Optimal. Leaf size=126 \[ -\frac{\left (b^2-a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^3 n}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{a^3 n \sqrt{b^2-4 a c}}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b x^{-n}}{a^2 n}-\frac{x^{-2 n}}{2 a n} \]
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Rubi [A] time = 0.170971, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1357, 709, 800, 634, 618, 206, 628} \[ -\frac{\left (b^2-a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^3 n}+\frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{a^3 n \sqrt{b^2-4 a c}}+\frac{\log (x) \left (b^2-a c\right )}{a^3}+\frac{b x^{-n}}{a^2 n}-\frac{x^{-2 n}}{2 a n} \]
Antiderivative was successfully verified.
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Rule 1357
Rule 709
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{-1-2 n}}{a+b x^n+c x^{2 n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x+c x^2\right )} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-2 n}}{2 a n}+\frac{\operatorname{Subst}\left (\int \frac{-b-c x}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^n\right )}{a n}\\ &=-\frac{x^{-2 n}}{2 a n}+\frac{\operatorname{Subst}\left (\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,x,x^n\right )}{a n}\\ &=-\frac{x^{-2 n}}{2 a n}+\frac{b x^{-n}}{a^2 n}+\frac{\left (b^2-a c\right ) \log (x)}{a^3}+\frac{\operatorname{Subst}\left (\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,x,x^n\right )}{a^3 n}\\ &=-\frac{x^{-2 n}}{2 a n}+\frac{b x^{-n}}{a^2 n}+\frac{\left (b^2-a c\right ) \log (x)}{a^3}-\frac{\left (b \left (b^2-3 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^n\right )}{2 a^3 n}-\frac{\left (b^2-a c\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^n\right )}{2 a^3 n}\\ &=-\frac{x^{-2 n}}{2 a n}+\frac{b x^{-n}}{a^2 n}+\frac{\left (b^2-a c\right ) \log (x)}{a^3}-\frac{\left (b^2-a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^3 n}+\frac{\left (b \left (b^2-3 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^n\right )}{a^3 n}\\ &=-\frac{x^{-2 n}}{2 a n}+\frac{b x^{-n}}{a^2 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 )}{a^3 \sqrt{b^2-4 a c} n}+\frac{\left (b^2-a c\right ) \log (x)}{a^3}-\frac{\left (b^2-a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^3 n}\\ \end{align*}
Mathematica [A] time = 0.339076, size = 112, normalized size = 0.89 \[ \frac{-a^2 x^{-2 n}-\left (b^2-a c\right ) \log \left (a+x^n \left (b+c x^n\right )\right )+\frac{2 b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}+2 n \log (x) \left (b^2-a c\right )+2 a b x^{-n}}{2 a^3 n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.135, size = 958, normalized size = 7.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, b x^{n} - a}{2 \, a^{2} n x^{2 \, n}} + \int \frac{b c x^{n} + b^{2} - a c}{a^{2} c x x^{2 \, n} + a^{2} b x x^{n} + a^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67542, size = 938, normalized size = 7.44 \begin{align*} \left [-\frac{a^{2} b^{2} - 4 \, a^{3} c - 2 \,{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} n x^{2 \, n} \log \left (x\right ) +{\left (b^{3} - 3 \, a b c\right )} \sqrt{b^{2} - 4 \, a c} x^{2 \, 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 ) +{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2 \, n} \log \left (c x^{2 \, n} + b x^{n} + a\right ) - 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{n}}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n x^{2 \, n}}, -\frac{a^{2} b^{2} - 4 \, a^{3} c - 2 \,{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} n x^{2 \, n} \log \left (x\right ) - 2 \,{\left (b^{3} - 3 \, a b c\right )} \sqrt{-b^{2} + 4 \, a c} x^{2 \, 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 ) +{\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2 \, n} \log \left (c x^{2 \, n} + b x^{n} + a\right ) - 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{n}}{2 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} n x^{2 \, n}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-2 \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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