Optimal. Leaf size=62 \[ \frac{x^{n/2} \sqrt{a+b x^n}}{b n}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{b^{3/2} n} \]
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Rubi [A] time = 0.0265925, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {355, 288, 206} \[ \frac{x^{n/2} \sqrt{a+b x^n}}{b n}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{b^{3/2} n} \]
Antiderivative was successfully verified.
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Rule 355
Rule 288
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{-1+\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx &=\frac{(2 a) \operatorname{Subst}\left (\int \frac{x^2}{\left (1-b x^2\right )^2} \, dx,x,\frac{x^{n/2}}{\sqrt{a+b x^n}}\right )}{n}\\ &=\frac{x^{n/2} \sqrt{a+b x^n}}{b n}-\frac{a \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^{n/2}}{\sqrt{a+b x^n}}\right )}{b n}\\ &=\frac{x^{n/2} \sqrt{a+b x^n}}{b n}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{b^{3/2} n}\\ \end{align*}
Mathematica [A] time = 0.0504219, size = 81, normalized size = 1.31 \[ \frac{\sqrt{b} x^{n/2} \left (a+b x^n\right )-a^{3/2} \sqrt{\frac{b x^n}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a}}\right )}{b^{3/2} n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 64, normalized size = 1. \begin{align*}{\frac{1}{nb}{{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}}\sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}}}-{\frac{a}{n}\ln \left ( \sqrt{b}{{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}}+\sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}} \right ){b}^{-{\frac{3}{2}}}} \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 [A] time = 1.46157, size = 285, normalized size = 4.6 \begin{align*} \left [\frac{2 \, \sqrt{b x^{n} + a} b x^{\frac{1}{2} \, n} + a \sqrt{b} \log \left (2 \, \sqrt{b x^{n} + a} \sqrt{b} x^{\frac{1}{2} \, n} - 2 \, b x^{n} - a\right )}{2 \, b^{2} n}, \frac{\sqrt{b x^{n} + a} b x^{\frac{1}{2} \, n} + a \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right )}{b^{2} n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 18.4395, size = 49, normalized size = 0.79 \begin{align*} \frac{\sqrt{a} x^{\frac{n}{2}} \sqrt{1 + \frac{b x^{n}}{a}}}{b n} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a}} \right )}}{b^{\frac{3}{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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