Optimal. Leaf size=53 \[ -\frac{\tanh ^{-1}\left (\frac{x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x^3+b x^5+c x^7}}\right )}{2 \sqrt{a}} \]
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Rubi [A] time = 0.0772904, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1997, 1913, 206} \[ -\frac{\tanh ^{-1}\left (\frac{x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x^3+b x^5+c x^7}}\right )}{2 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 1997
Rule 1913
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{\sqrt{x^3 \left (a+b x^2+c x^4\right )}} \, dx &=\int \frac{\sqrt{x}}{\sqrt{a x^3+b x^5+c x^7}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{x^{3/2} \left (2 a+b x^2\right )}{\sqrt{a x^3+b x^5+c x^7}}\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x^3+b x^5+c x^7}}\right )}{2 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0194707, size = 85, normalized size = 1.6 \[ -\frac{x^{3/2} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a} \sqrt{x^3 \left (a+b x^2+c x^4\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 74, normalized size = 1.4 \begin{align*} -{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{c{x}^{4}+b{x}^{2}+a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{{x}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}}{\frac{1}{\sqrt{a}}}} \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{{\left (c x^{4} + b x^{2} + a\right )} x^{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36241, size = 338, normalized size = 6.38 \begin{align*} \left [\frac{\log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{4} + 8 \, a^{2} x^{2} - 4 \, \sqrt{c x^{7} + b x^{5} + a x^{3}}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} \sqrt{x}}{x^{6}}\right )}{4 \, \sqrt{a}}, \frac{\sqrt{-a} \arctan \left (\frac{\sqrt{c x^{7} + b x^{5} + a x^{3}}{\left (b x^{2} + 2 \, a\right )} \sqrt{-a} \sqrt{x}}{2 \,{\left (a c x^{6} + a b x^{4} + a^{2} x^{2}\right )}}\right )}{2 \, a}\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{\sqrt{x}}{\sqrt{{\left (c x^{4} + b x^{2} + a\right )} x^{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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