Optimal. Leaf size=42 \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b^2 n}-\frac{2 a \sqrt{a+b x^n}}{b^2 n} \]
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Rubi [A] time = 0.0202429, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {266, 43} \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b^2 n}-\frac{2 a \sqrt{a+b x^n}}{b^2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{-1+2 n}}{\sqrt{a+b x^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x}} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{2 a \sqrt{a+b x^n}}{b^2 n}+\frac{2 \left (a+b x^n\right )^{3/2}}{3 b^2 n}\\ \end{align*}
Mathematica [A] time = 0.0130164, size = 30, normalized size = 0.71 \[ \frac{2 \left (b x^n-2 a\right ) \sqrt{a+b x^n}}{3 b^2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 28, normalized size = 0.7 \begin{align*} -{\frac{-2\,b{x}^{n}+4\,a}{3\,{b}^{2}n}\sqrt{a+b{x}^{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983788, size = 53, normalized size = 1.26 \begin{align*} \frac{2 \,{\left (b^{2} x^{2 \, n} - a b x^{n} - 2 \, a^{2}\right )}}{3 \, \sqrt{b x^{n} + a} b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03181, size = 58, normalized size = 1.38 \begin{align*} \frac{2 \, \sqrt{b x^{n} + a}{\left (b x^{n} - 2 \, a\right )}}{3 \, b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 22.0367, size = 275, normalized size = 6.55 \begin{align*} - \frac{4 a^{\frac{7}{2}} b^{\frac{3}{2}} x^{\frac{3 n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} n x^{n} + 3 a^{\frac{3}{2}} b^{4} n x^{2 n}} - \frac{2 a^{\frac{5}{2}} b^{\frac{5}{2}} x^{\frac{5 n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} n x^{n} + 3 a^{\frac{3}{2}} b^{4} n x^{2 n}} + \frac{2 a^{\frac{3}{2}} b^{\frac{7}{2}} x^{\frac{7 n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} n x^{n} + 3 a^{\frac{3}{2}} b^{4} n x^{2 n}} + \frac{4 a^{4} b x^{n}}{3 a^{\frac{5}{2}} b^{3} n x^{n} + 3 a^{\frac{3}{2}} b^{4} n x^{2 n}} + \frac{4 a^{3} b^{2} x^{2 n}}{3 a^{\frac{5}{2}} b^{3} n x^{n} + 3 a^{\frac{3}{2}} b^{4} n x^{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^{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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