Optimal. Leaf size=69 \[ \frac{2}{a^2 n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{5/2} n}+\frac{2}{3 a n \left (a+b x^n\right )^{3/2}} \]
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Rubi [A] time = 0.0362646, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{2}{a^2 n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{5/2} n}+\frac{2}{3 a n \left (a+b x^n\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^n\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{5/2}} \, dx,x,x^n\right )}{n}\\ &=\frac{2}{3 a n \left (a+b x^n\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,x^n\right )}{a n}\\ &=\frac{2}{3 a n \left (a+b x^n\right )^{3/2}}+\frac{2}{a^2 n \sqrt{a+b x^n}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{a^2 n}\\ &=\frac{2}{3 a n \left (a+b x^n\right )^{3/2}}+\frac{2}{a^2 n \sqrt{a+b x^n}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^n}\right )}{a^2 b n}\\ &=\frac{2}{3 a n \left (a+b x^n\right )^{3/2}}+\frac{2}{a^2 n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{5/2} n}\\ \end{align*}
Mathematica [C] time = 0.0096652, size = 39, normalized size = 0.57 \[ \frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b x^n}{a}+1\right )}{3 a n \left (a+b x^n\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 53, normalized size = 0.8 \begin{align*}{\frac{1}{n} \left ( 2\,{\frac{1}{{a}^{2}\sqrt{a+b{x}^{n}}}}+{\frac{2}{3\,a} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{1}{{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 = 1.05829, size = 517, normalized size = 7.49 \begin{align*} \left [\frac{3 \,{\left (\sqrt{a} b^{2} x^{2 \, n} + 2 \, a^{\frac{3}{2}} b x^{n} + a^{\frac{5}{2}}\right )} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \,{\left (3 \, a b x^{n} + 4 \, a^{2}\right )} \sqrt{b x^{n} + a}}{3 \,{\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}}, \frac{2 \,{\left (3 \,{\left (\sqrt{-a} b^{2} x^{2 \, n} + 2 \, \sqrt{-a} a b x^{n} + \sqrt{-a} a^{2}\right )} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) +{\left (3 \, a b x^{n} + 4 \, a^{2}\right )} \sqrt{b x^{n} + a}\right )}}{3 \,{\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.18551, size = 860, normalized size = 12.46 \begin{align*} \text{result too large to display} \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{1}{{\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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