Optimal. Leaf size=19 \[ \frac {\log \left (a+b x^n+c x^{2 n}\right )}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {1482, 642}
\begin {gather*} \frac {\log \left (a+b x^n+c x^{2 n}\right )}{n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 19, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 24, normalized size = 1.26
method | result | size |
norman | \(\frac {\ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}+c \,{\mathrm e}^{2 n \ln \left (x \right )}\right )}{n}\) | \(24\) |
risch | \(\frac {\ln \left (x^{2 n}+\frac {b \,x^{n}}{c}+\frac {a}{c}\right )}{n}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 23, normalized size = 1.21 \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.34, size = 19, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.38, size = 19, 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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.32, size = 121, normalized size = 6.37 \begin {gather*} -\frac {2\,b\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x^n}{\sqrt {4\,a\,c-b^2}}\right )-\ln \left (a+b\,x^n+c\,x^{2\,n}\right )\,\sqrt {4\,a\,c-b^2}}{n\,\sqrt {4\,a\,c-b^2}}-\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.
[In]
[Out]
________________________________________________________________________________________