Optimal. Leaf size=203 \[ -\frac{5 c^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-9 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{42 b^{13/4} \sqrt{b x^2+c x^4}}-\frac{5 \sqrt{b x^2+c x^4} (7 b B-9 A c)}{21 b^3 x^{5/2}}+\frac{7 b B-9 A c}{7 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}-\frac{2 A}{7 b x^{5/2} \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.327939, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2038, 2023, 2025, 2032, 329, 220} \[ -\frac{5 c^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (7 b B-9 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{42 b^{13/4} \sqrt{b x^2+c x^4}}-\frac{5 \sqrt{b x^2+c x^4} (7 b B-9 A c)}{21 b^3 x^{5/2}}+\frac{7 b B-9 A c}{7 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}-\frac{2 A}{7 b x^{5/2} \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2023
Rule 2025
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{2 A}{7 b x^{5/2} \sqrt{b x^2+c x^4}}-\frac{\left (2 \left (-\frac{7 b B}{2}+\frac{9 A c}{2}\right )\right ) \int \frac{\sqrt{x}}{\left (b x^2+c x^4\right )^{3/2}} \, dx}{7 b}\\ &=-\frac{2 A}{7 b x^{5/2} \sqrt{b x^2+c x^4}}+\frac{7 b B-9 A c}{7 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}+\frac{(5 (7 b B-9 A c)) \int \frac{1}{x^{3/2} \sqrt{b x^2+c x^4}} \, dx}{14 b^2}\\ &=-\frac{2 A}{7 b x^{5/2} \sqrt{b x^2+c x^4}}+\frac{7 b B-9 A c}{7 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}-\frac{5 (7 b B-9 A c) \sqrt{b x^2+c x^4}}{21 b^3 x^{5/2}}-\frac{(5 c (7 b B-9 A c)) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{42 b^3}\\ &=-\frac{2 A}{7 b x^{5/2} \sqrt{b x^2+c x^4}}+\frac{7 b B-9 A c}{7 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}-\frac{5 (7 b B-9 A c) \sqrt{b x^2+c x^4}}{21 b^3 x^{5/2}}-\frac{\left (5 c (7 b B-9 A c) x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{42 b^3 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 A}{7 b x^{5/2} \sqrt{b x^2+c x^4}}+\frac{7 b B-9 A c}{7 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}-\frac{5 (7 b B-9 A c) \sqrt{b x^2+c x^4}}{21 b^3 x^{5/2}}-\frac{\left (5 c (7 b B-9 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{21 b^3 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 A}{7 b x^{5/2} \sqrt{b x^2+c x^4}}+\frac{7 b B-9 A c}{7 b^2 \sqrt{x} \sqrt{b x^2+c x^4}}-\frac{5 (7 b B-9 A c) \sqrt{b x^2+c x^4}}{21 b^3 x^{5/2}}-\frac{5 c^{3/4} (7 b B-9 A c) x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{42 b^{13/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0457401, size = 79, normalized size = 0.39 \[ \frac{2 x^2 \sqrt{\frac{c x^2}{b}+1} (9 A c-7 b B) \, _2F_1\left (-\frac{3}{4},\frac{3}{2};\frac{1}{4};-\frac{c x^2}{b}\right )-6 A b}{21 b^2 x^{5/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 254, normalized size = 1.3 \begin{align*}{\frac{c{x}^{2}+b}{42\,{b}^{3}} \left ( 45\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{3}c-35\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{3}b+90\,A{c}^{2}{x}^{4}-70\,B{x}^{4}bc+36\,Abc{x}^{2}-28\,B{x}^{2}{b}^{2}-12\,A{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}} \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 (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{3}{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{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} \sqrt{x}}{c^{2} x^{10} + 2 \, b c x^{8} + b^{2} x^{6}}, 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 (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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