Optimal. Leaf size=45 \[ -\frac {a^2 x^{-4 n}}{4 n}-\frac {2 a b x^{-3 n}}{3 n}-\frac {b^2 x^{-2 n}}{2 n} \]
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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} \[ -\frac {a^2 x^{-4 n}}{4 n}-\frac {2 a b x^{-3 n}}{3 n}-\frac {b^2 x^{-2 n}}{2 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^5} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{x^5}+\frac {2 a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^2 x^{-4 n}}{4 n}-\frac {2 a b x^{-3 n}}{3 n}-\frac {b^2 x^{-2 n}}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.78 \[ -\frac {x^{-4 n} \left (3 a^2+8 a b x^n+6 b^2 x^{2 n}\right )}{12 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 35, normalized size = 0.78 \[ -\frac {6 \, b^{2} x^{2 \, n} + 8 \, a b x^{n} + 3 \, a^{2}}{12 \, n x^{4 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 35, normalized size = 0.78 \[ -\frac {6 \, b^{2} x^{2 \, n} + 8 \, a b x^{n} + 3 \, a^{2}}{12 \, n x^{4 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 45, normalized size = 1.00 \[ \left (-\frac {2 a b \,{\mathrm e}^{n \ln \relax (x )}}{3 n}-\frac {b^{2} {\mathrm e}^{2 n \ln \relax (x )}}{2 n}-\frac {a^{2}}{4 n}\right ) {\mathrm e}^{-4 n \ln \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 45, normalized size = 1.00 \[ -\frac {a^{2}}{4 \, n x^{4 \, n}} - \frac {2 \, a b}{3 \, n x^{3 \, n}} - \frac {b^{2}}{2 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 35, normalized size = 0.78 \[ -\frac {3\,a^2+6\,b^2\,x^{2\,n}+8\,a\,b\,x^n}{12\,n\,x^{4\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.47, size = 46, normalized size = 1.02 \[ \begin {cases} - \frac {a^{2} x^{- 4 n}}{4 n} - \frac {2 a b x^{- 3 n}}{3 n} - \frac {b^{2} x^{- 2 n}}{2 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{2} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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