Optimal. Leaf size=39 \[ \frac{x \left (a+b x^n\right )^{7/2} \, _2F_1\left (1,\frac{7}{2}+\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a} \]
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Rubi [A] time = 0.0109987, antiderivative size = 51, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {246, 245} \[ \frac{a^2 x \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{\sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b x^n\right )^{5/2} \, dx &=\frac{\left (a^2 \sqrt{a+b x^n}\right ) \int \left (1+\frac{b x^n}{a}\right )^{5/2} \, dx}{\sqrt{1+\frac{b x^n}{a}}}\\ &=\frac{a^2 x \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{\sqrt{1+\frac{b x^n}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0074471, size = 51, normalized size = 1.31 \[ \frac{a^2 x \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{\sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int \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{\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 = 14.0824, size = 41, normalized size = 1.05 \begin{align*} \frac{a^{\frac{5}{2}} x \Gamma \left (\frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac{1}{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{\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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