Optimal. Leaf size=85 \[ \frac{2 a^2 \sqrt{a+b x^n}}{n}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n}+\frac{2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 n} \]
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Rubi [A] time = 0.0414165, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ \frac{2 a^2 \sqrt{a+b x^n}}{n}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n}+\frac{2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^n\right )^{5/2}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{2 \left (a+b x^n\right )^{5/2}}{5 n}+\frac{a \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 n}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{2 a^2 \sqrt{a+b x^n}}{n}+\frac{2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 n}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{n}\\ &=\frac{2 a^2 \sqrt{a+b x^n}}{n}+\frac{2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 n}+\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^n}\right )}{b n}\\ &=\frac{2 a^2 \sqrt{a+b x^n}}{n}+\frac{2 a \left (a+b x^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b x^n\right )^{5/2}}{5 n}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0330789, size = 69, normalized size = 0.81 \[ \frac{2 \sqrt{a+b x^n} \left (23 a^2+11 a b x^n+3 b^2 x^{2 n}\right )-30 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{15 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 62, normalized size = 0.7 \begin{align*}{\frac{1}{n} \left ({\frac{2}{5} \left ( a+b{x}^{n} \right ) ^{{\frac{5}{2}}}}+{\frac{2\,a}{3} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}}+2\,{a}^{2}\sqrt{a+b{x}^{n}}-2\,{a}^{5/2}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) \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 = 1.08146, size = 339, normalized size = 3.99 \begin{align*} \left [\frac{15 \, a^{\frac{5}{2}} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \,{\left (3 \, b^{2} x^{2 \, n} + 11 \, a b x^{n} + 23 \, a^{2}\right )} \sqrt{b x^{n} + a}}{15 \, n}, \frac{2 \,{\left (15 \, \sqrt{-a} a^{2} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) +{\left (3 \, b^{2} x^{2 \, n} + 11 \, a b x^{n} + 23 \, a^{2}\right )} \sqrt{b x^{n} + a}\right )}}{15 \, n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.1493, size = 117, normalized size = 1.38 \begin{align*} \frac{46 a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{n}}{a}}}{15 n} + \frac{a^{\frac{5}{2}} \log{\left (\frac{b x^{n}}{a} \right )}}{n} - \frac{2 a^{\frac{5}{2}} \log{\left (\sqrt{1 + \frac{b x^{n}}{a}} + 1 \right )}}{n} + \frac{22 a^{\frac{3}{2}} b x^{n} \sqrt{1 + \frac{b x^{n}}{a}}}{15 n} + \frac{2 \sqrt{a} b^{2} x^{2 n} \sqrt{1 + \frac{b x^{n}}{a}}}{5 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{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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