Optimal. Leaf size=379 \[ -\frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-5 a B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right ),\frac{1}{2}\right )}{10 a^{11/4} e^{7/2} \sqrt{a+b x^2}}+\frac{3 \sqrt{a+b x^2} (7 A b-5 a B)}{5 a^3 e^3 \sqrt{e x}}-\frac{3 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (7 A b-5 a B)}{5 a^3 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{7 A b-5 a B}{5 a^2 e^3 \sqrt{e x} \sqrt{a+b x^2}}+\frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-5 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt{a+b x^2}}-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.298067, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {453, 290, 325, 329, 305, 220, 1196} \[ \frac{3 \sqrt{a+b x^2} (7 A b-5 a B)}{5 a^3 e^3 \sqrt{e x}}-\frac{3 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (7 A b-5 a B)}{5 a^3 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{7 A b-5 a B}{5 a^2 e^3 \sqrt{e x} \sqrt{a+b x^2}}-\frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 a^{11/4} e^{7/2} \sqrt{a+b x^2}}+\frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-5 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt{a+b x^2}}-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 290
Rule 325
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx &=-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}}-\frac{(7 A b-5 a B) \int \frac{1}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx}{5 a e^2}\\ &=-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}}-\frac{7 A b-5 a B}{5 a^2 e^3 \sqrt{e x} \sqrt{a+b x^2}}-\frac{(3 (7 A b-5 a B)) \int \frac{1}{(e x)^{3/2} \sqrt{a+b x^2}} \, dx}{10 a^2 e^2}\\ &=-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}}-\frac{7 A b-5 a B}{5 a^2 e^3 \sqrt{e x} \sqrt{a+b x^2}}+\frac{3 (7 A b-5 a B) \sqrt{a+b x^2}}{5 a^3 e^3 \sqrt{e x}}-\frac{(3 b (7 A b-5 a B)) \int \frac{\sqrt{e x}}{\sqrt{a+b x^2}} \, dx}{10 a^3 e^4}\\ &=-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}}-\frac{7 A b-5 a B}{5 a^2 e^3 \sqrt{e x} \sqrt{a+b x^2}}+\frac{3 (7 A b-5 a B) \sqrt{a+b x^2}}{5 a^3 e^3 \sqrt{e x}}-\frac{(3 b (7 A b-5 a B)) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 a^3 e^5}\\ &=-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}}-\frac{7 A b-5 a B}{5 a^2 e^3 \sqrt{e x} \sqrt{a+b x^2}}+\frac{3 (7 A b-5 a B) \sqrt{a+b x^2}}{5 a^3 e^3 \sqrt{e x}}-\frac{\left (3 \sqrt{b} (7 A b-5 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 a^{5/2} e^4}+\frac{\left (3 \sqrt{b} (7 A b-5 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 )}{5 a^{5/2} e^4}\\ &=-\frac{2 A}{5 a e (e x)^{5/2} \sqrt{a+b x^2}}-\frac{7 A b-5 a B}{5 a^2 e^3 \sqrt{e x} \sqrt{a+b x^2}}+\frac{3 (7 A b-5 a B) \sqrt{a+b x^2}}{5 a^3 e^3 \sqrt{e x}}-\frac{3 \sqrt{b} (7 A b-5 a B) \sqrt{e x} \sqrt{a+b x^2}}{5 a^3 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{3 \sqrt [4]{b} (7 A b-5 a B) \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 )}{5 a^{11/4} e^{7/2} \sqrt{a+b x^2}}-\frac{3 \sqrt [4]{b} (7 A b-5 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 )}{10 a^{11/4} e^{7/2} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0440894, size = 78, normalized size = 0.21 \[ \frac{x \left (2 x^2 \sqrt{\frac{b x^2}{a}+1} (7 A b-5 a B) \, _2F_1\left (-\frac{1}{4},\frac{3}{2};\frac{3}{4};-\frac{b x^2}{a}\right )-2 a A\right )}{5 a^2 (e x)^{7/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 417, normalized size = 1.1 \begin{align*} -{\frac{1}{10\,{x}^{2}{e}^{3}{a}^{3}} \left ( 42\,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 ){x}^{2}ab-21\,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 ){x}^{2}ab-30\,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 ){x}^{2}{a}^{2}+15\,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 ){x}^{2}{a}^{2}-42\,A{b}^{2}{x}^{4}+30\,B{x}^{4}ab-28\,aAb{x}^{2}+20\,B{x}^{2}{a}^{2}+4\,A{a}^{2} \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{7}{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^{4} x^{8} + 2 \, a b e^{4} x^{6} + a^{2} e^{4} x^{4}}, 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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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