Integrand size = 11, antiderivative size = 69 \[ \int \frac {1}{x^4 (a+b x)^2} \, dx=-\frac {1}{3 a^2 x^3}+\frac {b}{a^3 x^2}-\frac {3 b^2}{a^4 x}-\frac {b^3}{a^4 (a+b x)}-\frac {4 b^3 \log (x)}{a^5}+\frac {4 b^3 \log (a+b x)}{a^5} \]
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Time = 0.02 (sec) , antiderivative size = 69, 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} \[ \int \frac {1}{x^4 (a+b x)^2} \, dx=-\frac {4 b^3 \log (x)}{a^5}+\frac {4 b^3 \log (a+b x)}{a^5}-\frac {b^3}{a^4 (a+b x)}-\frac {3 b^2}{a^4 x}+\frac {b}{a^3 x^2}-\frac {1}{3 a^2 x^3} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^2 x^4}-\frac {2 b}{a^3 x^3}+\frac {3 b^2}{a^4 x^2}-\frac {4 b^3}{a^5 x}+\frac {b^4}{a^4 (a+b x)^2}+\frac {4 b^4}{a^5 (a+b x)}\right ) \, dx \\ & = -\frac {1}{3 a^2 x^3}+\frac {b}{a^3 x^2}-\frac {3 b^2}{a^4 x}-\frac {b^3}{a^4 (a+b x)}-\frac {4 b^3 \log (x)}{a^5}+\frac {4 b^3 \log (a+b x)}{a^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^4 (a+b x)^2} \, dx=-\frac {\frac {a \left (a^3-2 a^2 b x+6 a b^2 x^2+12 b^3 x^3\right )}{x^3 (a+b x)}+12 b^3 \log (x)-12 b^3 \log (a+b x)}{3 a^5} \]
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Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {1}{3 a^{2} x^{3}}+\frac {b}{a^{3} x^{2}}-\frac {3 b^{2}}{a^{4} x}-\frac {b^{3}}{a^{4} \left (b x +a \right )}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}+\frac {4 b^{3} \ln \left (b x +a \right )}{a^{5}}\) | \(68\) |
norman | \(\frac {\frac {4 b^{4} x^{4}}{a^{5}}-\frac {1}{3 a}+\frac {2 b x}{3 a^{2}}-\frac {2 b^{2} x^{2}}{a^{3}}}{x^{3} \left (b x +a \right )}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}+\frac {4 b^{3} \ln \left (b x +a \right )}{a^{5}}\) | \(72\) |
risch | \(\frac {-\frac {4 b^{3} x^{3}}{a^{4}}-\frac {2 b^{2} x^{2}}{a^{3}}+\frac {2 b x}{3 a^{2}}-\frac {1}{3 a}}{x^{3} \left (b x +a \right )}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}+\frac {4 b^{3} \ln \left (-b x -a \right )}{a^{5}}\) | \(75\) |
parallelrisch | \(-\frac {12 b^{4} \ln \left (x \right ) x^{4}-12 b^{4} \ln \left (b x +a \right ) x^{4}+12 \ln \left (x \right ) x^{3} a \,b^{3}-12 \ln \left (b x +a \right ) x^{3} a \,b^{3}-12 b^{4} x^{4}+6 a^{2} b^{2} x^{2}-2 a^{3} b x +a^{4}}{3 a^{5} x^{3} \left (b x +a \right )}\) | \(96\) |
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Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x^4 (a+b x)^2} \, dx=-\frac {12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4} - 12 \, {\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (x\right )}{3 \, {\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \]
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Time = 0.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^4 (a+b x)^2} \, dx=\frac {- a^{3} + 2 a^{2} b x - 6 a b^{2} x^{2} - 12 b^{3} x^{3}}{3 a^{5} x^{3} + 3 a^{4} b x^{4}} + \frac {4 b^{3} \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{5}} \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^4 (a+b x)^2} \, dx=-\frac {12 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}}{3 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac {4 \, b^{3} \log \left (b x + a\right )}{a^{5}} - \frac {4 \, b^{3} \log \left (x\right )}{a^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^4 (a+b x)^2} \, dx=-\frac {4 \, b^{3} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{5}} - \frac {b^{3}}{{\left (b x + a\right )} a^{4}} - \frac {\frac {30 \, a b^{3}}{b x + a} - \frac {18 \, a^{2} b^{3}}{{\left (b x + a\right )}^{2}} - 13 \, b^{3}}{3 \, a^{5} {\left (\frac {a}{b x + a} - 1\right )}^{3}} \]
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Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 (a+b x)^2} \, dx=\frac {8\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^5}-\frac {\frac {1}{3\,a}+\frac {2\,b^2\,x^2}{a^3}+\frac {4\,b^3\,x^3}{a^4}-\frac {2\,b\,x}{3\,a^2}}{b\,x^4+a\,x^3} \]
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