Integrand size = 11, antiderivative size = 157 \[ \int \frac {1}{x^4 (a+b x)^7} \, dx=-\frac {1}{3 a^7 x^3}+\frac {7 b}{2 a^8 x^2}-\frac {28 b^2}{a^9 x}-\frac {b^3}{6 a^4 (a+b x)^6}-\frac {4 b^3}{5 a^5 (a+b x)^5}-\frac {5 b^3}{2 a^6 (a+b x)^4}-\frac {20 b^3}{3 a^7 (a+b x)^3}-\frac {35 b^3}{2 a^8 (a+b x)^2}-\frac {56 b^3}{a^9 (a+b x)}-\frac {84 b^3 \log (x)}{a^{10}}+\frac {84 b^3 \log (a+b x)}{a^{10}} \]
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Time = 0.08 (sec) , antiderivative size = 157, 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 = {46} \[ \int \frac {1}{x^4 (a+b x)^7} \, dx=-\frac {84 b^3 \log (x)}{a^{10}}+\frac {84 b^3 \log (a+b x)}{a^{10}}-\frac {56 b^3}{a^9 (a+b x)}-\frac {28 b^2}{a^9 x}-\frac {35 b^3}{2 a^8 (a+b x)^2}+\frac {7 b}{2 a^8 x^2}-\frac {20 b^3}{3 a^7 (a+b x)^3}-\frac {1}{3 a^7 x^3}-\frac {5 b^3}{2 a^6 (a+b x)^4}-\frac {4 b^3}{5 a^5 (a+b x)^5}-\frac {b^3}{6 a^4 (a+b x)^6} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^7 x^4}-\frac {7 b}{a^8 x^3}+\frac {28 b^2}{a^9 x^2}-\frac {84 b^3}{a^{10} x}+\frac {b^4}{a^4 (a+b x)^7}+\frac {4 b^4}{a^5 (a+b x)^6}+\frac {10 b^4}{a^6 (a+b x)^5}+\frac {20 b^4}{a^7 (a+b x)^4}+\frac {35 b^4}{a^8 (a+b x)^3}+\frac {56 b^4}{a^9 (a+b x)^2}+\frac {84 b^4}{a^{10} (a+b x)}\right ) \, dx \\ & = -\frac {1}{3 a^7 x^3}+\frac {7 b}{2 a^8 x^2}-\frac {28 b^2}{a^9 x}-\frac {b^3}{6 a^4 (a+b x)^6}-\frac {4 b^3}{5 a^5 (a+b x)^5}-\frac {5 b^3}{2 a^6 (a+b x)^4}-\frac {20 b^3}{3 a^7 (a+b x)^3}-\frac {35 b^3}{2 a^8 (a+b x)^2}-\frac {56 b^3}{a^9 (a+b x)}-\frac {84 b^3 \log (x)}{a^{10}}+\frac {84 b^3 \log (a+b x)}{a^{10}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^4 (a+b x)^7} \, dx=-\frac {\frac {a \left (10 a^8-45 a^7 b x+360 a^6 b^2 x^2+6174 a^5 b^3 x^3+21924 a^4 b^4 x^4+35910 a^3 b^5 x^5+31080 a^2 b^6 x^6+13860 a b^7 x^7+2520 b^8 x^8\right )}{x^3 (a+b x)^6}+2520 b^3 \log (x)-2520 b^3 \log (a+b x)}{30 a^{10}} \]
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Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.81
method | result | size |
norman | \(\frac {-\frac {1}{3 a}+\frac {3 b x}{2 a^{2}}-\frac {12 b^{2} x^{2}}{a^{3}}+\frac {504 b^{4} x^{4}}{a^{5}}+\frac {1890 b^{5} x^{5}}{a^{6}}+\frac {3080 b^{6} x^{6}}{a^{7}}+\frac {2625 b^{7} x^{7}}{a^{8}}+\frac {5754 b^{8} x^{8}}{5 a^{9}}+\frac {1029 b^{9} x^{9}}{5 a^{10}}}{x^{3} \left (b x +a \right )^{6}}-\frac {84 b^{3} \ln \left (x \right )}{a^{10}}+\frac {84 b^{3} \ln \left (b x +a \right )}{a^{10}}\) | \(127\) |
risch | \(\frac {-\frac {84 b^{8} x^{8}}{a^{9}}-\frac {462 b^{7} x^{7}}{a^{8}}-\frac {1036 b^{6} x^{6}}{a^{7}}-\frac {1197 b^{5} x^{5}}{a^{6}}-\frac {3654 b^{4} x^{4}}{5 a^{5}}-\frac {1029 b^{3} x^{3}}{5 a^{4}}-\frac {12 b^{2} x^{2}}{a^{3}}+\frac {3 b x}{2 a^{2}}-\frac {1}{3 a}}{x^{3} \left (b x +a \right )^{6}}-\frac {84 b^{3} \ln \left (x \right )}{a^{10}}+\frac {84 b^{3} \ln \left (-b x -a \right )}{a^{10}}\) | \(130\) |
default | \(-\frac {1}{3 a^{7} x^{3}}+\frac {7 b}{2 a^{8} x^{2}}-\frac {28 b^{2}}{a^{9} x}-\frac {b^{3}}{6 a^{4} \left (b x +a \right )^{6}}-\frac {4 b^{3}}{5 a^{5} \left (b x +a \right )^{5}}-\frac {5 b^{3}}{2 a^{6} \left (b x +a \right )^{4}}-\frac {20 b^{3}}{3 a^{7} \left (b x +a \right )^{3}}-\frac {35 b^{3}}{2 a^{8} \left (b x +a \right )^{2}}-\frac {56 b^{3}}{a^{9} \left (b x +a \right )}-\frac {84 b^{3} \ln \left (x \right )}{a^{10}}+\frac {84 b^{3} \ln \left (b x +a \right )}{a^{10}}\) | \(144\) |
parallelrisch | \(-\frac {10 a^{9}+15120 \ln \left (x \right ) x^{8} a \,b^{8}-15120 \ln \left (b x +a \right ) x^{8} a \,b^{8}+37800 \ln \left (x \right ) x^{7} a^{2} b^{7}-37800 \ln \left (b x +a \right ) x^{7} a^{2} b^{7}+50400 \ln \left (x \right ) x^{6} a^{3} b^{6}+37800 \ln \left (x \right ) x^{5} a^{4} b^{5}+15120 \ln \left (x \right ) x^{4} a^{5} b^{4}+2520 \ln \left (x \right ) x^{3} a^{6} b^{3}-50400 \ln \left (b x +a \right ) x^{6} a^{3} b^{6}-37800 \ln \left (b x +a \right ) x^{5} a^{4} b^{5}-15120 \ln \left (b x +a \right ) x^{4} a^{5} b^{4}-2520 \ln \left (b x +a \right ) x^{3} a^{6} b^{3}-6174 b^{9} x^{9}-56700 a^{4} x^{5} b^{5}+2520 \ln \left (x \right ) x^{9} b^{9}-2520 \ln \left (b x +a \right ) x^{9} b^{9}-92400 x^{6} a^{3} b^{6}-34524 a \,x^{8} b^{8}-15120 a^{5} b^{4} x^{4}+360 a^{7} b^{2} x^{2}-45 a^{8} b x -78750 a^{2} x^{7} b^{7}}{30 a^{10} x^{3} \left (b x +a \right )^{6}}\) | \(301\) |
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Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (143) = 286\).
Time = 0.23 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.02 \[ \int \frac {1}{x^4 (a+b x)^7} \, dx=-\frac {2520 \, a b^{8} x^{8} + 13860 \, a^{2} b^{7} x^{7} + 31080 \, a^{3} b^{6} x^{6} + 35910 \, a^{4} b^{5} x^{5} + 21924 \, a^{5} b^{4} x^{4} + 6174 \, a^{6} b^{3} x^{3} + 360 \, a^{7} b^{2} x^{2} - 45 \, a^{8} b x + 10 \, a^{9} - 2520 \, {\left (b^{9} x^{9} + 6 \, a b^{8} x^{8} + 15 \, a^{2} b^{7} x^{7} + 20 \, a^{3} b^{6} x^{6} + 15 \, a^{4} b^{5} x^{5} + 6 \, a^{5} b^{4} x^{4} + a^{6} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 2520 \, {\left (b^{9} x^{9} + 6 \, a b^{8} x^{8} + 15 \, a^{2} b^{7} x^{7} + 20 \, a^{3} b^{6} x^{6} + 15 \, a^{4} b^{5} x^{5} + 6 \, a^{5} b^{4} x^{4} + a^{6} b^{3} x^{3}\right )} \log \left (x\right )}{30 \, {\left (a^{10} b^{6} x^{9} + 6 \, a^{11} b^{5} x^{8} + 15 \, a^{12} b^{4} x^{7} + 20 \, a^{13} b^{3} x^{6} + 15 \, a^{14} b^{2} x^{5} + 6 \, a^{15} b x^{4} + a^{16} x^{3}\right )}} \]
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Time = 0.44 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^4 (a+b x)^7} \, dx=\frac {- 10 a^{8} + 45 a^{7} b x - 360 a^{6} b^{2} x^{2} - 6174 a^{5} b^{3} x^{3} - 21924 a^{4} b^{4} x^{4} - 35910 a^{3} b^{5} x^{5} - 31080 a^{2} b^{6} x^{6} - 13860 a b^{7} x^{7} - 2520 b^{8} x^{8}}{30 a^{15} x^{3} + 180 a^{14} b x^{4} + 450 a^{13} b^{2} x^{5} + 600 a^{12} b^{3} x^{6} + 450 a^{11} b^{4} x^{7} + 180 a^{10} b^{5} x^{8} + 30 a^{9} b^{6} x^{9}} + \frac {84 b^{3} \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{10}} \]
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Time = 0.24 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^4 (a+b x)^7} \, dx=-\frac {2520 \, b^{8} x^{8} + 13860 \, a b^{7} x^{7} + 31080 \, a^{2} b^{6} x^{6} + 35910 \, a^{3} b^{5} x^{5} + 21924 \, a^{4} b^{4} x^{4} + 6174 \, a^{5} b^{3} x^{3} + 360 \, a^{6} b^{2} x^{2} - 45 \, a^{7} b x + 10 \, a^{8}}{30 \, {\left (a^{9} b^{6} x^{9} + 6 \, a^{10} b^{5} x^{8} + 15 \, a^{11} b^{4} x^{7} + 20 \, a^{12} b^{3} x^{6} + 15 \, a^{13} b^{2} x^{5} + 6 \, a^{14} b x^{4} + a^{15} x^{3}\right )}} + \frac {84 \, b^{3} \log \left (b x + a\right )}{a^{10}} - \frac {84 \, b^{3} \log \left (x\right )}{a^{10}} \]
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Time = 0.31 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^4 (a+b x)^7} \, dx=\frac {84 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{10}} - \frac {84 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{10}} - \frac {2520 \, a b^{8} x^{8} + 13860 \, a^{2} b^{7} x^{7} + 31080 \, a^{3} b^{6} x^{6} + 35910 \, a^{4} b^{5} x^{5} + 21924 \, a^{5} b^{4} x^{4} + 6174 \, a^{6} b^{3} x^{3} + 360 \, a^{7} b^{2} x^{2} - 45 \, a^{8} b x + 10 \, a^{9}}{30 \, {\left (b x + a\right )}^{6} a^{10} x^{3}} \]
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Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^4 (a+b x)^7} \, dx=\frac {168\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{10}}-\frac {\frac {1}{3\,a}+\frac {12\,b^2\,x^2}{a^3}+\frac {1029\,b^3\,x^3}{5\,a^4}+\frac {3654\,b^4\,x^4}{5\,a^5}+\frac {1197\,b^5\,x^5}{a^6}+\frac {1036\,b^6\,x^6}{a^7}+\frac {462\,b^7\,x^7}{a^8}+\frac {84\,b^8\,x^8}{a^9}-\frac {3\,b\,x}{2\,a^2}}{a^6\,x^3+6\,a^5\,b\,x^4+15\,a^4\,b^2\,x^5+20\,a^3\,b^3\,x^6+15\,a^2\,b^4\,x^7+6\,a\,b^5\,x^8+b^6\,x^9} \]
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