Optimal. Leaf size=58 \[ \frac{4 b x^{-n/2} \sqrt{a+b x^n}}{3 a^2 n}-\frac{2 x^{-3 n/2} \sqrt{a+b x^n}}{3 a n} \]
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Rubi [A] time = 0.0146927, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {271, 264} \[ \frac{4 b x^{-n/2} \sqrt{a+b x^n}}{3 a^2 n}-\frac{2 x^{-3 n/2} \sqrt{a+b x^n}}{3 a n} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{x^{-1-\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx &=-\frac{2 x^{-3 n/2} \sqrt{a+b x^n}}{3 a n}-\frac{(2 b) \int \frac{x^{-1-\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx}{3 a}\\ &=-\frac{2 x^{-3 n/2} \sqrt{a+b x^n}}{3 a n}+\frac{4 b x^{-n/2} \sqrt{a+b x^n}}{3 a^2 n}\\ \end{align*}
Mathematica [A] time = 0.0132765, size = 36, normalized size = 0.62 \[ -\frac{2 x^{-3 n/2} \left (a-2 b x^n\right ) \sqrt{a+b x^n}}{3 a^2 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1-{\frac{3\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{3}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.8277, size = 51, normalized size = 0.88 \begin{align*} - \frac{2 \sqrt{b} x^{- n} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a n} + \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{3}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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