Optimal. Leaf size=214 \[ \frac{4 a^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (11 A b-a B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right ),\frac{1}{2}\right )}{77 b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \left (a+b x^2\right )^{3/2} (11 A b-a B)}{77 b e}+\frac{4 a \sqrt{e x} \sqrt{a+b x^2} (11 A b-a B)}{77 b e}+\frac{2 B \sqrt{e x} \left (a+b x^2\right )^{5/2}}{11 b e} \]
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Rubi [A] time = 0.136428, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {459, 279, 329, 220} \[ \frac{4 a^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (11 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{77 b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \left (a+b x^2\right )^{3/2} (11 A b-a B)}{77 b e}+\frac{4 a \sqrt{e x} \sqrt{a+b x^2} (11 A b-a B)}{77 b e}+\frac{2 B \sqrt{e x} \left (a+b x^2\right )^{5/2}}{11 b e} \]
Antiderivative was successfully verified.
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Rule 459
Rule 279
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{\sqrt{e x}} \, dx &=\frac{2 B \sqrt{e x} \left (a+b x^2\right )^{5/2}}{11 b e}-\frac{\left (2 \left (-\frac{11 A b}{2}+\frac{a B}{2}\right )\right ) \int \frac{\left (a+b x^2\right )^{3/2}}{\sqrt{e x}} \, dx}{11 b}\\ &=\frac{2 (11 A b-a B) \sqrt{e x} \left (a+b x^2\right )^{3/2}}{77 b e}+\frac{2 B \sqrt{e x} \left (a+b x^2\right )^{5/2}}{11 b e}+\frac{(6 a (11 A b-a B)) \int \frac{\sqrt{a+b x^2}}{\sqrt{e x}} \, dx}{77 b}\\ &=\frac{4 a (11 A b-a B) \sqrt{e x} \sqrt{a+b x^2}}{77 b e}+\frac{2 (11 A b-a B) \sqrt{e x} \left (a+b x^2\right )^{3/2}}{77 b e}+\frac{2 B \sqrt{e x} \left (a+b x^2\right )^{5/2}}{11 b e}+\frac{\left (4 a^2 (11 A b-a B)\right ) \int \frac{1}{\sqrt{e x} \sqrt{a+b x^2}} \, dx}{77 b}\\ &=\frac{4 a (11 A b-a B) \sqrt{e x} \sqrt{a+b x^2}}{77 b e}+\frac{2 (11 A b-a B) \sqrt{e x} \left (a+b x^2\right )^{3/2}}{77 b e}+\frac{2 B \sqrt{e x} \left (a+b x^2\right )^{5/2}}{11 b e}+\frac{\left (8 a^2 (11 A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{77 b e}\\ &=\frac{4 a (11 A b-a B) \sqrt{e x} \sqrt{a+b x^2}}{77 b e}+\frac{2 (11 A b-a B) \sqrt{e x} \left (a+b x^2\right )^{3/2}}{77 b e}+\frac{2 B \sqrt{e x} \left (a+b x^2\right )^{5/2}}{11 b e}+\frac{4 a^{7/4} (11 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 )}{77 b^{5/4} \sqrt{e} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0754132, size = 96, normalized size = 0.45 \[ \frac{2 x \sqrt{a+b x^2} \left (a (11 A b-a B) \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+B \sqrt{\frac{b x^2}{a}+1} \left (a+b x^2\right )^2\right )}{11 b \sqrt{e x} \sqrt{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 272, normalized size = 1.3 \begin{align*}{\frac{2}{77\,{b}^{2}} \left ( 7\,B{x}^{7}{b}^{4}+22\,A\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{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-ab}{a}^{2}b+11\,A{x}^{5}{b}^{4}-2\,B\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{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-ab}{a}^{3}+20\,B{x}^{5}a{b}^{3}+44\,A{x}^{3}a{b}^{3}+17\,B{x}^{3}{a}^{2}{b}^{2}+33\,Ax{a}^{2}{b}^{2}+4\,Bx{a}^{3}b \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{{\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 (\frac{{\left (B b x^{4} +{\left (B a + A b\right )} x^{2} + A a\right )} \sqrt{b x^{2} + a} \sqrt{e x}}{e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 12.2051, size = 199, normalized size = 0.93 \begin{align*} \frac{A a^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{A \sqrt{a} 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{e} \Gamma \left (\frac{9}{4}\right )} + \frac{B a^{\frac{3}{2}} 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{e} \Gamma \left (\frac{9}{4}\right )} + \frac{B \sqrt{a} b x^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{13}{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{{\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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