Optimal. Leaf size=377 \[ \frac{4 a^{9/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (13 A b-3 a B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right ),\frac{1}{2}\right )}{195 b^{7/4} \sqrt{a+b x^2}}+\frac{8 a^2 \sqrt{e x} \sqrt{a+b x^2} (13 A b-3 a B)}{195 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{8 a^{9/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (13 A b-3 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 b^{7/4} \sqrt{a+b x^2}}+\frac{2 (e x)^{3/2} \left (a+b x^2\right )^{3/2} (13 A b-3 a B)}{117 b e}+\frac{4 a (e x)^{3/2} \sqrt{a+b x^2} (13 A b-3 a B)}{195 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e} \]
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Rubi [A] time = 0.289959, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {459, 279, 329, 305, 220, 1196} \[ \frac{8 a^2 \sqrt{e x} \sqrt{a+b x^2} (13 A b-3 a B)}{195 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{4 a^{9/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (13 A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 b^{7/4} \sqrt{a+b x^2}}-\frac{8 a^{9/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (13 A b-3 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 b^{7/4} \sqrt{a+b x^2}}+\frac{2 (e x)^{3/2} \left (a+b x^2\right )^{3/2} (13 A b-3 a B)}{117 b e}+\frac{4 a (e x)^{3/2} \sqrt{a+b x^2} (13 A b-3 a B)}{195 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e} \]
Antiderivative was successfully verified.
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Rule 459
Rule 279
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \sqrt{e x} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx &=\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}-\frac{\left (2 \left (-\frac{13 A b}{2}+\frac{3 a B}{2}\right )\right ) \int \sqrt{e x} \left (a+b x^2\right )^{3/2} \, dx}{13 b}\\ &=\frac{2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}+\frac{(2 a (13 A b-3 a B)) \int \sqrt{e x} \sqrt{a+b x^2} \, dx}{39 b}\\ &=\frac{4 a (13 A b-3 a B) (e x)^{3/2} \sqrt{a+b x^2}}{195 b e}+\frac{2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}+\frac{\left (4 a^2 (13 A b-3 a B)\right ) \int \frac{\sqrt{e x}}{\sqrt{a+b x^2}} \, dx}{195 b}\\ &=\frac{4 a (13 A b-3 a B) (e x)^{3/2} \sqrt{a+b x^2}}{195 b e}+\frac{2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}+\frac{\left (8 a^2 (13 A b-3 a B)\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{195 b e}\\ &=\frac{4 a (13 A b-3 a B) (e x)^{3/2} \sqrt{a+b x^2}}{195 b e}+\frac{2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}+\frac{\left (8 a^{5/2} (13 A b-3 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{195 b^{3/2}}-\frac{\left (8 a^{5/2} (13 A b-3 a B)\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a} e}}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{195 b^{3/2}}\\ &=\frac{4 a (13 A b-3 a B) (e x)^{3/2} \sqrt{a+b x^2}}{195 b e}+\frac{8 a^2 (13 A b-3 a B) \sqrt{e x} \sqrt{a+b x^2}}{195 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}-\frac{8 a^{9/4} (13 A b-3 a B) \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 b^{7/4} \sqrt{a+b x^2}}+\frac{4 a^{9/4} (13 A b-3 a B) \sqrt{e} \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 )}{195 b^{7/4} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.108196, size = 97, normalized size = 0.26 \[ \frac{2 x \sqrt{e x} \sqrt{a+b x^2} \left (a (13 A b-3 a B) \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+3 B \sqrt{\frac{b x^2}{a}+1} \left (a+b x^2\right )^2\right )}{39 b \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 438, normalized size = 1.2 \begin{align*}{\frac{2}{585\,{b}^{2}x}\sqrt{ex} \left ( 45\,B{x}^{8}{b}^{4}+65\,A{x}^{6}{b}^{4}+120\,B{x}^{6}a{b}^{3}+156\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{3}b-78\,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 ){a}^{3}b-36\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{4}+18\,B\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 ){a}^{4}+208\,A{x}^{4}a{b}^{3}+87\,B{x}^{4}{a}^{2}{b}^{2}+143\,A{x}^{2}{a}^{2}{b}^{2}+12\,B{x}^{2}{a}^{3}b \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \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^{2} + A\right )}{\left (b x^{2} + a\right )}^{\frac{3}{2}} \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 ({\left (B b x^{4} +{\left (B a + A b\right )} x^{2} + A a\right )} \sqrt{b x^{2} + a} \sqrt{e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 12.5934, size = 197, normalized size = 0.52 \begin{align*} \frac{A a^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 e \Gamma \left (\frac{7}{4}\right )} + \frac{A \sqrt{a} b \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{B a^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{B \sqrt{a} b \left (e x\right )^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{5} \Gamma \left (\frac{15}{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 (B x^{2} + A\right )}{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{e x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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