + 6 \, a^{7} b x + a^{8}\right )} \log \left (b x + a\right )}{30 \, {\left (b^{15} x^{6} + 6 \, a b^{14} x^{5} + 15 \, a^{2} b^{13} x^{4} + 20 \, a^{3} b^{12} x^{3} + 15 \, a^{4} b^{11} x^{2} + 6 \, a^{5} b^{10} x + a^{6} b^{9}\right )}} \]
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Time = 0.39 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.29 \[ \int \frac {x^8}{(a+b x)^7} \, dx=\frac {28 a^{2} \log {\left (a + b x \right )}}{b^{9}} - \frac {7 a x}{b^{8}} + \frac {1023 a^{8} + 5508 a^{7} b x + 11970 a^{6} b^{2} x^{2} + 13160 a^{5} b^{3} x^{3} + 7350 a^{4} b^{4} x^{4} + 1680 a^{3} b^{5} x^{5}}{30 a^{6} b^{9} + 180 a^{5} b^{10} x + 450 a^{4} b^{11} x^{2} + 600 a^{3} b^{12} x^{3} + 450 a^{2} b^{13} x^{4} + 180 a b^{14} x^{5} + 30 b^{15} x^{6}} + \frac {x^{2}}{2 b^{7}} \]
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Time = 0.20 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.23 \[ \int \frac {x^8}{(a+b x)^7} \, dx=\frac {1680 \, a^{3} b^{5} x^{5} + 7350 \, a^{4} b^{4} x^{4} + 13160 \, a^{5} b^{3} x^{3} + 11970 \, a^{6} b^{2} x^{2} + 5508 \, a^{7} b x + 1023 \, a^{8}}{30 \, {\left (b^{15} x^{6} + 6 \, a b^{14} x^{5} + 15 \, a^{2} b^{13} x^{4} + 20 \, a^{3} b^{12} x^{3} + 15 \, a^{4} b^{11} x^{2} + 6 \, a^{5} b^{10} x + a^{6} b^{9}\right )}} + \frac {28 \, a^{2} \log \left (b x + a\right )}{b^{9}} + \frac {b x^{2} - 14 \, a x}{2 \, b^{8}} \]
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Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.82 \[ \int \frac {x^8}{(a+b x)^7} \, dx=\frac {28 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{9}} + \frac {b^{7} x^{2} - 14 \, a b^{6} x}{2 \, b^{14}} + \frac {1680 \, a^{3} b^{5} x^{5} + 7350 \, a^{4} b^{4} x^{4} + 13160 \, a^{5} b^{3} x^{3} + 11970 \, a^{6} b^{2} x^{2} + 5508 \, a^{7} b x + 1023 \, a^{8}}{30 \, {\left (b x + a\right )}^{6} b^{9}} \]
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Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int \frac {x^8}{(a+b x)^7} \, dx=\frac {\frac {{\left (a+b\,x\right )}^2}{2}+\frac {56\,a^3}{a+b\,x}-\frac {35\,a^4}{{\left (a+b\,x\right )}^2}+\frac {56\,a^5}{3\,{\left (a+b\,x\right )}^3}-\frac {7\,a^6}{{\left (a+b\,x\right )}^4}+\frac {8\,a^7}{5\,{\left (a+b\,x\right )}^5}-\frac {a^8}{6\,{\left (a+b\,x\right )}^6}+28\,a^2\,\ln \left (a+b\,x\right )-8\,a\,b\,x}{b^9} \]
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