Optimal. Leaf size=71 \[ \frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c} \sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.0366343, antiderivative size = 71, 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 = {1914, 621, 206} \[ \frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c} \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx &=\frac{\left (x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{\sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\left (2 x \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{\sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.0412742, size = 66, normalized size = 0.93 \[ \frac{x \sqrt{a+b x+c x^2} \log \left (2 \sqrt{c} \sqrt{a+b x+c x^2}+b+2 c x\right )}{\sqrt{c} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 65, normalized size = 0.9 \begin{align*}{x\sqrt{c{x}^{2}+bx+a}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b \right ){\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}}}}{\frac{1}{\sqrt{c}}}} \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{x}{\sqrt{c x^{4} + b x^{3} + a x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54597, size = 298, normalized size = 4.2 \begin{align*} \left [\frac{\log \left (-\frac{8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{c} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right )}{2 \, \sqrt{c}}, -\frac{\sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right )}{c}\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{x}{\sqrt{x^{2} \left (a + b x + c x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15887, size = 50, normalized size = 0.7 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}} - \frac{\sqrt{a}}{x}}{\sqrt{-c}}\right )}{\sqrt{-c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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