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 = 0.0166628, antiderivative size = 45, normalized size of antiderivative = 1., 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 266
Rule 43
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*}
Mathematica [A] time = 0.0129687, 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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Maple [A] time = 0.015, size = 45, normalized size = 1. \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ( -{\frac{{a}^{2}}{4\,n}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}}-{\frac{2\,a{{\rm e}^{n\ln \left ( x \right ) }}b}{3\,n}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27133, size = 74, normalized size = 1.64 \begin{align*} -\frac{6 \, b^{2} x^{2 \, n} + 8 \, a b x^{n} + 3 \, a^{2}}{12 \, n x^{4 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.3454, size = 46, normalized size = 1.02 \begin{align*} \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{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16439, size = 47, normalized size = 1.04 \begin{align*} -\frac{6 \, b^{2} x^{2 \, n} + 8 \, a b x^{n} + 3 \, a^{2}}{12 \, n x^{4 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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