Optimal. Leaf size=71 \[ \frac {16 \log \left (b x^n+2\right )}{b^5 n}-\frac {8 x^n}{b^4 n}+\frac {2 x^{2 n}}{b^3 n}-\frac {2 x^{3 n}}{3 b^2 n}+\frac {x^{4 n}}{4 b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 71, 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} \[ -\frac {8 x^n}{b^4 n}+\frac {2 x^{2 n}}{b^3 n}-\frac {2 x^{3 n}}{3 b^2 n}+\frac {16 \log \left (b x^n+2\right )}{b^5 n}+\frac {x^{4 n}}{4 b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1+5 n}}{2+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{2+b x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {8}{b^4}+\frac {4 x}{b^3}-\frac {2 x^2}{b^2}+\frac {x^3}{b}+\frac {16}{b^4 (2+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {8 x^n}{b^4 n}+\frac {2 x^{2 n}}{b^3 n}-\frac {2 x^{3 n}}{3 b^2 n}+\frac {x^{4 n}}{4 b n}+\frac {16 \log \left (2+b x^n\right )}{b^5 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 54, normalized size = 0.76 \[ \frac {b x^n \left (3 b^3 x^{3 n}-8 b^2 x^{2 n}+24 b x^n-96\right )+192 \log \left (b x^n+2\right )}{12 b^5 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 55, normalized size = 0.77 \[ \frac {3 \, b^{4} x^{4 \, n} - 8 \, b^{3} x^{3 \, n} + 24 \, b^{2} x^{2 \, n} - 96 \, b x^{n} + 192 \, \log \left (b x^{n} + 2\right )}{12 \, b^{5} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5 \, n - 1}}{b x^{n} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 78, normalized size = 1.10 \[ \frac {{\mathrm e}^{4 n \ln \relax (x )}}{4 b n}-\frac {2 \,{\mathrm e}^{3 n \ln \relax (x )}}{3 b^{2} n}+\frac {2 \,{\mathrm e}^{2 n \ln \relax (x )}}{b^{3} n}-\frac {8 \,{\mathrm e}^{n \ln \relax (x )}}{b^{4} n}+\frac {16 \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+2\right )}{b^{5} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 63, normalized size = 0.89 \[ \frac {3 \, b^{3} x^{4 \, n} - 8 \, b^{2} x^{3 \, n} + 24 \, b x^{2 \, n} - 96 \, x^{n}}{12 \, b^{4} n} + \frac {16 \, \log \left (\frac {b x^{n} + 2}{b}\right )}{b^{5} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{5\,n-1}}{b\,x^n+2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 76.52, size = 78, normalized size = 1.10 \[ \begin {cases} \frac {\log {\relax (x )}}{2} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{b + 2} & \text {for}\: n = 0 \\\frac {x^{5 n}}{10 n} & \text {for}\: b = 0 \\\frac {x^{4 n}}{4 b n} - \frac {2 x^{3 n}}{3 b^{2} n} + \frac {2 x^{2 n}}{b^{3} n} - \frac {8 x^{n}}{b^{4} n} + \frac {16 \log {\left (x^{n} + \frac {2}{b} \right )}}{b^{5} n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________