Optimal. Leaf size=36 \[ \frac {2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
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Rubi [A] time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1114, 636} \[ \frac {2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 636
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 36, normalized size = 1.00 \[ \frac {2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 67, normalized size = 1.86 \[ \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 44, normalized size = 1.22 \[ \frac {\frac {b x^{2}}{b^{2} - 4 \, a c} + \frac {2 \, a}{b^{2} - 4 \, a c}}{\sqrt {c x^{4} + b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 1.06 \[ -\frac {b \,x^{2}+2 a}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.47, size = 37, normalized size = 1.03 \[ -\frac {b\,x^2+2\,a}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^4+b\,x^2+a}} \]
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 \[ \int \frac {x^{3}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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