Integrand size = 11, antiderivative size = 114 \[ \int \frac {x^8}{(a+b x)^4} \, dx=\frac {35 a^4 x}{b^8}-\frac {10 a^3 x^2}{b^7}+\frac {10 a^2 x^3}{3 b^6}-\frac {a x^4}{b^5}+\frac {x^5}{5 b^4}-\frac {a^8}{3 b^9 (a+b x)^3}+\frac {4 a^7}{b^9 (a+b x)^2}-\frac {28 a^6}{b^9 (a+b x)}-\frac {56 a^5 \log (a+b x)}{b^9} \]
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Time = 0.06 (sec) , antiderivative size = 114, 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} \[ \int \frac {x^8}{(a+b x)^4} \, dx=-\frac {a^8}{3 b^9 (a+b x)^3}+\frac {4 a^7}{b^9 (a+b x)^2}-\frac {28 a^6}{b^9 (a+b x)}-\frac {56 a^5 \log (a+b x)}{b^9}+\frac {35 a^4 x}{b^8}-\frac {10 a^3 x^2}{b^7}+\frac {10 a^2 x^3}{3 b^6}-\frac {a x^4}{b^5}+\frac {x^5}{5 b^4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {35 a^4}{b^8}-\frac {20 a^3 x}{b^7}+\frac {10 a^2 x^2}{b^6}-\frac {4 a x^3}{b^5}+\frac {x^4}{b^4}+\frac {a^8}{b^8 (a+b x)^4}-\frac {8 a^7}{b^8 (a+b x)^3}+\frac {28 a^6}{b^8 (a+b x)^2}-\frac {56 a^5}{b^8 (a+b x)}\right ) \, dx \\ & = \frac {35 a^4 x}{b^8}-\frac {10 a^3 x^2}{b^7}+\frac {10 a^2 x^3}{3 b^6}-\frac {a x^4}{b^5}+\frac {x^5}{5 b^4}-\frac {a^8}{3 b^9 (a+b x)^3}+\frac {4 a^7}{b^9 (a+b x)^2}-\frac {28 a^6}{b^9 (a+b x)}-\frac {56 a^5 \log (a+b x)}{b^9} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int \frac {x^8}{(a+b x)^4} \, dx=\frac {525 a^4 b x-150 a^3 b^2 x^2+50 a^2 b^3 x^3-15 a b^4 x^4+3 b^5 x^5-\frac {5 a^8}{(a+b x)^3}+\frac {60 a^7}{(a+b x)^2}-\frac {420 a^6}{a+b x}-840 a^5 \log (a+b x)}{15 b^9} \]
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Time = 0.18 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.87
method | result | size |
risch | \(\frac {x^{5}}{5 b^{4}}-\frac {a \,x^{4}}{b^{5}}+\frac {10 a^{2} x^{3}}{3 b^{6}}-\frac {10 a^{3} x^{2}}{b^{7}}+\frac {35 a^{4} x}{b^{8}}+\frac {-28 a^{6} b \,x^{2}-52 a^{7} x -\frac {73 a^{8}}{3 b}}{b^{8} \left (b x +a \right )^{3}}-\frac {56 a^{5} \ln \left (b x +a \right )}{b^{9}}\) | \(99\) |
norman | \(\frac {\frac {x^{8}}{5 b}-\frac {2 a \,x^{7}}{5 b^{2}}+\frac {14 a^{2} x^{6}}{15 b^{3}}-\frac {14 a^{3} x^{5}}{5 b^{4}}-\frac {308 a^{8}}{3 b^{9}}+\frac {14 a^{4} x^{4}}{b^{5}}-\frac {168 a^{6} x^{2}}{b^{7}}-\frac {252 a^{7} x}{b^{8}}}{\left (b x +a \right )^{3}}-\frac {56 a^{5} \ln \left (b x +a \right )}{b^{9}}\) | \(103\) |
default | \(\frac {\frac {1}{5} b^{4} x^{5}-a \,b^{3} x^{4}+\frac {10}{3} a^{2} b^{2} x^{3}-10 a^{3} b \,x^{2}+35 a^{4} x}{b^{8}}-\frac {56 a^{5} \ln \left (b x +a \right )}{b^{9}}-\frac {a^{8}}{3 b^{9} \left (b x +a \right )^{3}}+\frac {4 a^{7}}{b^{9} \left (b x +a \right )^{2}}-\frac {28 a^{6}}{b^{9} \left (b x +a \right )}\) | \(109\) |
parallelrisch | \(-\frac {-3 b^{8} x^{8}+6 a \,x^{7} b^{7}-14 a^{2} x^{6} b^{6}+42 a^{3} x^{5} b^{5}+840 \ln \left (b x +a \right ) x^{3} a^{5} b^{3}-210 a^{4} x^{4} b^{4}+2520 \ln \left (b x +a \right ) x^{2} a^{6} b^{2}+2520 \ln \left (b x +a \right ) x \,a^{7} b +2520 a^{6} x^{2} b^{2}+840 \ln \left (b x +a \right ) a^{8}+3780 a^{7} x b +1540 a^{8}}{15 b^{9} \left (b x +a \right )^{3}}\) | \(145\) |
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none
Time = 0.22 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.42 \[ \int \frac {x^8}{(a+b x)^4} \, dx=\frac {3 \, b^{8} x^{8} - 6 \, a b^{7} x^{7} + 14 \, a^{2} b^{6} x^{6} - 42 \, a^{3} b^{5} x^{5} + 210 \, a^{4} b^{4} x^{4} + 1175 \, a^{5} b^{3} x^{3} + 1005 \, a^{6} b^{2} x^{2} - 255 \, a^{7} b x - 365 \, a^{8} - 840 \, {\left (a^{5} b^{3} x^{3} + 3 \, a^{6} b^{2} x^{2} + 3 \, a^{7} b x + a^{8}\right )} \log \left (b x + a\right )}{15 \, {\left (b^{12} x^{3} + 3 \, a b^{11} x^{2} + 3 \, a^{2} b^{10} x + a^{3} b^{9}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.15 \[ \int \frac {x^8}{(a+b x)^4} \, dx=- \frac {56 a^{5} \log {\left (a + b x \right )}}{b^{9}} + \frac {35 a^{4} x}{b^{8}} - \frac {10 a^{3} x^{2}}{b^{7}} + \frac {10 a^{2} x^{3}}{3 b^{6}} - \frac {a x^{4}}{b^{5}} + \frac {- 73 a^{8} - 156 a^{7} b x - 84 a^{6} b^{2} x^{2}}{3 a^{3} b^{9} + 9 a^{2} b^{10} x + 9 a b^{11} x^{2} + 3 b^{12} x^{3}} + \frac {x^{5}}{5 b^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.10 \[ \int \frac {x^8}{(a+b x)^4} \, dx=-\frac {84 \, a^{6} b^{2} x^{2} + 156 \, a^{7} b x + 73 \, a^{8}}{3 \, {\left (b^{12} x^{3} + 3 \, a b^{11} x^{2} + 3 \, a^{2} b^{10} x + a^{3} b^{9}\right )}} - \frac {56 \, a^{5} \log \left (b x + a\right )}{b^{9}} + \frac {3 \, b^{4} x^{5} - 15 \, a b^{3} x^{4} + 50 \, a^{2} b^{2} x^{3} - 150 \, a^{3} b x^{2} + 525 \, a^{4} x}{15 \, b^{8}} \]
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Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.93 \[ \int \frac {x^8}{(a+b x)^4} \, dx=-\frac {56 \, a^{5} \log \left ({\left | b x + a \right |}\right )}{b^{9}} - \frac {84 \, a^{6} b^{2} x^{2} + 156 \, a^{7} b x + 73 \, a^{8}}{3 \, {\left (b x + a\right )}^{3} b^{9}} + \frac {3 \, b^{16} x^{5} - 15 \, a b^{15} x^{4} + 50 \, a^{2} b^{14} x^{3} - 150 \, a^{3} b^{13} x^{2} + 525 \, a^{4} b^{12} x}{15 \, b^{20}} \]
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Time = 0.40 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {x^8}{(a+b x)^4} \, dx=-\frac {2\,a\,{\left (a+b\,x\right )}^4-\frac {{\left (a+b\,x\right )}^5}{5}-\frac {28\,a^2\,{\left (a+b\,x\right )}^3}{3}+28\,a^3\,{\left (a+b\,x\right )}^2+\frac {28\,a^6}{a+b\,x}-\frac {4\,a^7}{{\left (a+b\,x\right )}^2}+\frac {a^8}{3\,{\left (a+b\,x\right )}^3}+56\,a^5\,\ln \left (a+b\,x\right )-70\,a^4\,b\,x}{b^9} \]
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