Optimal. Leaf size=45 \[ -\frac {a^2 x^{-5 n}}{5 n}-\frac {a b x^{-4 n}}{2 n}-\frac {b^2 x^{-3 n}}{3 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 45, 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} \[ -\frac {a^2 x^{-5 n}}{5 n}-\frac {a b x^{-4 n}}{2 n}-\frac {b^2 x^{-3 n}}{3 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-5 n} \left (a+b x^n\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^6} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{x^6}+\frac {2 a b}{x^5}+\frac {b^2}{x^4}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^2 x^{-5 n}}{5 n}-\frac {a b x^{-4 n}}{2 n}-\frac {b^2 x^{-3 n}}{3 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 35, normalized size = 0.78 \[ -\frac {x^{-5 n} \left (6 a^2+15 a b x^n+10 b^2 x^{2 n}\right )}{30 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 35, normalized size = 0.78 \[ -\frac {10 \, b^{2} x^{2 \, n} + 15 \, a b x^{n} + 6 \, a^{2}}{30 \, n x^{5 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 35, normalized size = 0.78 \[ -\frac {10 \, b^{2} x^{2 \, n} + 15 \, a b x^{n} + 6 \, a^{2}}{30 \, n x^{5 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 45, normalized size = 1.00 \[ \left (-\frac {a b \,{\mathrm e}^{n \ln \relax (x )}}{2 n}-\frac {b^{2} {\mathrm e}^{2 n \ln \relax (x )}}{3 n}-\frac {a^{2}}{5 n}\right ) {\mathrm e}^{-5 n \ln \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.46, size = 45, normalized size = 1.00 \[ -\frac {a^{2}}{5 \, n x^{5 \, n}} - \frac {a b}{2 \, n x^{4 \, n}} - \frac {b^{2}}{3 \, n x^{3 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.28, size = 35, normalized size = 0.78 \[ -\frac {6\,a^2+10\,b^2\,x^{2\,n}+15\,a\,b\,x^n}{30\,n\,x^{5\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 11.33, size = 44, normalized size = 0.98 \[ \begin {cases} - \frac {a^{2} x^{- 5 n}}{5 n} - \frac {a b x^{- 4 n}}{2 n} - \frac {b^{2} x^{- 3 n}}{3 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{2} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________