Optimal. Leaf size=77 \[ \frac{b \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{2 a^{3/2}}-\frac{\sqrt{a x^2+b x^3+c x^4}}{a x^2} \]
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Rubi [A] time = 0.0540936, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1927, 1904, 206} \[ \frac{b \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{2 a^{3/2}}-\frac{\sqrt{a x^2+b x^3+c x^4}}{a x^2} \]
Antiderivative was successfully verified.
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Rule 1927
Rule 1904
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{a x^2+b x^3+c x^4}} \, dx &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{a x^2}-\frac{b \int \frac{1}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{2 a}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{a x^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{x (2 a+b x)}{\sqrt{a x^2+b x^3+c x^4}}\right )}{a}\\ &=-\frac{\sqrt{a x^2+b x^3+c x^4}}{a x^2}+\frac{b \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{2 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0555978, size = 89, normalized size = 1.16 \[ \frac{b x \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{a} (a+x (b+c x))}{2 a^{3/2} \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 88, normalized size = 1.1 \begin{align*} -{\frac{1}{2}\sqrt{c{x}^{2}+bx+a} \left ( 2\,\sqrt{c{x}^{2}+bx+a}{a}^{3/2}-b\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ) ax \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}}}}{a}^{-{\frac{5}{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{1}{\sqrt{c x^{4} + b x^{3} + a x^{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73191, size = 444, normalized size = 5.77 \begin{align*} \left [\frac{\sqrt{a} b x^{2} \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^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}{4 \, a^{2} x^{2}}, -\frac{\sqrt{-a} b x^{2} \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}{2 \, a^{2} x^{2}}\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{1}{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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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