Optimal. Leaf size=68 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{c n \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x^n+c x^{2 n}\right )}{2 c n} \]
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Rubi [A] time = 0.0505273, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1357, 634, 618, 206, 628} \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{c n \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x^n+c x^{2 n}\right )}{2 c n} \]
Antiderivative was successfully verified.
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Rule 1357
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{x}{a+b x+c x^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^n\right )}{2 c n}-\frac{b \operatorname{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 \operatorname{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*}
Mathematica [A] time = 0.0673643, size = 62, normalized size = 0.91 \[ \frac{\frac{2 b \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}+\log \left (a+x^n \left (b+c x^n\right )\right )}{2 c n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.076, size = 402, normalized size = 5.9 \begin{align*}{\frac{\ln \left ( x \right ) }{c}}-4\,{\frac{{n}^{2}\ln \left ( x \right ) ac}{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}}}+2\,{\frac{a}{ \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}-1/2\,{\frac{-{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}}}{bc}} \right ) }-{\frac{{b}^{2}}{2\,c \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}-{\frac{1}{2\,bc} \left ( -{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) }+{\frac{1}{2\,c \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}-{\frac{1}{2\,bc} \left ( -{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) \sqrt{-4\,a{b}^{2}c+{b}^{4}}}+2\,{\frac{a}{ \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}+1/2\,{\frac{{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}}}{bc}} \right ) }-{\frac{{b}^{2}}{2\,c \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}+{\frac{1}{2\,bc} \left ({b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) }-{\frac{1}{2\,c \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}+{\frac{1}{2\,bc} \left ({b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) \sqrt{-4\,a{b}^{2}c+{b}^{4}}} \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{\log \left (x\right )}{c} - \int \frac{b x^{n} + a}{c^{2} x x^{2 \, n} + b c x x^{n} + a c x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62732, size = 517, normalized size = 7.6 \begin{align*} \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 ] \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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