Optimal. Leaf size=47 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x} (2 a+b x)}{2 \sqrt{a} \sqrt{a x+b x^2+c x^3}}\right )}{\sqrt{a}} \]
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Rubi [A] time = 0.0756258, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1997, 1913, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x} (2 a+b x)}{2 \sqrt{a} \sqrt{a x+b x^2+c x^3}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 1997
Rule 1913
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \sqrt{x \left (a+b x+c x^2\right )}} \, dx &=\int \frac{1}{\sqrt{x} \sqrt{a x+b x^2+c x^3}} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{\sqrt{x} (2 a+b x)}{\sqrt{a x+b x^2+c x^3}}\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{x} (2 a+b x)}{2 \sqrt{a} \sqrt{a x+b x^2+c x^3}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.033077, size = 72, normalized size = 1.53 \[ -\frac{\sqrt{x} \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{\sqrt{a} \sqrt{x (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 64, normalized size = 1.4 \begin{align*} -{\sqrt{x}\sqrt{c{x}^{2}+bx+a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{x \left ( c{x}^{2}+bx+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{1}{\sqrt{{\left (c x^{2} + b x + a\right )} x} \sqrt{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70371, size = 315, normalized size = 6.7 \begin{align*} \left [\frac{\log \left (\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{3} + b x^{2} + a x}{\left (b x + 2 \, a\right )} \sqrt{a} \sqrt{x}}{x^{3}}\right )}{2 \, \sqrt{a}}, \frac{\sqrt{-a} \arctan \left (\frac{\sqrt{c x^{3} + b x^{2} + a x}{\left (b x + 2 \, a\right )} \sqrt{-a} \sqrt{x}}{2 \,{\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right )}{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 [A] time = 1.14507, size = 47, normalized size = 1. \begin{align*} \frac{2 \, \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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