Optimal. Leaf size=145 \[ \frac{a^2 A \sqrt{a+c x^2} (e x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{e (m+1) \sqrt{\frac{c x^2}{a}+1}}+\frac{a^2 B \sqrt{a+c x^2} (e x)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{e^2 (m+2) \sqrt{\frac{c x^2}{a}+1}} \]
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Rubi [A] time = 0.0762838, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {808, 365, 364} \[ \frac{a^2 A \sqrt{a+c x^2} (e x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{e (m+1) \sqrt{\frac{c x^2}{a}+1}}+\frac{a^2 B \sqrt{a+c x^2} (e x)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{e^2 (m+2) \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 808
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (e x)^m (A+B x) \left (a+c x^2\right )^{5/2} \, dx &=A \int (e x)^m \left (a+c x^2\right )^{5/2} \, dx+\frac{B \int (e x)^{1+m} \left (a+c x^2\right )^{5/2} \, dx}{e}\\ &=\frac{\left (a^2 A \sqrt{a+c x^2}\right ) \int (e x)^m \left (1+\frac{c x^2}{a}\right )^{5/2} \, dx}{\sqrt{1+\frac{c x^2}{a}}}+\frac{\left (a^2 B \sqrt{a+c x^2}\right ) \int (e x)^{1+m} \left (1+\frac{c x^2}{a}\right )^{5/2} \, dx}{e \sqrt{1+\frac{c x^2}{a}}}\\ &=\frac{a^2 A (e x)^{1+m} \sqrt{a+c x^2} \, _2F_1\left (-\frac{5}{2},\frac{1+m}{2};\frac{3+m}{2};-\frac{c x^2}{a}\right )}{e (1+m) \sqrt{1+\frac{c x^2}{a}}}+\frac{a^2 B (e x)^{2+m} \sqrt{a+c x^2} \, _2F_1\left (-\frac{5}{2},\frac{2+m}{2};\frac{4+m}{2};-\frac{c x^2}{a}\right )}{e^2 (2+m) \sqrt{1+\frac{c x^2}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0575547, size = 111, normalized size = 0.77 \[ \frac{a^2 x \sqrt{a+c x^2} (e x)^m \left (A (m+2) \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )+B (m+1) x \, _2F_1\left (-\frac{5}{2},\frac{m}{2}+1;\frac{m}{2}+2;-\frac{c x^2}{a}\right )\right )}{(m+1) (m+2) \sqrt{\frac{c x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( Bx+A \right ) \left ( c{x}^{2}+a \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 (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )} \left (e 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 (B c^{2} x^{5} + A c^{2} x^{4} + 2 \, B a c x^{3} + 2 \, A a c x^{2} + B a^{2} x + A a^{2}\right )} \sqrt{c x^{2} + a} \left (e x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 54.6884, size = 360, normalized size = 2.48 \begin{align*} \frac{A a^{\frac{5}{2}} e^{m} x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{A a^{\frac{3}{2}} c e^{m} x^{3} x^{m} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{A \sqrt{a} c^{2} e^{m} x^{5} x^{m} \Gamma \left (\frac{m}{2} + \frac{5}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + \frac{5}{2} \\ \frac{m}{2} + \frac{7}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + \frac{7}{2}\right )} + \frac{B a^{\frac{5}{2}} e^{m} x^{2} x^{m} \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + 2\right )} + \frac{B a^{\frac{3}{2}} c e^{m} x^{4} x^{m} \Gamma \left (\frac{m}{2} + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac{m}{2} + 3\right )} + \frac{B \sqrt{a} c^{2} e^{m} x^{6} x^{m} \Gamma \left (\frac{m}{2} + 3\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{2} + 3 \\ \frac{m}{2} + 4 \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + 4\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 (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )} \left (e x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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