Optimal. Leaf size=293 \[ -\frac{\sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-5 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{5 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{2 x^{3/2} \left (b+c x^2\right ) (3 b B-5 A c)}{5 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{2 \sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-5 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{2 B \sqrt{x} \sqrt{b x^2+c x^4}}{5 c} \]
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Rubi [A] time = 0.31129, antiderivative size = 293, 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 = {2039, 2032, 329, 305, 220, 1196} \[ -\frac{2 x^{3/2} \left (b+c x^2\right ) (3 b B-5 A c)}{5 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{\sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{2 \sqrt [4]{b} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (3 b B-5 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{2 B \sqrt{x} \sqrt{b x^2+c x^4}}{5 c} \]
Antiderivative was successfully verified.
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Rule 2039
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^{3/2} \left (A+B x^2\right )}{\sqrt{b x^2+c x^4}} \, dx &=\frac{2 B \sqrt{x} \sqrt{b x^2+c x^4}}{5 c}-\frac{\left (2 \left (\frac{3 b B}{2}-\frac{5 A c}{2}\right )\right ) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{5 c}\\ &=\frac{2 B \sqrt{x} \sqrt{b x^2+c x^4}}{5 c}-\frac{\left (2 \left (\frac{3 b B}{2}-\frac{5 A c}{2}\right ) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{5 c \sqrt{b x^2+c x^4}}\\ &=\frac{2 B \sqrt{x} \sqrt{b x^2+c x^4}}{5 c}-\frac{\left (4 \left (\frac{3 b B}{2}-\frac{5 A c}{2}\right ) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{5 c \sqrt{b x^2+c x^4}}\\ &=\frac{2 B \sqrt{x} \sqrt{b x^2+c x^4}}{5 c}-\frac{\left (4 \sqrt{b} \left (\frac{3 b B}{2}-\frac{5 A c}{2}\right ) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{5 c^{3/2} \sqrt{b x^2+c x^4}}+\frac{\left (4 \sqrt{b} \left (\frac{3 b B}{2}-\frac{5 A c}{2}\right ) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{b}}}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{5 c^{3/2} \sqrt{b x^2+c x^4}}\\ &=-\frac{2 (3 b B-5 A c) x^{3/2} \left (b+c x^2\right )}{5 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{2 B \sqrt{x} \sqrt{b x^2+c x^4}}{5 c}+\frac{2 \sqrt [4]{b} (3 b B-5 A c) x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{\sqrt [4]{b} (3 b B-5 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 )}{5 c^{7/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0878932, size = 81, normalized size = 0.28 \[ \frac{2 x^{5/2} \left (\sqrt{\frac{c x^2}{b}+1} (5 A c-3 b B) \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right )+3 B \left (b+c x^2\right )\right )}{15 c \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 378, normalized size = 1.3 \begin{align*}{\frac{1}{5\,{c}^{2}}\sqrt{x} \left ( 10\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) bc-5\,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 ) bc-6\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{2}+3\,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 ){b}^{2}+2\,B{c}^{2}{x}^{4}+2\,B{x}^{2}bc \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \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 )} x^{\frac{3}{2}}}{\sqrt{c x^{4} + b x^{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 x^{3} + b x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}} \left (A + B x^{2}\right )}{\sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \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 )} x^{\frac{3}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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