Integrand size = 22, antiderivative size = 122 \[ \int \frac {1}{(a+b x)^2 \left (a^2-b^2 x^2\right )^3} \, dx=\frac {1}{64 a^5 b (a-b x)^2}+\frac {5}{64 a^6 b (a-b x)}-\frac {1}{32 a^3 b (a+b x)^4}-\frac {1}{16 a^4 b (a+b x)^3}-\frac {3}{32 a^5 b (a+b x)^2}-\frac {5}{32 a^6 b (a+b x)}+\frac {15 \text {arctanh}\left (\frac {b x}{a}\right )}{64 a^7 b} \]
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Time = 0.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {641, 46, 214} \[ \int \frac {1}{(a+b x)^2 \left (a^2-b^2 x^2\right )^3} \, dx=\frac {15 \text {arctanh}\left (\frac {b x}{a}\right )}{64 a^7 b}+\frac {5}{64 a^6 b (a-b x)}-\frac {5}{32 a^6 b (a+b x)}+\frac {1}{64 a^5 b (a-b x)^2}-\frac {3}{32 a^5 b (a+b x)^2}-\frac {1}{16 a^4 b (a+b x)^3}-\frac {1}{32 a^3 b (a+b x)^4} \]
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Rule 46
Rule 214
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a-b x)^3 (a+b x)^5} \, dx \\ & = \int \left (\frac {1}{32 a^5 (a-b x)^3}+\frac {5}{64 a^6 (a-b x)^2}+\frac {1}{8 a^3 (a+b x)^5}+\frac {3}{16 a^4 (a+b x)^4}+\frac {3}{16 a^5 (a+b x)^3}+\frac {5}{32 a^6 (a+b x)^2}+\frac {15}{64 a^6 \left (a^2-b^2 x^2\right )}\right ) \, dx \\ & = \frac {1}{64 a^5 b (a-b x)^2}+\frac {5}{64 a^6 b (a-b x)}-\frac {1}{32 a^3 b (a+b x)^4}-\frac {1}{16 a^4 b (a+b x)^3}-\frac {3}{32 a^5 b (a+b x)^2}-\frac {5}{32 a^6 b (a+b x)}+\frac {15 \int \frac {1}{a^2-b^2 x^2} \, dx}{64 a^6} \\ & = \frac {1}{64 a^5 b (a-b x)^2}+\frac {5}{64 a^6 b (a-b x)}-\frac {1}{32 a^3 b (a+b x)^4}-\frac {1}{16 a^4 b (a+b x)^3}-\frac {3}{32 a^5 b (a+b x)^2}-\frac {5}{32 a^6 b (a+b x)}+\frac {15 \tanh ^{-1}\left (\frac {b x}{a}\right )}{64 a^7 b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(a+b x)^2 \left (a^2-b^2 x^2\right )^3} \, dx=\frac {\frac {2 a \left (-16 a^5+17 a^4 b x+50 a^3 b^2 x^2+10 a^2 b^3 x^3-30 a b^4 x^4-15 b^5 x^5\right )}{(a-b x)^2 (a+b x)^4}-15 \log (a-b x)+15 \log (a+b x)}{128 a^7 b} \]
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Time = 2.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.85
method | result | size |
norman | \(\frac {-\frac {1}{4 b a}+\frac {25 b \,x^{2}}{32 a^{3}}-\frac {15 b^{3} x^{4}}{32 a^{5}}-\frac {15 b^{4} x^{5}}{64 a^{6}}+\frac {5 b^{2} x^{3}}{32 a^{4}}+\frac {17 x}{64 a^{2}}}{\left (b x +a \right )^{4} \left (-b x +a \right )^{2}}-\frac {15 \ln \left (-b x +a \right )}{128 a^{7} b}+\frac {15 \ln \left (b x +a \right )}{128 a^{7} b}\) | \(104\) |
risch | \(\frac {-\frac {1}{4 b a}+\frac {25 b \,x^{2}}{32 a^{3}}-\frac {15 b^{3} x^{4}}{32 a^{5}}-\frac {15 b^{4} x^{5}}{64 a^{6}}+\frac {5 b^{2} x^{3}}{32 a^{4}}+\frac {17 x}{64 a^{2}}}{\left (b x +a \right )^{2} \left (-b^{2} x^{2}+a^{2}\right )^{2}}-\frac {15 \ln \left (-b x +a \right )}{128 a^{7} b}+\frac {15 \ln \left (b x +a \right )}{128 a^{7} b}\) | \(110\) |
default | \(\frac {15 \ln \left (b x +a \right )}{128 a^{7} b}-\frac {5}{32 a^{6} b \left (b x +a \right )}-\frac {3}{32 a^{5} b \left (b x +a \right )^{2}}-\frac {1}{16 a^{4} b \left (b x +a \right )^{3}}-\frac {1}{32 a^{3} b \left (b x +a \right )^{4}}-\frac {15 \ln \left (-b x +a \right )}{128 a^{7} b}+\frac {5}{64 a^{6} b \left (-b x +a \right )}+\frac {1}{64 a^{5} b \left (-b x +a \right )^{2}}\) | \(123\) |
parallelrisch | \(-\frac {32 a^{6} b^{5}+60 \ln \left (b x +a \right ) x^{3} a^{3} b^{8}-15 \ln \left (b x -a \right ) x^{2} a^{4} b^{7}+15 \ln \left (b x +a \right ) x^{2} a^{4} b^{7}+30 \ln \left (b x -a \right ) x \,a^{5} b^{6}-30 \ln \left (b x +a \right ) x \,a^{5} b^{6}-15 \ln \left (b x +a \right ) x^{6} b^{11}+15 \ln \left (b x -a \right ) a^{6} b^{5}-15 \ln \left (b x +a \right ) a^{6} b^{5}+30 x^{5} a \,b^{10}+60 x^{4} a^{2} b^{9}-20 x^{3} a^{3} b^{8}-100 x^{2} a^{4} b^{7}-34 x \,a^{5} b^{6}+15 \ln \left (b x -a \right ) x^{6} b^{11}+30 \ln \left (b x -a \right ) x^{5} a \,b^{10}-30 \ln \left (b x +a \right ) x^{5} a \,b^{10}-15 \ln \left (b x -a \right ) x^{4} a^{2} b^{9}+15 \ln \left (b x +a \right ) x^{4} a^{2} b^{9}-60 \ln \left (b x -a \right ) x^{3} a^{3} b^{8}}{128 a^{7} b^{6} \left (b x +a \right )^{2} \left (b^{2} x^{2}-a^{2}\right )^{2}}\) | \(323\) |
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Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (110) = 220\).
Time = 0.28 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.18 \[ \int \frac {1}{(a+b x)^2 \left (a^2-b^2 x^2\right )^3} \, dx=-\frac {30 \, a b^{5} x^{5} + 60 \, a^{2} b^{4} x^{4} - 20 \, a^{3} b^{3} x^{3} - 100 \, a^{4} b^{2} x^{2} - 34 \, a^{5} b x + 32 \, a^{6} - 15 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} - a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{3} - a^{4} b^{2} x^{2} + 2 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right ) + 15 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} - a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{3} - a^{4} b^{2} x^{2} + 2 \, a^{5} b x + a^{6}\right )} \log \left (b x - a\right )}{128 \, {\left (a^{7} b^{7} x^{6} + 2 \, a^{8} b^{6} x^{5} - a^{9} b^{5} x^{4} - 4 \, a^{10} b^{4} x^{3} - a^{11} b^{3} x^{2} + 2 \, a^{12} b^{2} x + a^{13} b\right )}} \]
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Time = 0.37 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(a+b x)^2 \left (a^2-b^2 x^2\right )^3} \, dx=- \frac {16 a^{5} - 17 a^{4} b x - 50 a^{3} b^{2} x^{2} - 10 a^{2} b^{3} x^{3} + 30 a b^{4} x^{4} + 15 b^{5} x^{5}}{64 a^{12} b + 128 a^{11} b^{2} x - 64 a^{10} b^{3} x^{2} - 256 a^{9} b^{4} x^{3} - 64 a^{8} b^{5} x^{4} + 128 a^{7} b^{6} x^{5} + 64 a^{6} b^{7} x^{6}} - \frac {\frac {15 \log {\left (- \frac {a}{b} + x \right )}}{128} - \frac {15 \log {\left (\frac {a}{b} + x \right )}}{128}}{a^{7} b} \]
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Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.28 \[ \int \frac {1}{(a+b x)^2 \left (a^2-b^2 x^2\right )^3} \, dx=-\frac {15 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} - 50 \, a^{3} b^{2} x^{2} - 17 \, a^{4} b x + 16 \, a^{5}}{64 \, {\left (a^{6} b^{7} x^{6} + 2 \, a^{7} b^{6} x^{5} - a^{8} b^{5} x^{4} - 4 \, a^{9} b^{4} x^{3} - a^{10} b^{3} x^{2} + 2 \, a^{11} b^{2} x + a^{12} b\right )}} + \frac {15 \, \log \left (b x + a\right )}{128 \, a^{7} b} - \frac {15 \, \log \left (b x - a\right )}{128 \, a^{7} b} \]
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Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(a+b x)^2 \left (a^2-b^2 x^2\right )^3} \, dx=-\frac {15 \, \log \left ({\left | -\frac {2 \, a}{b x + a} + 1 \right |}\right )}{128 \, a^{7} b} + \frac {\frac {24 \, a}{b x + a} - 11}{256 \, a^{7} b {\left (\frac {2 \, a}{b x + a} - 1\right )}^{2}} - \frac {\frac {5 \, a^{6} b^{11}}{b x + a} + \frac {3 \, a^{7} b^{11}}{{\left (b x + a\right )}^{2}} + \frac {2 \, a^{8} b^{11}}{{\left (b x + a\right )}^{3}} + \frac {a^{9} b^{11}}{{\left (b x + a\right )}^{4}}}{32 \, a^{12} b^{12}} \]
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Time = 9.94 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+b x)^2 \left (a^2-b^2 x^2\right )^3} \, dx=\frac {\frac {17\,x}{64\,a^2}-\frac {1}{4\,a\,b}+\frac {25\,b\,x^2}{32\,a^3}+\frac {5\,b^2\,x^3}{32\,a^4}-\frac {15\,b^3\,x^4}{32\,a^5}-\frac {15\,b^4\,x^5}{64\,a^6}}{a^6+2\,a^5\,b\,x-a^4\,b^2\,x^2-4\,a^3\,b^3\,x^3-a^2\,b^4\,x^4+2\,a\,b^5\,x^5+b^6\,x^6}+\frac {15\,\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{64\,a^7\,b} \]
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