Optimal. Leaf size=318 \[ -\frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{2 b^{7/4} c^{3/4} \sqrt{b x^2+c x^4}}+\frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{7/4} c^{3/4} \sqrt{b x^2+c x^4}}+\frac{x^{5/2} (b B-3 A c)}{b^2 \sqrt{b x^2+c x^4}}-\frac{x^{3/2} \left (b+c x^2\right ) (b B-3 A c)}{b^2 \sqrt{c} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 A \sqrt{x}}{b \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.389588, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2038, 2023, 2032, 329, 305, 220, 1196} \[ -\frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 b^{7/4} c^{3/4} \sqrt{b x^2+c x^4}}+\frac{x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{b^{7/4} c^{3/4} \sqrt{b x^2+c x^4}}+\frac{x^{5/2} (b B-3 A c)}{b^2 \sqrt{b x^2+c x^4}}-\frac{x^{3/2} \left (b+c x^2\right ) (b B-3 A c)}{b^2 \sqrt{c} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 A \sqrt{x}}{b \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2023
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 )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{2 A \sqrt{x}}{b \sqrt{b x^2+c x^4}}-\frac{\left (2 \left (-\frac{b B}{2}+\frac{3 A c}{2}\right )\right ) \int \frac{x^{7/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx}{b}\\ &=-\frac{2 A \sqrt{x}}{b \sqrt{b x^2+c x^4}}+\frac{(b B-3 A c) x^{5/2}}{b^2 \sqrt{b x^2+c x^4}}-\frac{(b B-3 A c) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{2 b^2}\\ &=-\frac{2 A \sqrt{x}}{b \sqrt{b x^2+c x^4}}+\frac{(b B-3 A c) x^{5/2}}{b^2 \sqrt{b x^2+c x^4}}-\frac{\left ((b B-3 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{2 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 A \sqrt{x}}{b \sqrt{b x^2+c x^4}}+\frac{(b B-3 A c) x^{5/2}}{b^2 \sqrt{b x^2+c x^4}}-\frac{\left ((b B-3 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 A \sqrt{x}}{b \sqrt{b x^2+c x^4}}+\frac{(b B-3 A c) x^{5/2}}{b^2 \sqrt{b x^2+c x^4}}-\frac{\left ((b B-3 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 )}{b^{3/2} \sqrt{c} \sqrt{b x^2+c x^4}}+\frac{\left ((b B-3 A c) 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 )}{b^{3/2} \sqrt{c} \sqrt{b x^2+c x^4}}\\ &=-\frac{2 A \sqrt{x}}{b \sqrt{b x^2+c x^4}}+\frac{(b B-3 A c) x^{5/2}}{b^2 \sqrt{b x^2+c x^4}}-\frac{(b B-3 A c) x^{3/2} \left (b+c x^2\right )}{b^2 \sqrt{c} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{(b B-3 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 )}{b^{7/4} c^{3/4} \sqrt{b x^2+c x^4}}-\frac{(b B-3 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 )}{2 b^{7/4} c^{3/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0469074, size = 77, normalized size = 0.24 \[ \frac{2 \sqrt{x} \left (x^2 \sqrt{\frac{c x^2}{b}+1} (b B-3 A c) \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{c x^2}{b}\right )-3 A b\right )}{3 b^2 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 392, normalized size = 1.2 \begin{align*}{\frac{c{x}^{2}+b}{2\,{b}^{2}c}{x}^{{\frac{5}{2}}} \left ( 6\,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-3\,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-2\,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}+B\sqrt{{ \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{2}\sqrt{{ \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}}{\it EllipticF} \left ( \sqrt{{ \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ){b}^{2}-6\,A{x}^{2}{c}^{2}+2\,B{x}^{2}bc-4\,Abc \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{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}}}{{\left (c x^{4} + b x^{2}\right )}^{\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^{7} + 2 \, b c x^{5} + b^{2} 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{{\left (B x^{2} + A\right )} x^{\frac{3}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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