Integrand size = 20, antiderivative size = 74 \[ \int \frac {1}{x \left (a+b x^n+c x^{2 n}\right )} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c} n}+\frac {\log (x)}{a}-\frac {\log \left (a+b x^n+c x^{2 n}\right )}{2 a n} \]
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Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1371, 719, 29, 648, 632, 212, 642} \[ \int \frac {1}{x \left (a+b x^n+c x^{2 n}\right )} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{a n \sqrt {b^2-4 a c}}-\frac {\log \left (a+b x^n+c x^{2 n}\right )}{2 a n}+\frac {\log (x)}{a} \]
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Rule 29
Rule 212
Rule 632
Rule 642
Rule 648
Rule 719
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{a n}+\frac {\text {Subst}\left (\int \frac {-b-c x}{a+b x+c x^2} \, dx,x,x^n\right )}{a n} \\ & = \frac {\log (x)}{a}-\frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^n\right )}{2 a n}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^n\right )}{2 a n} \\ & = \frac {\log (x)}{a}-\frac {\log \left (a+b x^n+c x^{2 n}\right )}{2 a n}+\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^n\right )}{a n} \\ & = \frac {b \tanh ^{-1}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c} n}+\frac {\log (x)}{a}-\frac {\log \left (a+b x^n+c x^{2 n}\right )}{2 a n} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \left (a+b x^n+c x^{2 n}\right )} \, dx=-\frac {\frac {2 b \arctan \left (\frac {b+2 c x^n}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-2 \log \left (x^n\right )+\log \left (a+x^n \left (b+c x^n\right )\right )}{2 a n} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(396\) vs. \(2(68)=136\).
Time = 0.19 (sec) , antiderivative size = 397, normalized size of antiderivative = 5.36
method | result | size |
risch | \(\frac {4 n^{2} \ln \left (x \right ) a c}{4 a^{2} c \,n^{2}-a \,b^{2} n^{2}}-\frac {n^{2} \ln \left (x \right ) b^{2}}{4 a^{2} c \,n^{2}-a \,b^{2} n^{2}}-\frac {2 \ln \left (x^{n}-\frac {-b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) c}{\left (4 a c -b^{2}\right ) n}+\frac {\ln \left (x^{n}-\frac {-b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) b^{2}}{2 a \left (4 a c -b^{2}\right ) n}+\frac {\ln \left (x^{n}-\frac {-b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) \sqrt {-4 a \,b^{2} c +b^{4}}}{2 a \left (4 a c -b^{2}\right ) n}-\frac {2 \ln \left (x^{n}+\frac {b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) c}{\left (4 a c -b^{2}\right ) n}+\frac {\ln \left (x^{n}+\frac {b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) b^{2}}{2 a \left (4 a c -b^{2}\right ) n}-\frac {\ln \left (x^{n}+\frac {b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) \sqrt {-4 a \,b^{2} c +b^{4}}}{2 a \left (4 a c -b^{2}\right ) n}\) | \(397\) |
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none
Time = 0.28 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.50 \[ \int \frac {1}{x \left (a+b x^n+c x^{2 n}\right )} \, dx=\left [\frac {2 \, {\left (b^{2} - 4 \, a c\right )} n \log \left (x\right ) + \sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} x^{2 \, n} + b^{2} - 2 \, a c + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} x^{n} + \sqrt {b^{2} - 4 \, a c} b}{c x^{2 \, n} + b x^{n} + a}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2 \, n} + b x^{n} + a\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} n}, \frac {2 \, {\left (b^{2} - 4 \, a c\right )} n \log \left (x\right ) + 2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {2 \, \sqrt {-b^{2} + 4 \, a c} c x^{n} + \sqrt {-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2 \, n} + b x^{n} + a\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} n}\right ] \]
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Timed out. \[ \int \frac {1}{x \left (a+b x^n+c x^{2 n}\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{x \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x} \,d x } \]
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\[ \int \frac {1}{x \left (a+b x^n+c x^{2 n}\right )} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x} \,d x } \]
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Time = 8.49 (sec) , antiderivative size = 224, normalized size of antiderivative = 3.03 \[ \int \frac {1}{x \left (a+b x^n+c x^{2 n}\right )} \, dx=\frac {\ln \left (-\frac {1}{c\,x}-\frac {\left (2\,a\,n+b\,n\,x^n\right )\,\left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2\right )}{2\,c\,x\,\left (a\,b^2\,n-4\,a^2\,c\,n\right )}\right )\,\left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2\right )}{2\,\left (a\,b^2\,n-4\,a^2\,c\,n\right )}-\frac {\ln \left (\frac {\left (2\,a\,n+b\,n\,x^n\right )\,\left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2\right )}{2\,c\,x\,\left (a\,b^2\,n-4\,a^2\,c\,n\right )}-\frac {1}{c\,x}\right )\,\left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2\right )}{2\,\left (a\,b^2\,n-4\,a^2\,c\,n\right )}+\frac {\ln \left (x\right )\,\left (n-1\right )}{a\,n} \]
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