Optimal. Leaf size=30 \[ -\frac{a^2 x^{-n}}{n}+2 a b \log (x)+\frac{b^2 x^n}{n} \]
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Rubi [A] time = 0.0152437, antiderivative size = 30, 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} \[ -\frac{a^2 x^{-n}}{n}+2 a b \log (x)+\frac{b^2 x^n}{n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-n} \left (a+b x^n\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^2+\frac{a^2}{x^2}+\frac{2 a b}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^2 x^{-n}}{n}+\frac{b^2 x^n}{n}+2 a b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0164571, size = 29, normalized size = 0.97 \[ \frac{-a^2 x^{-n}+2 a b n \log (x)+b^2 x^n}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 43, normalized size = 1.4 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ({\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}+2\,ab\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}-{\frac{{a}^{2}}{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 = 1.04035, size = 68, normalized size = 2.27 \begin{align*} \frac{2 \, a b n x^{n} \log \left (x\right ) + b^{2} x^{2 \, n} - a^{2}}{n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 39.3753, size = 175, normalized size = 5.83 \begin{align*} \begin{cases} a^{2} x + 2 a b \log{\left (x \right )} - \frac{b^{2}}{x} & \text{for}\: n = -1 \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{a^{2} n}{n^{2} x^{n} + n x^{n}} - \frac{a^{2}}{n^{2} x^{n} + n x^{n}} + \frac{2 a b n^{2} x^{n} \log{\left (x \right )}}{n^{2} x^{n} + n x^{n}} + \frac{2 a b n x^{n} \log{\left (x \right )}}{n^{2} x^{n} + n x^{n}} + \frac{2 a b n x^{n}}{n^{2} x^{n} + n x^{n}} + \frac{b^{2} n x^{2 n}}{n^{2} x^{n} + n x^{n}} + \frac{b^{2} x^{2 n}}{n^{2} x^{n} + n x^{n}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18877, size = 46, normalized size = 1.53 \begin{align*} \frac{2 \, a b n x^{n} \log \left (x\right ) + b^{2} x^{2 \, n} - a^{2}}{n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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