Optimal. Leaf size=22 \[ \frac {x^2}{b \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2014} \begin {gather*} \frac {x^2}{b \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int \frac {x^3}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac {x^2}{b \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \frac {x^2}{b \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 28, normalized size = 1.27 \begin {gather*} \frac {\sqrt {b x^2+c x^4}}{b \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.91, size = 26, normalized size = 1.18 \begin {gather*} \frac {\sqrt {c x^{4} + b x^{2}}}{b c x^{2} + b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 35, normalized size = 1.59 \begin {gather*} \frac {1}{{\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )} \sqrt {c} + b\right )} \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 1.27 \begin {gather*} \frac {\left (c \,x^{2}+b \right ) x^{4}}{\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 20, normalized size = 0.91 \begin {gather*} \frac {x^{2}}{\sqrt {c x^{4} + b x^{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.13, size = 26, normalized size = 1.18 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}}{b\,\left (c\,x^2+b\right )} \end {gather*}
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 \begin {gather*} \int \frac {x^{3}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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