Optimal. Leaf size=80 \[ \frac{d^4 \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
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Rubi [A] time = 0.0248633, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {365, 364} \[ \frac{d^4 \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx &=\frac{\left (d^4 \sqrt{d^2-e^2 x^2}\right ) \int (g x)^m \left (1-\frac{e^2 x^2}{d^2}\right )^{5/2} \, dx}{\sqrt{1-\frac{e^2 x^2}{d^2}}}\\ &=\frac{d^4 (g x)^{1+m} \sqrt{d^2-e^2 x^2} \, _2F_1\left (-\frac{5}{2},\frac{1+m}{2};\frac{3+m}{2};\frac{e^2 x^2}{d^2}\right )}{g (1+m) \sqrt{1-\frac{e^2 x^2}{d^2}}}\\ \end{align*}
Mathematica [A] time = 0.0192638, size = 78, normalized size = 0.98 \[ \frac{d^4 x \sqrt{d^2-e^2 x^2} (g x)^m \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+1}{2}+1;\frac{e^2 x^2}{d^2}\right )}{(m+1) \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.365, size = 0, normalized size = 0. \begin{align*} \int \left ( gx \right ) ^{m} \left ( -{x}^{2}{e}^{2}+{d}^{2} \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 (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} \left (g x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e^{4} x^{4} - 2 \, d^{2} e^{2} x^{2} + d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 47.7421, size = 61, normalized size = 0.76 \begin{align*} \frac{d^{5} g^{m} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\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 (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} \left (g x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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