Integrand size = 29, antiderivative size = 139 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=-\frac {(e f-d g)^2}{8 d e^3 (d+e x)^4}-\frac {(e f-d g) (e f+3 d g)}{12 d^2 e^3 (d+e x)^3}-\frac {(e f+d g)^2}{16 d^3 e^3 (d+e x)^2}-\frac {(e f+d g)^2}{16 d^4 e^3 (d+e x)}+\frac {(e f+d g)^2 \text {arctanh}\left (\frac {e x}{d}\right )}{16 d^5 e^3} \]
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Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {862, 90, 214} \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=\frac {\text {arctanh}\left (\frac {e x}{d}\right ) (d g+e f)^2}{16 d^5 e^3}-\frac {(d g+e f)^2}{16 d^4 e^3 (d+e x)}-\frac {(d g+e f)^2}{16 d^3 e^3 (d+e x)^2}-\frac {(3 d g+e f) (e f-d g)}{12 d^2 e^3 (d+e x)^3}-\frac {(e f-d g)^2}{8 d e^3 (d+e x)^4} \]
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Rule 90
Rule 214
Rule 862
Rubi steps \begin{align*} \text {integral}& = \int \frac {(f+g x)^2}{(d-e x) (d+e x)^5} \, dx \\ & = \int \left (\frac {(-e f+d g)^2}{2 d e^2 (d+e x)^5}+\frac {(e f-d g) (e f+3 d g)}{4 d^2 e^2 (d+e x)^4}+\frac {(e f+d g)^2}{8 d^3 e^2 (d+e x)^3}+\frac {(e f+d g)^2}{16 d^4 e^2 (d+e x)^2}+\frac {(e f+d g)^2}{16 d^4 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx \\ & = -\frac {(e f-d g)^2}{8 d e^3 (d+e x)^4}-\frac {(e f-d g) (e f+3 d g)}{12 d^2 e^3 (d+e x)^3}-\frac {(e f+d g)^2}{16 d^3 e^3 (d+e x)^2}-\frac {(e f+d g)^2}{16 d^4 e^3 (d+e x)}+\frac {(e f+d g)^2 \int \frac {1}{d^2-e^2 x^2} \, dx}{16 d^4 e^2} \\ & = -\frac {(e f-d g)^2}{8 d e^3 (d+e x)^4}-\frac {(e f-d g) (e f+3 d g)}{12 d^2 e^3 (d+e x)^3}-\frac {(e f+d g)^2}{16 d^3 e^3 (d+e x)^2}-\frac {(e f+d g)^2}{16 d^4 e^3 (d+e x)}+\frac {(e f+d g)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{16 d^5 e^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.02 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=-\frac {\frac {12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac {8 d^3 \left (e^2 f^2+2 d e f g-3 d^2 g^2\right )}{(d+e x)^3}+\frac {6 d^2 (e f+d g)^2}{(d+e x)^2}+\frac {6 d (e f+d g)^2}{d+e x}+3 (e f+d g)^2 \log (d-e x)-3 (e f+d g)^2 \log (d+e x)}{96 d^5 e^3} \]
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Time = 0.47 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.43
method | result | size |
norman | \(\frac {-\frac {\left (3 d^{2} g^{2}-26 d e f g -61 e^{2} f^{2}\right ) x^{3}}{48 d^{4}}-\frac {\left (d^{2} g^{2}-2 d e f g -7 e^{2} f^{2}\right ) x^{2}}{4 e \,d^{3}}+\frac {e^{2} \left (d f g +2 e \,f^{2}\right ) x^{4}}{6 d^{5}}-\frac {\left (d^{2} g^{2}+2 d e f g -15 e^{2} f^{2}\right ) x}{16 d^{2} e^{2}}}{\left (e x +d \right )^{4}}-\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{32 e^{3} d^{5}}+\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (e x +d \right )}{32 e^{3} d^{5}}\) | \(199\) |
default | \(\frac {\left (-d^{2} g^{2}-2 d e f g -e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{32 e^{3} d^{5}}-\frac {-3 d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{12 d^{2} e^{3} \left (e x +d \right )^{3}}-\frac {d^{2} g^{2}-2 d e f g +e^{2} f^{2}}{8 e^{3} d \left (e x +d \right )^{4}}+\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (e x +d \right )}{32 e^{3} d^{5}}-\frac {d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{16 e^{3} d^{4} \left (e x +d \right )}-\frac {d^{2} g^{2}+2 d e f g +e^{2} f^{2}}{16 e^{3} d^{3} \left (e x +d \right )^{2}}\) | \(220\) |
risch | \(\frac {-\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) x^{3}}{16 d^{4}}-\frac {\left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) x^{2}}{4 d^{3} e}-\frac {\left (3 d^{2} g^{2}+38 d e f g +19 e^{2} f^{2}\right ) x}{48 d^{2} e^{2}}-\frac {f \left (d g +2 e f \right )}{6 e^{2} d}}{\left (e x +d \right )^{4}}-\frac {\ln \left (-e x +d \right ) g^{2}}{32 e^{3} d^{3}}-\frac {\ln \left (-e x +d \right ) f g}{16 e^{2} d^{4}}-\frac {\ln \left (-e x +d \right ) f^{2}}{32 e \,d^{5}}+\frac {\ln \left (e x +d \right ) g^{2}}{32 e^{3} d^{3}}+\frac {\ln \left (e x +d \right ) f g}{16 e^{2} d^{4}}+\frac {\ln \left (e x +d \right ) f^{2}}{32 e \,d^{5}}\) | \(224\) |
parallelrisch | \(-\frac {-12 \ln \left (e x +d \right ) x \,d^{3} e^{3} f^{2}+6 \ln \left (e x -d \right ) d^{5} e f g -6 \ln \left (e x +d \right ) d^{5} e f g +3 \ln \left (e x -d \right ) x^{4} d^{2} e^{4} g^{2}-3 \ln \left (e x +d \right ) x^{4} d^{2} e^{4} g^{2}+12 \ln \left (e x -d \right ) x^{3} d^{3} e^{3} g^{2}+12 \ln \left (e x -d \right ) x^{3} d \,e^{5} f^{2}-12 \ln \left (e x +d \right ) x^{3} d^{3} e^{3} g^{2}-12 \ln \left (e x +d \right ) x^{3} d \,e^{5} f^{2}+18 \ln \left (e x -d \right ) x^{2} d^{4} e^{2} g^{2}+18 \ln \left (e x -d \right ) x^{2} d^{2} e^{4} f^{2}-18 \ln \left (e x +d \right ) x^{2} d^{4} e^{2} g^{2}-18 \ln \left (e x +d \right ) x^{2} d^{2} e^{4} f^{2}+12 \ln \left (e x -d \right ) x \,d^{5} e \,g^{2}+12 \ln \left (e x -d \right ) x \,d^{3} e^{3} f^{2}-12 \ln \left (e x +d \right ) x \,d^{5} e \,g^{2}-32 e^{6} f^{2} x^{4}+3 \ln \left (e x -d \right ) d^{4} e^{2} f^{2}-3 \ln \left (e x +d \right ) d^{4} e^{2} f^{2}+3 \ln \left (e x -d \right ) x^{4} e^{6} f^{2}-3 \ln \left (e x +d \right ) x^{4} e^{6} f^{2}+6 d^{5} e \,g^{2} x -90 d^{3} e^{3} f^{2} x +6 d^{3} e^{3} g^{2} x^{3}-122 d \,e^{5} f^{2} x^{3}+24 d^{4} e^{2} g^{2} x^{2}-168 d^{2} e^{4} f^{2} x^{2}-52 d^{2} e^{4} f g \,x^{3}-48 d^{3} e^{3} f g \,x^{2}+12 d^{4} e^{2} f g x -16 d \,e^{5} f g \,x^{4}+3 \ln \left (e x -d \right ) d^{6} g^{2}-3 \ln \left (e x +d \right ) d^{6} g^{2}-24 \ln \left (e x +d \right ) x^{3} d^{2} e^{4} f g -36 \ln \left (e x +d \right ) x^{2} d^{3} e^{3} f g +24 \ln \left (e x -d \right ) x \,d^{4} e^{2} f g -24 \ln \left (e x +d \right ) x \,d^{4} e^{2} f g +24 \ln \left (e x -d \right ) x^{3} d^{2} e^{4} f g -6 \ln \left (e x +d \right ) x^{4} d \,e^{5} f g +6 \ln \left (e x -d \right ) x^{4} d \,e^{5} f g +36 \ln \left (e x -d \right ) x^{2} d^{3} e^{3} f g}{96 e^{3} d^{5} \left (e x +d \right )^{4}}\) | \(714\) |
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Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (129) = 258\).
Time = 0.30 (sec) , antiderivative size = 511, normalized size of antiderivative = 3.68 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=-\frac {32 \, d^{4} e^{2} f^{2} + 16 \, d^{5} e f g + 6 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 24 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (19 \, d^{3} e^{3} f^{2} + 38 \, d^{4} e^{2} f g + 3 \, d^{5} e g^{2}\right )} x - 3 \, {\left (d^{4} e^{2} f^{2} + 2 \, d^{5} e f g + d^{6} g^{2} + {\left (e^{6} f^{2} + 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} + 4 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 6 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 4 \, {\left (d^{3} e^{3} f^{2} + 2 \, d^{4} e^{2} f g + d^{5} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \, {\left (d^{4} e^{2} f^{2} + 2 \, d^{5} e f g + d^{6} g^{2} + {\left (e^{6} f^{2} + 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} + 4 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 6 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 4 \, {\left (d^{3} e^{3} f^{2} + 2 \, d^{4} e^{2} f g + d^{5} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{96 \, {\left (d^{5} e^{7} x^{4} + 4 \, d^{6} e^{6} x^{3} + 6 \, d^{7} e^{5} x^{2} + 4 \, d^{8} e^{4} x + d^{9} e^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (122) = 244\).
Time = 0.68 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.03 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=- \frac {8 d^{4} f g + 16 d^{3} e f^{2} + x^{3} \cdot \left (3 d^{2} e^{2} g^{2} + 6 d e^{3} f g + 3 e^{4} f^{2}\right ) + x^{2} \cdot \left (12 d^{3} e g^{2} + 24 d^{2} e^{2} f g + 12 d e^{3} f^{2}\right ) + x \left (3 d^{4} g^{2} + 38 d^{3} e f g + 19 d^{2} e^{2} f^{2}\right )}{48 d^{8} e^{2} + 192 d^{7} e^{3} x + 288 d^{6} e^{4} x^{2} + 192 d^{5} e^{5} x^{3} + 48 d^{4} e^{6} x^{4}} - \frac {\left (d g + e f\right )^{2} \log {\left (- \frac {d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{32 d^{5} e^{3}} + \frac {\left (d g + e f\right )^{2} \log {\left (\frac {d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{32 d^{5} e^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.70 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=-\frac {16 \, d^{3} e f^{2} + 8 \, d^{4} f g + 3 \, {\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{3} + 12 \, {\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x^{2} + {\left (19 \, d^{2} e^{2} f^{2} + 38 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} x}{48 \, {\left (d^{4} e^{6} x^{4} + 4 \, d^{5} e^{5} x^{3} + 6 \, d^{6} e^{4} x^{2} + 4 \, d^{7} e^{3} x + d^{8} e^{2}\right )}} + \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{32 \, d^{5} e^{3}} - \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{32 \, d^{5} e^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.53 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=\frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{32 \, d^{5} e^{3}} - \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left ({\left | e x - d \right |}\right )}{32 \, d^{5} e^{3}} - \frac {16 \, d^{4} e^{2} f^{2} + 8 \, d^{5} e f g + 3 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 12 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + {\left (19 \, d^{3} e^{3} f^{2} + 38 \, d^{4} e^{2} f g + 3 \, d^{5} e g^{2}\right )} x}{48 \, {\left (e x + d\right )}^{4} d^{5} e^{3}} \]
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Time = 0.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.29 \[ \int \frac {(f+g x)^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )} \, dx=\frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,{\left (d\,g+e\,f\right )}^2}{16\,d^5\,e^3}-\frac {\frac {x^3\,\left (d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{16\,d^4}+\frac {2\,e\,f^2+d\,g\,f}{6\,d\,e^2}+\frac {x\,\left (3\,d^2\,g^2+38\,d\,e\,f\,g+19\,e^2\,f^2\right )}{48\,d^2\,e^2}+\frac {x^2\,\left (d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{4\,d^3\,e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]
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