Optimal. Leaf size=176 \[ -\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-3 a B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right ),\frac{1}{2}\right )}{6 a^{9/4} \sqrt [4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{\sqrt{e x} (5 A b-3 a B)}{3 a^2 e^3 \sqrt{a+b x^2}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.113333, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {453, 290, 329, 220} \[ -\frac{\sqrt{e x} (5 A b-3 a B)}{3 a^2 e^3 \sqrt{a+b x^2}}-\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 a^{9/4} \sqrt [4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 290
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{A+B x^2}{(e x)^{5/2} \left (a+b x^2\right )^{3/2}} \, dx &=-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}}-\frac{(5 A b-3 a B) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{3/2}} \, dx}{3 a e^2}\\ &=-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}}-\frac{(5 A b-3 a B) \sqrt{e x}}{3 a^2 e^3 \sqrt{a+b x^2}}-\frac{(5 A b-3 a B) \int \frac{1}{\sqrt{e x} \sqrt{a+b x^2}} \, dx}{6 a^2 e^2}\\ &=-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}}-\frac{(5 A b-3 a B) \sqrt{e x}}{3 a^2 e^3 \sqrt{a+b x^2}}-\frac{(5 A b-3 a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3 a^2 e^3}\\ &=-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}}-\frac{(5 A b-3 a B) \sqrt{e x}}{3 a^2 e^3 \sqrt{a+b x^2}}-\frac{(5 A b-3 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 )}{6 a^{9/4} \sqrt [4]{b} e^{5/2} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0549425, size = 91, normalized size = 0.52 \[ \frac{x \left (x^2 \sqrt{\frac{b x^2}{a}+1} (3 a B-5 A b) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^2}{a}\right )-2 a A+3 a B x^2-5 A b x^2\right )}{3 a^2 (e x)^{5/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 232, normalized size = 1.3 \begin{align*} -{\frac{1}{6\,{a}^{2}bx{e}^{2}} \left ( 5\,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}xb-3\,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 ) \sqrt{-ab}xa+10\,A{x}^{2}{b}^{2}-6\,B{x}^{2}ab+4\,Aab \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}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{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^{2} e^{3} x^{7} + 2 \, a b e^{3} x^{5} + a^{2} e^{3} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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