Optimal. Leaf size=139 \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right ),\frac{1}{2}\right )}{3 \sqrt [4]{a} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 B \sqrt{e x} \sqrt{a+b x^2}}{3 b e} \]
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Rubi [A] time = 0.0811609, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {459, 329, 220} \[ \frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 B \sqrt{e x} \sqrt{a+b x^2}}{3 b e} \]
Antiderivative was successfully verified.
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Rule 459
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{A+B x^2}{\sqrt{e x} \sqrt{a+b x^2}} \, dx &=\frac{2 B \sqrt{e x} \sqrt{a+b x^2}}{3 b e}-\frac{\left (2 \left (-\frac{3 A b}{2}+\frac{a B}{2}\right )\right ) \int \frac{1}{\sqrt{e x} \sqrt{a+b x^2}} \, dx}{3 b}\\ &=\frac{2 B \sqrt{e x} \sqrt{a+b x^2}}{3 b e}+\frac{(2 (3 A b-a B)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3 b e}\\ &=\frac{2 B \sqrt{e x} \sqrt{a+b x^2}}{3 b e}+\frac{(3 A b-a B) \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} b^{5/4} \sqrt{e} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0554849, size = 79, normalized size = 0.57 \[ \frac{2 x \left (\sqrt{\frac{b x^2}{a}+1} (3 A b-a B) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right )+B \left (a+b x^2\right )\right )}{3 b \sqrt{e x} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 214, normalized size = 1.5 \begin{align*}{\frac{1}{3\,{b}^{2}} \left ( 3\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}b-B\sqrt{{ \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{ \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{ \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}a+2\,{b}^{2}B{x}^{3}+2\,Bxab \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \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{B x^{2} + A}{\sqrt{b x^{2} + a} \sqrt{e x}}\,{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^{2} + A\right )} \sqrt{b x^{2} + a} \sqrt{e x}}{b e x^{3} + a e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.37705, size = 94, normalized size = 0.68 \begin{align*} \frac{A \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{B x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \sqrt{e} \Gamma \left (\frac{9}{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 \frac{B x^{2} + A}{\sqrt{b x^{2} + a} \sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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