Optimal. Leaf size=21 \[ \frac {\log \left (a-b x^n-c x^{2 n}\right )}{n} \]
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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {1482, 642}
\begin {gather*} \frac {\log \left (a-b x^n-c x^{2 n}\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 642
Rule 1482
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 {\text {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*}
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Mathematica [A]
time = 0.04, size = 21, normalized size = 1.00 \begin {gather*} \frac {\log \left (a-b x^n-c x^{2 n}\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 26, normalized size = 1.24
method | result | size |
norman | \(\frac {\ln \left (-c \,{\mathrm e}^{2 n \ln \left (x \right )}-b \,{\mathrm e}^{n \ln \left (x \right )}+a \right )}{n}\) | \(26\) |
risch | \(\frac {\ln \left (x^{2 n}+\frac {b \,x^{n}}{c}-\frac {a}{c}\right )}{n}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 25, normalized size = 1.19 \begin {gather*} \frac {\log \left (\frac {c x^{2 \, n} + b x^{n} - a}{c}\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 21, normalized size = 1.00 \begin {gather*} \frac {\log \left (c x^{2 \, n} + b x^{n} - a\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.71, size = 21, normalized size = 1.00 \begin {gather*} \frac {\log \left (c x^{2 \, n} + b x^{n} - a\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.68, size = 199, normalized size = 9.48 \begin {gather*} \ln \left (\frac {2\,c\,x^n}{n}-\left (\frac {1}{n}+\frac {b\,\sqrt {b^2+4\,a\,c}}{n\,b^2+4\,a\,c\,n}\right )\,\left (b+2\,c\,x^n\right )\right )\,\left (\frac {1}{n}+\frac {b\,\sqrt {b^2+4\,a\,c}}{n\,b^2+4\,a\,c\,n}\right )+\ln \left (\frac {2\,c\,x^n}{n}-\left (\frac {1}{n}-\frac {b\,\sqrt {b^2+4\,a\,c}}{n\,b^2+4\,a\,c\,n}\right )\,\left (b+2\,c\,x^n\right )\right )\,\left (\frac {1}{n}-\frac {b\,\sqrt {b^2+4\,a\,c}}{n\,b^2+4\,a\,c\,n}\right )-\frac {2\,b\,\mathrm {atanh}\left (\frac {b+2\,c\,x^n}{\sqrt {b^2+4\,a\,c}}\right )}{n\,\sqrt {b^2+4\,a\,c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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