Integrand size = 24, antiderivative size = 68 \[ \int \frac {x^{-1+2 n}}{a+b x^n+c x^{2 n}} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c} n}+\frac {\log \left (a+b x^n+c x^{2 n}\right )}{2 c n} \]
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Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1371, 648, 632, 212, 642} \[ \int \frac {x^{-1+2 n}}{a+b x^n+c x^{2 n}} \, dx=\frac {b \text {arctanh}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{c n \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x^n+c x^{2 n}\right )}{2 c n} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x}{a+b x+c x^2} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^n\right )}{2 c n}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^n\right )}{2 c n} \\ & = \frac {\log \left (a+b x^n+c x^{2 n}\right )}{2 c n}+\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^n\right )}{c n} \\ & = \frac {b \tanh ^{-1}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c} n}+\frac {\log \left (a+b x^n+c x^{2 n}\right )}{2 c n} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97 \[ \int \frac {x^{-1+2 n}}{a+b x^n+c x^{2 n}} \, dx=\frac {-\frac {2 b \arctan \left (\frac {b+2 c x^n}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\log \left (a+x^n \left (b+c x^n\right )\right )}{2 c n} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(401\) vs. \(2(62)=124\).
Time = 0.24 (sec) , antiderivative size = 402, normalized size of antiderivative = 5.91
method | result | size |
risch | \(\frac {\ln \left (x \right )}{c}-\frac {4 n^{2} \ln \left (x \right ) a c}{4 a \,c^{2} n^{2}-b^{2} c \,n^{2}}+\frac {n^{2} \ln \left (x \right ) b^{2}}{4 a \,c^{2} n^{2}-b^{2} c \,n^{2}}+\frac {2 \ln \left (x^{n}-\frac {-b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) a}{\left (4 a c -b^{2}\right ) n}-\frac {\ln \left (x^{n}-\frac {-b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) b^{2}}{2 c \left (4 a c -b^{2}\right ) n}+\frac {\ln \left (x^{n}-\frac {-b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) \sqrt {-4 a \,b^{2} c +b^{4}}}{2 c \left (4 a c -b^{2}\right ) n}+\frac {2 \ln \left (x^{n}+\frac {b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) a}{\left (4 a c -b^{2}\right ) n}-\frac {\ln \left (x^{n}+\frac {b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) b^{2}}{2 c \left (4 a c -b^{2}\right ) n}-\frac {\ln \left (x^{n}+\frac {b^{2}+\sqrt {-4 a \,b^{2} c +b^{4}}}{2 b c}\right ) \sqrt {-4 a \,b^{2} c +b^{4}}}{2 c \left (4 a c -b^{2}\right ) n}\) | \(402\) |
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none
Time = 0.26 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.40 \[ \int \frac {x^{-1+2 n}}{a+b x^n+c x^{2 n}} \, dx=\left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} x^{2 \, n} + b^{2} - 2 \, a c + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} x^{n} + \sqrt {b^{2} - 4 \, a c} b}{c x^{2 \, n} + b x^{n} + a}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2 \, n} + b x^{n} + a\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} n}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {2 \, \sqrt {-b^{2} + 4 \, a c} c x^{n} + \sqrt {-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2 \, n} + b x^{n} + a\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} n}\right ] \]
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Timed out. \[ \int \frac {x^{-1+2 n}}{a+b x^n+c x^{2 n}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^{-1+2 n}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{2 \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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\[ \int \frac {x^{-1+2 n}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{2 \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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Timed out. \[ \int \frac {x^{-1+2 n}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{2\,n-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \]
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