Integrand size = 24, antiderivative size = 164 \[ \int \frac {x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx=-\frac {x^{-3 n}}{3 a n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (b^2-a c\right ) x^{-n}}{a^3 n}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) \text {arctanh}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c} n}-\frac {b \left (b^2-2 a c\right ) \log (x)}{a^4}+\frac {b \left (b^2-2 a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^4 n} \]
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Time = 0.17 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1371, 723, 814, 648, 632, 212, 642} \[ \int \frac {x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx=\frac {b \left (b^2-2 a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^4 n}-\frac {b \log (x) \left (b^2-2 a c\right )}{a^4}-\frac {x^{-n} \left (b^2-a c\right )}{a^3 n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \text {arctanh}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{a^4 n \sqrt {b^2-4 a c}}-\frac {x^{-3 n}}{3 a n} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 723
Rule 814
Rule 1371
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^4 \left (a+b x+c x^2\right )} \, dx,x,x^n\right )}{n} \\ & = -\frac {x^{-3 n}}{3 a n}+\frac {\text {Subst}\left (\int \frac {-b-c x}{x^3 \left (a+b x+c x^2\right )} \, dx,x,x^n\right )}{a n} \\ & = -\frac {x^{-3 n}}{3 a n}+\frac {\text {Subst}\left (\int \left (-\frac {b}{a x^3}+\frac {b^2-a c}{a^2 x^2}+\frac {-b^3+2 a b c}{a^3 x}+\frac {b^4-3 a b^2 c+a^2 c^2+b c \left (b^2-2 a c\right ) x}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^n\right )}{a n} \\ & = -\frac {x^{-3 n}}{3 a n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (b^2-a c\right ) x^{-n}}{a^3 n}-\frac {b \left (b^2-2 a c\right ) \log (x)}{a^4}+\frac {\text {Subst}\left (\int \frac {b^4-3 a b^2 c+a^2 c^2+b c \left (b^2-2 a c\right ) x}{a+b x+c x^2} \, dx,x,x^n\right )}{a^4 n} \\ & = -\frac {x^{-3 n}}{3 a n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (b^2-a c\right ) x^{-n}}{a^3 n}-\frac {b \left (b^2-2 a c\right ) \log (x)}{a^4}+\frac {\left (b \left (b^2-2 a c\right )\right ) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^n\right )}{2 a^4 n}+\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^n\right )}{2 a^4 n} \\ & = -\frac {x^{-3 n}}{3 a n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (b^2-a c\right ) x^{-n}}{a^3 n}-\frac {b \left (b^2-2 a c\right ) \log (x)}{a^4}+\frac {b \left (b^2-2 a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^4 n}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^n\right )}{a^4 n} \\ & = -\frac {x^{-3 n}}{3 a n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (b^2-a c\right ) x^{-n}}{a^3 n}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c} n}-\frac {b \left (b^2-2 a c\right ) \log (x)}{a^4}+\frac {b \left (b^2-2 a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^4 n} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.90 \[ \int \frac {x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx=\frac {a x^{-3 n} \left (-2 a^2-6 b^2 x^{2 n}+3 a x^n \left (b+2 c x^n\right )\right )+\frac {6 \left (b^4-4 a b^2 c+2 a^2 c^2\right ) \arctan \left (\frac {b+2 c x^n}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-6 \left (b^3-2 a b c\right ) \log \left (x^n\right )+3 \left (b^3-2 a b c\right ) \log \left (a+x^n \left (b+c x^n\right )\right )}{6 a^4 n} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1299\) vs. \(2(158)=316\).
Time = 0.37 (sec) , antiderivative size = 1300, normalized size of antiderivative = 7.93
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none
Time = 0.28 (sec) , antiderivative size = 522, normalized size of antiderivative = 3.18 \[ \int \frac {x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx=\left [-\frac {2 \, a^{3} b^{2} - 8 \, a^{4} c + 6 \, {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} n x^{3 \, n} \log \left (x\right ) - 3 \, {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} x^{3 \, n} \log \left (\frac {2 \, c^{2} x^{2 \, n} + b^{2} - 2 \, a c + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} x^{n} - \sqrt {b^{2} - 4 \, a c} b}{c x^{2 \, n} + b x^{n} + a}\right ) - 3 \, {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} x^{3 \, n} \log \left (c x^{2 \, n} + b x^{n} + a\right ) + 6 \, {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} x^{2 \, n} - 3 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{n}}{6 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} n x^{3 \, n}}, -\frac {2 \, a^{3} b^{2} - 8 \, a^{4} c + 6 \, {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} n x^{3 \, n} \log \left (x\right ) + 6 \, {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} x^{3 \, n} \arctan \left (-\frac {2 \, \sqrt {-b^{2} + 4 \, a c} c x^{n} + \sqrt {-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) - 3 \, {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} x^{3 \, n} \log \left (c x^{2 \, n} + b x^{n} + a\right ) + 6 \, {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} x^{2 \, n} - 3 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{n}}{6 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} n x^{3 \, n}}\right ] \]
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Timed out. \[ \int \frac {x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-3 \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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\[ \int \frac {x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{-3 \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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Timed out. \[ \int \frac {x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {1}{x^{3\,n+1}\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \]
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