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 = 0.04, antiderivative size = 40, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.62, size = 73, normalized size = 1.82 \[ \frac {2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + {\left (b^{3} - 4 \, a b c\right )} x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 45, normalized size = 1.12 \[ \frac {2 \, {\left (\frac {b}{b^{2} - 4 \, a c} + \frac {2 \, a}{{\left (b^{2} - 4 \, a c\right )} x}\right )}}{\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 53, normalized size = 1.32 \[ -\frac {2 \left (c \,x^{2}+b x +a \right ) \left (b x +2 a \right ) x^{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 75, normalized size = 1.88 \[ -\frac {\left (\frac {4\,a\,c}{4\,a\,c^2-b^2\,c}+\frac {2\,b\,c\,x}{4\,a\,c^2-b^2\,c}\right )\,\sqrt {c\,x^4+b\,x^3+a\,x^2}}{x\,\left (c\,x^2+b\,x+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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