Optimal. Leaf size=28 \[ -\frac {b+2 c x^2}{b^2 \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2013, 613} \begin {gather*} -\frac {b+2 c x^2}{b^2 \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 2013
Rubi steps
\begin {align*} \int \frac {x}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {b+2 c x^2}{b^2 \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.04 \begin {gather*} \frac {-b-2 c x^2}{b^2 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 41, normalized size = 1.46 \begin {gather*} \frac {\left (-b-2 c x^2\right ) \sqrt {b x^2+c x^4}}{b^2 x^2 \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.48, size = 41, normalized size = 1.46 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}} {\left (2 \, c x^{2} + b\right )}}{b^{2} c x^{4} + b^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 28, normalized size = 1.00 \begin {gather*} -\frac {\frac {2 \, c x^{2}}{b^{2}} + \frac {1}{b}}{\sqrt {c x^{4} + b x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 1.32 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (2 c \,x^{2}+b \right ) x^{2}}{\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 41, normalized size = 1.46 \begin {gather*} -\frac {2 \, c x^{2}}{\sqrt {c x^{4} + b x^{2}} b^{2}} - \frac {1}{\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 = 0.93 \begin {gather*} -\frac {2\,c\,x^2+b}{b^2\,\sqrt {c\,x^4+b\,x^2}} \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}{\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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