Optimal. Leaf size=48 \[ \frac{x^2 \, _2F_1\left (1,\frac{2}{n}-\frac{3}{2};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a \left (a+b x^n\right )^{3/2}} \]
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Rubi [A] time = 0.0170939, antiderivative size = 60, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {365, 364} \[ \frac{x^2 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{5}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^2 \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b x^n\right )^{5/2}} \, dx &=\frac{\sqrt{1+\frac{b x^n}{a}} \int \frac{x}{\left (1+\frac{b x^n}{a}\right )^{5/2}} \, dx}{a^2 \sqrt{a+b x^n}}\\ &=\frac{x^2 \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{5}{2},\frac{2}{n};\frac{2+n}{n};-\frac{b x^n}{a}\right )}{2 a^2 \sqrt{a+b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0154599, size = 60, normalized size = 1.25 \[ \frac{x^2 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{5}{2},\frac{2}{n};1+\frac{2}{n};-\frac{b x^n}{a}\right )}{2 a^2 \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b{x}^{n} \right ) ^{-{\frac{5}{2}}}}\, dx \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}{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.6058, size = 41, normalized size = 0.85 \begin{align*} \frac{x^{2} \Gamma \left (\frac{2}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, \frac{2}{n} \\ 1 + \frac{2}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{a^{\frac{5}{2}} n \Gamma \left (1 + \frac{2}{n}\right )} \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}{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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