Integrand size = 11, antiderivative size = 70 \[ \int \frac {1}{x^2 (a+b x)^4} \, dx=-\frac {1}{a^4 x}-\frac {b}{3 a^2 (a+b x)^3}-\frac {b}{a^3 (a+b x)^2}-\frac {3 b}{a^4 (a+b x)}-\frac {4 b \log (x)}{a^5}+\frac {4 b \log (a+b x)}{a^5} \]
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Time = 0.03 (sec) , antiderivative size = 70, 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} \[ \int \frac {1}{x^2 (a+b x)^4} \, dx=-\frac {4 b \log (x)}{a^5}+\frac {4 b \log (a+b x)}{a^5}-\frac {3 b}{a^4 (a+b x)}-\frac {1}{a^4 x}-\frac {b}{a^3 (a+b x)^2}-\frac {b}{3 a^2 (a+b x)^3} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^4 x^2}-\frac {4 b}{a^5 x}+\frac {b^2}{a^2 (a+b x)^4}+\frac {2 b^2}{a^3 (a+b x)^3}+\frac {3 b^2}{a^4 (a+b x)^2}+\frac {4 b^2}{a^5 (a+b x)}\right ) \, dx \\ & = -\frac {1}{a^4 x}-\frac {b}{3 a^2 (a+b x)^3}-\frac {b}{a^3 (a+b x)^2}-\frac {3 b}{a^4 (a+b x)}-\frac {4 b \log (x)}{a^5}+\frac {4 b \log (a+b x)}{a^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^2 (a+b x)^4} \, dx=-\frac {\frac {a \left (3 a^3+22 a^2 b x+30 a b^2 x^2+12 b^3 x^3\right )}{x (a+b x)^3}+12 b \log (x)-12 b \log (a+b x)}{3 a^5} \]
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Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {1}{a^{4} x}-\frac {b}{3 a^{2} \left (b x +a \right )^{3}}-\frac {b}{a^{3} \left (b x +a \right )^{2}}-\frac {3 b}{a^{4} \left (b x +a \right )}-\frac {4 b \ln \left (x \right )}{a^{5}}+\frac {4 b \ln \left (b x +a \right )}{a^{5}}\) | \(69\) |
risch | \(\frac {-\frac {4 b^{3} x^{3}}{a^{4}}-\frac {10 b^{2} x^{2}}{a^{3}}-\frac {22 b x}{3 a^{2}}-\frac {1}{a}}{x \left (b x +a \right )^{3}}-\frac {4 b \ln \left (x \right )}{a^{5}}+\frac {4 b \ln \left (-b x -a \right )}{a^{5}}\) | \(71\) |
norman | \(\frac {-\frac {1}{a}+\frac {12 b^{2} x^{2}}{a^{3}}+\frac {18 b^{3} x^{3}}{a^{4}}+\frac {22 b^{4} x^{4}}{3 a^{5}}}{x \left (b x +a \right )^{3}}-\frac {4 b \ln \left (x \right )}{a^{5}}+\frac {4 b \ln \left (b x +a \right )}{a^{5}}\) | \(72\) |
parallelrisch | \(-\frac {12 b^{4} \ln \left (x \right ) x^{4}-12 b^{4} \ln \left (b x +a \right ) x^{4}+36 \ln \left (x \right ) x^{3} a \,b^{3}-36 \ln \left (b x +a \right ) x^{3} a \,b^{3}-22 b^{4} x^{4}+36 \ln \left (x \right ) x^{2} a^{2} b^{2}-36 \ln \left (b x +a \right ) x^{2} a^{2} b^{2}-54 a \,b^{3} x^{3}+12 \ln \left (x \right ) x \,a^{3} b -12 \ln \left (b x +a \right ) x \,a^{3} b -36 a^{2} b^{2} x^{2}+3 a^{4}}{3 a^{5} x \left (b x +a \right )^{3}}\) | \(152\) |
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (68) = 136\).
Time = 0.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.19 \[ \int \frac {1}{x^2 (a+b x)^4} \, dx=-\frac {12 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 22 \, a^{3} b x + 3 \, a^{4} - 12 \, {\left (b^{4} x^{4} + 3 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + a^{3} b x\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} x^{4} + 3 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + a^{3} b x\right )} \log \left (x\right )}{3 \, {\left (a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{3} + 3 \, a^{7} b x^{2} + a^{8} x\right )}} \]
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Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^2 (a+b x)^4} \, dx=\frac {- 3 a^{3} - 22 a^{2} b x - 30 a b^{2} x^{2} - 12 b^{3} x^{3}}{3 a^{7} x + 9 a^{6} b x^{2} + 9 a^{5} b^{2} x^{3} + 3 a^{4} b^{3} x^{4}} + \frac {4 b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^2 (a+b x)^4} \, dx=-\frac {12 \, b^{3} x^{3} + 30 \, a b^{2} x^{2} + 22 \, a^{2} b x + 3 \, a^{3}}{3 \, {\left (a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{3} + 3 \, a^{6} b x^{2} + a^{7} x\right )}} + \frac {4 \, b \log \left (b x + a\right )}{a^{5}} - \frac {4 \, b \log \left (x\right )}{a^{5}} \]
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Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^2 (a+b x)^4} \, dx=\frac {4 \, b \log \left ({\left | b x + a \right |}\right )}{a^{5}} - \frac {4 \, b \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {12 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 22 \, a^{3} b x + 3 \, a^{4}}{3 \, {\left (b x + a\right )}^{3} a^{5} x} \]
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Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^2 (a+b x)^4} \, dx=\frac {8\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^5}-\frac {\frac {1}{a}+\frac {10\,b^2\,x^2}{a^3}+\frac {4\,b^3\,x^3}{a^4}+\frac {22\,b\,x}{3\,a^2}}{a^3\,x+3\,a^2\,b\,x^2+3\,a\,b^2\,x^3+b^3\,x^4} \]
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