Optimal. Leaf size=19 \[ \frac{\log \left (a+b x^n+c x^{2 n}\right )}{n} \]
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Rubi [A] time = 0.0269621, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1468, 628} \[ \frac{\log \left (a+b x^n+c x^{2 n}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 1468
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{-1+n} \left (b+2 c x^n\right )}{a+b x^n+c x^{2 n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\log \left (a+b x^n+c x^{2 n}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.109311, size = 19, normalized size = 1. \[ \frac{\log \left (a+b x^n+c x^{2 n}\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 24, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }}+c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2} \right ) }{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16091, size = 31, normalized size = 1.63 \begin{align*} \frac{\log \left (\frac{c x^{2 \, n} + b x^{n} + a}{c}\right )}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.10865, size = 41, normalized size = 2.16 \begin{align*} \frac{\log \left (c x^{2 \, n} + b x^{n} + a\right )}{n} \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 [A] time = 1.08196, size = 26, normalized size = 1.37 \begin{align*} \frac{\log \left (c x^{2 \, n} + b x^{n} + a\right )}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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