Optimal. Leaf size=139 \[ \frac{A \sqrt{\frac{c x^2}{a}+1} (e x)^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{e (m+1) \sqrt{a+c x^2}}+\frac{B \sqrt{\frac{c x^2}{a}+1} (e x)^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{e^2 (m+2) \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.0631845, antiderivative size = 139, 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} \[ \frac{A \sqrt{\frac{c x^2}{a}+1} (e x)^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )}{e (m+1) \sqrt{a+c x^2}}+\frac{B \sqrt{\frac{c x^2}{a}+1} (e x)^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{c x^2}{a}\right )}{e^2 (m+2) \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 808
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m (A+B x)}{\sqrt{a+c x^2}} \, dx &=A \int \frac{(e x)^m}{\sqrt{a+c x^2}} \, dx+\frac{B \int \frac{(e x)^{1+m}}{\sqrt{a+c x^2}} \, dx}{e}\\ &=\frac{\left (A \sqrt{1+\frac{c x^2}{a}}\right ) \int \frac{(e x)^m}{\sqrt{1+\frac{c x^2}{a}}} \, dx}{\sqrt{a+c x^2}}+\frac{\left (B \sqrt{1+\frac{c x^2}{a}}\right ) \int \frac{(e x)^{1+m}}{\sqrt{1+\frac{c x^2}{a}}} \, dx}{e \sqrt{a+c x^2}}\\ &=\frac{A (e x)^{1+m} \sqrt{1+\frac{c x^2}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};-\frac{c x^2}{a}\right )}{e (1+m) \sqrt{a+c x^2}}+\frac{B (e x)^{2+m} \sqrt{1+\frac{c x^2}{a}} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};-\frac{c x^2}{a}\right )}{e^2 (2+m) \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0489435, size = 108, normalized size = 0.78 \[ \frac{x \sqrt{\frac{c x^2}{a}+1} (e x)^m \left (A (m+2) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{a}\right )+B (m+1) x \, _2F_1\left (\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-\frac{c x^2}{a}\right )\right )}{(m+1) (m+2) \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex \right ) ^{m} \left ( Bx+A \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}}\, 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 + A\right )} \left (e x\right )^{m}}{\sqrt{c x^{2} + a}}\,{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 (\frac{{\left (B x + A\right )} \left (e x\right )^{m}}{\sqrt{c x^{2} + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.14802, size = 112, normalized size = 0.81 \begin{align*} \frac{A 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 \sqrt{a} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{B 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 \sqrt{a} \Gamma \left (\frac{m}{2} + 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 \frac{{\left (B x + A\right )} \left (e x\right )^{m}}{\sqrt{c x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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