Optimal. Leaf size=51 \[ -\frac{\left (a+b x^n\right )^{7/2} \, _2F_1\left (1,\frac{7}{2}-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2} \]
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Rubi [A] time = 0.0192186, antiderivative size = 63, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ -\frac{a^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 x^2 \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (a+b x^n\right )^{5/2}}{x^3} \, dx &=\frac{\left (a^2 \sqrt{a+b x^n}\right ) \int \frac{\left (1+\frac{b x^n}{a}\right )^{5/2}}{x^3} \, dx}{\sqrt{1+\frac{b x^n}{a}}}\\ &=-\frac{a^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 x^2 \sqrt{1+\frac{b x^n}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0127004, size = 60, normalized size = 1.18 \[ -\frac{a^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},-\frac{2}{n};1-\frac{2}{n};-\frac{b x^n}{a}\right )}{2 x^2 \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x^{3}}\,{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 = 11.7735, size = 46, normalized size = 0.9 \begin{align*} \frac{a^{\frac{5}{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 )}}{n x^{2} \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{{\left (b x^{n} + a\right )}^{\frac{5}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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