Optimal. Leaf size=45 \[ -\frac {a^2 x^{-6 n}}{6 n}-\frac {2 a b x^{-5 n}}{5 n}-\frac {b^2 x^{-4 n}}{4 n} \]
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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} \[ -\frac {a^2 x^{-6 n}}{6 n}-\frac {2 a b x^{-5 n}}{5 n}-\frac {b^2 x^{-4 n}}{4 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-6 n} \left (a+b x^n\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^7} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{x^7}+\frac {2 a b}{x^6}+\frac {b^2}{x^5}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^2 x^{-6 n}}{6 n}-\frac {2 a b x^{-5 n}}{5 n}-\frac {b^2 x^{-4 n}}{4 n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.78 \[ -\frac {x^{-6 n} \left (10 a^2+24 a b x^n+15 b^2 x^{2 n}\right )}{60 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 35, normalized size = 0.78 \[ -\frac {15 \, b^{2} x^{2 \, n} + 24 \, a b x^{n} + 10 \, a^{2}}{60 \, n x^{6 \, n}} \]
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 \[ -\frac {15 \, b^{2} x^{2 \, n} + 24 \, a b x^{n} + 10 \, a^{2}}{60 \, n x^{6 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 40, normalized size = 0.89 \[ -\frac {a^{2} x^{-6 n}}{6 n}-\frac {2 a b \,x^{-5 n}}{5 n}-\frac {b^{2} x^{-4 n}}{4 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 45, normalized size = 1.00 \[ -\frac {a^{2}}{6 \, n x^{6 \, n}} - \frac {2 \, a b}{5 \, n x^{5 \, n}} - \frac {b^{2}}{4 \, n x^{4 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 35, normalized size = 0.78 \[ -\frac {10\,a^2+15\,b^2\,x^{2\,n}+24\,a\,b\,x^n}{60\,n\,x^{6\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.68, size = 46, normalized size = 1.02 \[ \begin {cases} - \frac {a^{2} x^{- 6 n}}{6 n} - \frac {2 a b x^{- 5 n}}{5 n} - \frac {b^{2} x^{- 4 n}}{4 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{2} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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