Optimal. Leaf size=73 \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{c n}+\frac{2 a \sqrt{a+b x^n}}{c n}+\frac{2 \left (a+b x^n\right )^{3/2}}{3 c n} \]
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Rubi [A] time = 0.0408936, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {12, 266, 50, 63, 208} \[ -\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{c n}+\frac{2 a \sqrt{a+b x^n}}{c n}+\frac{2 \left (a+b x^n\right )^{3/2}}{3 c n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^n\right )^{3/2}}{c x} \, dx &=\frac{\int \frac{\left (a+b x^n\right )^{3/2}}{x} \, dx}{c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,x^n\right )}{c n}\\ &=\frac{2 \left (a+b x^n\right )^{3/2}}{3 c n}+\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^n\right )}{c n}\\ &=\frac{2 a \sqrt{a+b x^n}}{c n}+\frac{2 \left (a+b x^n\right )^{3/2}}{3 c n}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{c n}\\ &=\frac{2 a \sqrt{a+b x^n}}{c n}+\frac{2 \left (a+b x^n\right )^{3/2}}{3 c n}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^n}\right )}{b c n}\\ &=\frac{2 a \sqrt{a+b x^n}}{c n}+\frac{2 \left (a+b x^n\right )^{3/2}}{3 c n}-\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{c n}\\ \end{align*}
Mathematica [A] time = 0.0257814, size = 58, normalized size = 0.79 \[ \frac{2 \sqrt{a+b x^n} \left (4 a+b x^n\right )-6 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{3 c n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 51, normalized size = 0.7 \begin{align*}{\frac{1}{cn} \left ({\frac{2}{3} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}}+2\,a\sqrt{a+b{x}^{n}}-2\,{a}^{3/2}{\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 = 0.967679, size = 277, normalized size = 3.79 \begin{align*} \left [\frac{3 \, a^{\frac{3}{2}} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \,{\left (b x^{n} + 4 \, a\right )} \sqrt{b x^{n} + a}}{3 \, c n}, \frac{2 \,{\left (3 \, \sqrt{-a} a \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) +{\left (b x^{n} + 4 \, a\right )} \sqrt{b x^{n} + a}\right )}}{3 \, c n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.75678, size = 88, normalized size = 1.21 \begin{align*} \frac{\frac{8 a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{n}}{a}}}{3 n} + \frac{a^{\frac{3}{2}} \log{\left (\frac{b x^{n}}{a} \right )}}{n} - \frac{2 a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b x^{n}}{a}} + 1 \right )}}{n} + \frac{2 \sqrt{a} b x^{n} \sqrt{1 + \frac{b x^{n}}{a}}}{3 n}}{c} \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{3}{2}}}{c x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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