Optimal. Leaf size=121 \[ \frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.0463985, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1914, 1103} \[ \frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a x+b x^3+c x^5}} \]
Antiderivative was successfully verified.
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Rule 1914
Rule 1103
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{\sqrt{a x+b x^3+c x^5}} \, dx &=\frac{\left (\sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{a x+b x^3+c x^5}}\\ &=\frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a x+b x^3+c x^5}}\\ \end{align*}
Mathematica [C] time = 0.126265, size = 193, normalized size = 1.6 \[ -\frac{i \sqrt{x} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left (a+b x^2+c x^4\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 177, normalized size = 1.5 \begin{align*}{\frac{1}{2\,c{x}^{4}+2\,b{x}^{2}+2\,a}\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) }\sqrt{-2\,{\frac{{x}^{2}\sqrt{-4\,ac+{b}^{2}}-b{x}^{2}-2\,a}{a}}}\sqrt{{\frac{1}{a} \left ({x}^{2}\sqrt{-4\,ac+{b}^{2}}+b{x}^{2}+2\,a \right ) }}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{\sqrt{2}}{2}\sqrt{{\frac{1}{ac} \left ( b\sqrt{-4\,ac+{b}^{2}}-2\,ac+{b}^{2} \right ) }}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \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{x}}{\sqrt{c x^{5} + b x^{3} + a 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{x}}{\sqrt{c x^{5} + b x^{3} + a 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}}{\sqrt{x \left (a + b x^{2} + c x^{4}\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{\sqrt{x}}{\sqrt{c x^{5} + b x^{3} + a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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