Optimal. Leaf size=55 \[ \frac{x^{m+1} \left (a+b x^n\right )^{5/2} \, _2F_1\left (1,\frac{m+1}{n}+\frac{5}{2};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)} \]
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Rubi [A] time = 0.0227152, antiderivative size = 65, normalized size of antiderivative = 1.18, 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} \[ \frac{a x^{m+1} \sqrt{a+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{(m+1) \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int x^m \left (a+b x^n\right )^{3/2} \, dx &=\frac{\left (a \sqrt{a+b x^n}\right ) \int x^m \left (1+\frac{b x^n}{a}\right )^{3/2} \, dx}{\sqrt{1+\frac{b x^n}{a}}}\\ &=\frac{a x^{1+m} \sqrt{a+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{(1+m) \sqrt{1+\frac{b x^n}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0205355, size = 66, normalized size = 1.2 \[ \frac{a x^{m+1} \sqrt{a+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{m+1}{n};\frac{m+1}{n}+1;-\frac{b x^n}{a}\right )}{(m+1) \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.078, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{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{3}{2}} x^{m}\,{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 = 63.054, size = 58, normalized size = 1.05 \begin{align*} \frac{a^{\frac{3}{2}} x x^{m} \Gamma \left (\frac{m}{n} + \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{m}{n} + \frac{1}{n} \\ \frac{m}{n} + 1 + \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac{m}{n} + 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{3}{2}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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