Optimal. Leaf size=332 \[ \frac{\sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-3 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{5 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{2 \sqrt{c} x^{3/2} \left (b+c x^2\right ) (5 b B-3 A c)}{5 b^2 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (5 b B-3 A c)}{5 b^2 x^{3/2}}-\frac{2 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-3 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{2 A \sqrt{b x^2+c x^4}}{5 b x^{7/2}} \]
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Rubi [A] time = 0.378017, antiderivative size = 332, 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, 2025, 2032, 329, 305, 220, 1196} \[ \frac{2 \sqrt{c} x^{3/2} \left (b+c x^2\right ) (5 b B-3 A c)}{5 b^2 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (5 b B-3 A c)}{5 b^2 x^{3/2}}+\frac{\sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-3 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{2 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-3 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{7/4} \sqrt{b x^2+c x^4}}-\frac{2 A \sqrt{b x^2+c x^4}}{5 b x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2025
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^{5/2} \sqrt{b x^2+c x^4}} \, dx &=-\frac{2 A \sqrt{b x^2+c x^4}}{5 b x^{7/2}}-\frac{\left (2 \left (-\frac{5 b B}{2}+\frac{3 A c}{2}\right )\right ) \int \frac{1}{\sqrt{x} \sqrt{b x^2+c x^4}} \, dx}{5 b}\\ &=-\frac{2 A \sqrt{b x^2+c x^4}}{5 b x^{7/2}}-\frac{2 (5 b B-3 A c) \sqrt{b x^2+c x^4}}{5 b^2 x^{3/2}}+\frac{(c (5 b B-3 A c)) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{5 b^2}\\ &=-\frac{2 A \sqrt{b x^2+c x^4}}{5 b x^{7/2}}-\frac{2 (5 b B-3 A c) \sqrt{b x^2+c x^4}}{5 b^2 x^{3/2}}+\frac{\left (c (5 b B-3 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{5 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 A \sqrt{b x^2+c x^4}}{5 b x^{7/2}}-\frac{2 (5 b B-3 A c) \sqrt{b x^2+c x^4}}{5 b^2 x^{3/2}}+\frac{\left (2 c (5 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 )}{5 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 A \sqrt{b x^2+c x^4}}{5 b x^{7/2}}-\frac{2 (5 b B-3 A c) \sqrt{b x^2+c x^4}}{5 b^2 x^{3/2}}+\frac{\left (2 \sqrt{c} (5 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 )}{5 b^{3/2} \sqrt{b x^2+c x^4}}-\frac{\left (2 \sqrt{c} (5 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 )}{5 b^{3/2} \sqrt{b x^2+c x^4}}\\ &=\frac{2 \sqrt{c} (5 b B-3 A c) x^{3/2} \left (b+c x^2\right )}{5 b^2 \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 A \sqrt{b x^2+c x^4}}{5 b x^{7/2}}-\frac{2 (5 b B-3 A c) \sqrt{b x^2+c x^4}}{5 b^2 x^{3/2}}-\frac{2 \sqrt [4]{c} (5 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 )}{5 b^{7/4} \sqrt{b x^2+c x^4}}+\frac{\sqrt [4]{c} (5 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 )}{5 b^{7/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0446758, size = 83, normalized size = 0.25 \[ -\frac{2 \left (x^2 \sqrt{\frac{c x^2}{b}+1} (5 b B-3 A c) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{c x^2}{b}\right )+A \left (b+c x^2\right )\right )}{5 b x^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 413, normalized size = 1.2 \begin{align*} -{\frac{1}{5\,{b}^{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 ){x}^{2}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 ){x}^{2}bc-10\,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 ){x}^{2}{b}^{2}+5\,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 ){x}^{2}{b}^{2}-6\,A{c}^{2}{x}^{4}+10\,B{x}^{4}bc-4\,Abc{x}^{2}+10\,B{x}^{2}{b}^{2}+2\,A{b}^{2} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{x}^{-{\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{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\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{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} \sqrt{x}}{c x^{7} + b x^{5}}, 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{A + B x^{2}}{x^{\frac{5}{2}} \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{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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