Optimal. Leaf size=40 \[ \frac{2 x (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.0399566, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1916} \[ \frac{2 x (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}} \]
Antiderivative was successfully verified.
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Rule 1916
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx &=\frac{2 x (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.0741147, size = 37, normalized size = 0.92 \[ \frac{2 x (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 53, normalized size = 1.3 \begin{align*} -2\,{\frac{ \left ( c{x}^{2}+bx+a \right ) \left ( bx+2\,a \right ){x}^{3}}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{3/2}}} \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{x^{4}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98544, size = 150, normalized size = 3.75 \begin{align*} \frac{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} +{\left (b^{3} - 4 \, a b c\right )} x^{2} +{\left (a b^{2} - 4 \, a^{2} c\right )} x} \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^{4}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19208, size = 61, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (\frac{b}{b^{2} - 4 \, a c} + \frac{2 \, a}{{\left (b^{2} - 4 \, a c\right )} x}\right )}}{\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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