Optimal. Leaf size=21 \[ \frac{\log \left (a-b x^n-c x^{2 n}\right )}{n} \]
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Rubi [A] time = 0.0288613, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, 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 = 0.11436, size = 21, 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 = 0.02, size = 26, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( -c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-b{{\rm e}^{n\ln \left ( x \right ) }}+a \right ) }{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14785, size = 34, normalized size = 1.62 \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 = 1.20768, size = 41, normalized size = 1.95 \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 = 1.09546, size = 28, normalized size = 1.33 \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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