Optimal. Leaf size=43 \[ \frac {4 \log \left (b x^n+2\right )}{b^3 n}-\frac {2 x^n}{b^2 n}+\frac {x^{2 n}}{2 b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ -\frac {2 x^n}{b^2 n}+\frac {4 \log \left (b x^n+2\right )}{b^3 n}+\frac {x^{2 n}}{2 b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1+3 n}}{2+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{2+b x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {2}{b^2}+\frac {x}{b}+\frac {4}{b^2 (2+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {2 x^n}{b^2 n}+\frac {x^{2 n}}{2 b n}+\frac {4 \log \left (2+b x^n\right )}{b^3 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 33, normalized size = 0.77 \[ \frac {b x^n \left (b x^n-4\right )+8 \log \left (b x^n+2\right )}{2 b^3 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 34, normalized size = 0.79 \[ \frac {b^{2} x^{2 \, n} - 4 \, b x^{n} + 8 \, \log \left (b x^{n} + 2\right )}{2 \, b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3 \, n - 1}}{b x^{n} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 48, normalized size = 1.12 \[ \frac {{\mathrm e}^{2 n \ln \relax (x )}}{2 b n}-\frac {2 \,{\mathrm e}^{n \ln \relax (x )}}{b^{2} n}+\frac {4 \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+2\right )}{b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.54, size = 42, normalized size = 0.98 \[ \frac {b x^{2 \, n} - 4 \, x^{n}}{2 \, b^{2} n} + \frac {4 \, \log \left (\frac {b x^{n} + 2}{b}\right )}{b^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^{3\,n-1}}{b\,x^n+2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 19.57, size = 53, normalized size = 1.23 \[ \begin {cases} \frac {\log {\relax (x )}}{2} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{b + 2} & \text {for}\: n = 0 \\\frac {x^{3 n}}{6 n} & \text {for}\: b = 0 \\\frac {x^{2 n}}{2 b n} - \frac {2 x^{n}}{b^{2} n} + \frac {4 \log {\left (x^{n} + \frac {2}{b} \right )}}{b^{3} n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________