Integrand size = 22, antiderivative size = 104 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )} \, dx=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b \left (b^2-3 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log (x)}{a^3}-\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3} \]
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Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1599, 723, 814, 648, 632, 212, 642} \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )} \, dx=\frac {b \left (b^2-3 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c}}-\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\log (x) \left (b^2-a c\right )}{a^3}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 723
Rule 814
Rule 1599
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^3 \left (a+b x+c x^2\right )} \, dx \\ & = -\frac {1}{2 a x^2}+\frac {\int \frac {-b-c x}{x^2 \left (a+b x+c x^2\right )} \, dx}{a} \\ & = -\frac {1}{2 a x^2}+\frac {\int \left (-\frac {b}{a x^2}+\frac {b^2-a c}{a^2 x}+\frac {-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx}{a} \\ & = -\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {\left (b^2-a c\right ) \log (x)}{a^3}+\frac {\int \frac {-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x}{a+b x+c x^2} \, dx}{a^3} \\ & = -\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {\left (b^2-a c\right ) \log (x)}{a^3}-\frac {\left (b \left (b^2-3 a c\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 a^3}-\frac {\left (b^2-a c\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^3} \\ & = -\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {\left (b^2-a c\right ) \log (x)}{a^3}-\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\left (b \left (b^2-3 a c\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^3} \\ & = -\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log (x)}{a^3}-\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )} \, dx=\frac {-\frac {a^2}{x^2}+\frac {2 a b}{x}-\frac {2 b \left (b^2-3 a c\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+2 \left (b^2-a c\right ) \log (x)+\left (-b^2+a c\right ) \log (a+x (b+c x))}{2 a^3} \]
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Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.23
method | result | size |
default | \(-\frac {1}{2 a \,x^{2}}+\frac {\left (-a c +b^{2}\right ) \ln \left (x \right )}{a^{3}}+\frac {b}{a^{2} x}+\frac {\frac {\left (a \,c^{2}-b^{2} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (2 a b c -b^{3}-\frac {\left (a \,c^{2}-b^{2} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{a^{3}}\) | \(128\) |
risch | \(\frac {\frac {b x}{a^{2}}-\frac {1}{2 a}}{x^{2}}-\frac {\ln \left (x \right ) c}{a^{2}}+\frac {\ln \left (x \right ) b^{2}}{a^{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 c \,a^{4}-a^{3} b^{2}\right ) \textit {\_Z}^{2}+\left (-4 a^{2} c^{2}+5 a \,b^{2} c -b^{4}\right ) \textit {\_Z} +c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (6 c \,a^{5}-2 b^{2} a^{4}\right ) \textit {\_R}^{2}+\left (-3 a^{3} c^{2}+2 a^{2} b^{2} c \right ) \textit {\_R} +b^{2} c^{2}\right ) x -a^{5} b \,\textit {\_R}^{2}+\left (-2 a^{3} b c +a^{2} b^{3}\right ) \textit {\_R} -a b \,c^{2}+b^{3} c \right )\right )\) | \(176\) |
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Time = 0.30 (sec) , antiderivative size = 358, normalized size of antiderivative = 3.44 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )} \, dx=\left [-\frac {{\left (b^{3} - 3 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} x^{2} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + a^{2} b^{2} - 4 \, a^{3} c + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \left (x\right ) - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}}, \frac {2 \, {\left (b^{3} - 3 \, a b c\right )} \sqrt {-b^{2} + 4 \, a c} x^{2} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - a^{2} b^{2} + 4 \, a^{3} c - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \left (c x^{2} + b x + a\right ) + 2 \, {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \left (x\right ) + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}}\right ] \]
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Timed out. \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.33 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )} \, dx=-\frac {{\left (b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} + \frac {{\left (b^{2} - a c\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} a^{3}} + \frac {2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \]
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Time = 0.34 (sec) , antiderivative size = 447, normalized size of antiderivative = 4.30 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )} \, dx=\frac {\ln \left (2\,a\,b^4+2\,b^5\,x+6\,a^3\,c^2+2\,a\,b^3\,\sqrt {b^2-4\,a\,c}+2\,b^4\,x\,\sqrt {b^2-4\,a\,c}-9\,a^2\,b^2\,c-10\,a\,b^3\,c\,x-3\,a^2\,b\,c\,\sqrt {b^2-4\,a\,c}+9\,a^2\,b\,c^2\,x+3\,a^2\,c^2\,x\,\sqrt {b^2-4\,a\,c}-6\,a\,b^2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {b^4}{2}-a\,\left (\frac {5\,b^2\,c}{2}+\frac {3\,b\,c\,\sqrt {b^2-4\,a\,c}}{2}\right )+\frac {b^3\,\sqrt {b^2-4\,a\,c}}{2}+2\,a^2\,c^2\right )}{4\,a^4\,c-a^3\,b^2}-\frac {\ln \left (2\,a\,b^4+2\,b^5\,x+6\,a^3\,c^2-2\,a\,b^3\,\sqrt {b^2-4\,a\,c}-2\,b^4\,x\,\sqrt {b^2-4\,a\,c}-9\,a^2\,b^2\,c-10\,a\,b^3\,c\,x+3\,a^2\,b\,c\,\sqrt {b^2-4\,a\,c}+9\,a^2\,b\,c^2\,x-3\,a^2\,c^2\,x\,\sqrt {b^2-4\,a\,c}+6\,a\,b^2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (a\,\left (\frac {5\,b^2\,c}{2}-\frac {3\,b\,c\,\sqrt {b^2-4\,a\,c}}{2}\right )-\frac {b^4}{2}+\frac {b^3\,\sqrt {b^2-4\,a\,c}}{2}-2\,a^2\,c^2\right )}{4\,a^4\,c-a^3\,b^2}-\frac {\frac {1}{2\,a}-\frac {b\,x}{a^2}}{x^2}-\frac {\ln \left (x\right )\,\left (a\,c-b^2\right )}{a^3} \]
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