Integrand size = 24, antiderivative size = 111 \[ \int \frac {x^{-1+4 n}}{a+b x^n+c x^{2 n}} \, dx=-\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n}+\frac {b \left (b^2-3 a c\right ) \text {arctanh}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} n}+\frac {\left (b^2-a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 c^3 n} \]
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Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1371, 715, 648, 632, 212, 642} \[ \int \frac {x^{-1+4 n}}{a+b x^n+c x^{2 n}} \, dx=\frac {b \left (b^2-3 a c\right ) \text {arctanh}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{c^3 n \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 c^3 n}-\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 715
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{a+b x+c x^2} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {b}{c^2}+\frac {x}{c}+\frac {a b+\left (b^2-a c\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^n\right )}{n} \\ & = -\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n}+\frac {\text {Subst}\left (\int \frac {a b+\left (b^2-a c\right ) x}{a+b x+c x^2} \, dx,x,x^n\right )}{c^2 n} \\ & = -\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n}-\frac {\left (b \left (b^2-3 a c\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^n\right )}{2 c^3 n}+\frac {\left (b^2-a c\right ) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^n\right )}{2 c^3 n} \\ & = -\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n}+\frac {\left (b^2-a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 c^3 n}+\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^n\right )}{c^3 n} \\ & = -\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c} n}+\frac {\left (b^2-a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 c^3 n} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87 \[ \int \frac {x^{-1+4 n}}{a+b x^n+c x^{2 n}} \, dx=\frac {c x^n \left (-2 b+c x^n\right )-\frac {2 b \left (b^2-3 a c\right ) \arctan \left (\frac {b+2 c x^n}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\left (b^2-a c\right ) \log \left (a+x^n \left (b+c x^n\right )\right )}{2 c^3 n} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(972\) vs. \(2(103)=206\).
Time = 0.35 (sec) , antiderivative size = 973, normalized size of antiderivative = 8.77
method | result | size |
risch | \(-\frac {\ln \left (x \right ) a}{c^{2}}+\frac {\ln \left (x \right ) b^{2}}{c^{3}}+\frac {x^{2 n}}{2 c n}-\frac {b \,x^{n}}{c^{2} n}+\frac {4 n^{2} \ln \left (x \right ) a^{2} c^{2}}{4 a \,c^{4} n^{2}-b^{2} c^{3} n^{2}}-\frac {5 n^{2} \ln \left (x \right ) a \,b^{2} c}{4 a \,c^{4} n^{2}-b^{2} c^{3} n^{2}}+\frac {n^{2} \ln \left (x \right ) b^{4}}{4 a \,c^{4} n^{2}-b^{2} c^{3} n^{2}}-\frac {2 \ln \left (x^{n}+\frac {3 a \,b^{2} c -b^{4}+\sqrt {-36 a^{3} b^{2} c^{3}+33 a^{2} b^{4} c^{2}-10 a \,b^{6} c +b^{8}}}{2 c b \left (3 a c -b^{2}\right )}\right ) a^{2}}{c \left (4 a c -b^{2}\right ) n}+\frac {5 \ln \left (x^{n}+\frac {3 a \,b^{2} c -b^{4}+\sqrt {-36 a^{3} b^{2} c^{3}+33 a^{2} b^{4} c^{2}-10 a \,b^{6} c +b^{8}}}{2 c b \left (3 a c -b^{2}\right )}\right ) a \,b^{2}}{2 c^{2} \left (4 a c -b^{2}\right ) n}-\frac {\ln \left (x^{n}+\frac {3 a \,b^{2} c -b^{4}+\sqrt {-36 a^{3} b^{2} c^{3}+33 a^{2} b^{4} c^{2}-10 a \,b^{6} c +b^{8}}}{2 c b \left (3 a c -b^{2}\right )}\right ) b^{4}}{2 c^{3} \left (4 a c -b^{2}\right ) n}+\frac {\ln \left (x^{n}+\frac {3 a \,b^{2} c -b^{4}+\sqrt {-36 a^{3} b^{2} c^{3}+33 a^{2} b^{4} c^{2}-10 a \,b^{6} c +b^{8}}}{2 c b \left (3 a c -b^{2}\right )}\right ) \sqrt {-36 a^{3} b^{2} c^{3}+33 a^{2} b^{4} c^{2}-10 a \,b^{6} c +b^{8}}}{2 c^{3} \left (4 a c -b^{2}\right ) n}-\frac {2 \ln \left (x^{n}-\frac {-3 a \,b^{2} c +b^{4}+\sqrt {-36 a^{3} b^{2} c^{3}+33 a^{2} b^{4} c^{2}-10 a \,b^{6} c +b^{8}}}{2 c b \left (3 a c -b^{2}\right )}\right ) a^{2}}{c \left (4 a c -b^{2}\right ) n}+\frac {5 \ln \left (x^{n}-\frac {-3 a \,b^{2} c +b^{4}+\sqrt {-36 a^{3} b^{2} c^{3}+33 a^{2} b^{4} c^{2}-10 a \,b^{6} c +b^{8}}}{2 c b \left (3 a c -b^{2}\right )}\right ) a \,b^{2}}{2 c^{2} \left (4 a c -b^{2}\right ) n}-\frac {\ln \left (x^{n}-\frac {-3 a \,b^{2} c +b^{4}+\sqrt {-36 a^{3} b^{2} c^{3}+33 a^{2} b^{4} c^{2}-10 a \,b^{6} c +b^{8}}}{2 c b \left (3 a c -b^{2}\right )}\right ) b^{4}}{2 c^{3} \left (4 a c -b^{2}\right ) n}-\frac {\ln \left (x^{n}-\frac {-3 a \,b^{2} c +b^{4}+\sqrt {-36 a^{3} b^{2} c^{3}+33 a^{2} b^{4} c^{2}-10 a \,b^{6} c +b^{8}}}{2 c b \left (3 a c -b^{2}\right )}\right ) \sqrt {-36 a^{3} b^{2} c^{3}+33 a^{2} b^{4} c^{2}-10 a \,b^{6} c +b^{8}}}{2 c^{3} \left (4 a c -b^{2}\right ) n}\) | \(973\) |
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none
Time = 0.27 (sec) , antiderivative size = 353, normalized size of antiderivative = 3.18 \[ \int \frac {x^{-1+4 n}}{a+b x^n+c x^{2 n}} \, dx=\left [-\frac {{\left (b^{3} - 3 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2 \, n} + b^{2} - 2 \, a c + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} x^{n} - \sqrt {b^{2} - 4 \, a c} b}{c x^{2 \, n} + b x^{n} + a}\right ) - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2 \, n} + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{n} - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{2 \, n} + b x^{n} + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} n}, \frac {2 \, {\left (b^{3} - 3 \, a b c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {2 \, \sqrt {-b^{2} + 4 \, a c} c x^{n} + \sqrt {-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2 \, n} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{n} + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{2 \, n} + b x^{n} + a\right )}{2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} n}\right ] \]
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Timed out. \[ \int \frac {x^{-1+4 n}}{a+b x^n+c x^{2 n}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^{-1+4 n}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{4 \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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\[ \int \frac {x^{-1+4 n}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{4 \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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Timed out. \[ \int \frac {x^{-1+4 n}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{4\,n-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \]
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