Optimal. Leaf size=326 \[ -\frac{2 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{3/2} \left (b+c x^2\right ) (b B-3 A c)}{15 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{x} \sqrt{b x^2+c x^4} (b B-3 A c)}{15 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}} \]
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Rubi [A] time = 0.374683, antiderivative size = 326, 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 = {2039, 2021, 2032, 329, 305, 220, 1196} \[ -\frac{2 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (b B-3 A c) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{3/2} \left (b+c x^2\right ) (b B-3 A c)}{15 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{x} \sqrt{b x^2+c x^4} (b B-3 A c)}{15 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2039
Rule 2021
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{\sqrt{x}} \, dx &=\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}-\frac{\left (2 \left (\frac{3 b B}{2}-\frac{9 A c}{2}\right )\right ) \int \frac{\sqrt{b x^2+c x^4}}{\sqrt{x}} \, dx}{9 c}\\ &=-\frac{2 (b B-3 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{15 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}-\frac{(2 b (b B-3 A c)) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx}{15 c}\\ &=-\frac{2 (b B-3 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{15 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}-\frac{\left (2 b (b B-3 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{15 c \sqrt{b x^2+c x^4}}\\ &=-\frac{2 (b B-3 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{15 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}-\frac{\left (4 b (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 )}{15 c \sqrt{b x^2+c x^4}}\\ &=-\frac{2 (b B-3 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{15 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}-\frac{\left (4 b^{3/2} (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 )}{15 c^{3/2} \sqrt{b x^2+c x^4}}+\frac{\left (4 b^{3/2} (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 )}{15 c^{3/2} \sqrt{b x^2+c x^4}}\\ &=-\frac{4 b (b B-3 A c) x^{3/2} \left (b+c x^2\right )}{15 c^{3/2} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 (b B-3 A c) \sqrt{x} \sqrt{b x^2+c x^4}}{15 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}+\frac{4 b^{5/4} (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 )}{15 c^{7/4} \sqrt{b x^2+c x^4}}-\frac{2 b^{5/4} (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 )}{15 c^{7/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0823903, size = 94, normalized size = 0.29 \[ \frac{2 \sqrt{x} \sqrt{x^2 \left (b+c x^2\right )} \left ((3 A c-b B) \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right )+B \sqrt{\frac{c x^2}{b}+1} \left (b+c x^2\right )\right )}{9 c \sqrt{\frac{c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 422, normalized size = 1.3 \begin{align*}{\frac{2}{ \left ( 45\,c{x}^{2}+45\,b \right ){c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 5\,B{c}^{3}{x}^{6}+18\,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 ){b}^{2}c-9\,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 ){b}^{2}c-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}^{3}+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}^{3}+9\,A{x}^{4}{c}^{3}+7\,B{x}^{4}b{c}^{2}+9\,A{x}^{2}b{c}^{2}+2\,B{x}^{2}{b}^{2}c \right ){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{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{\sqrt{x}}\,{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}}, 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{\sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{\sqrt{x}}\, 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{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{\sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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