Optimal. Leaf size=21 \[ -\frac {x}{c \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.02, antiderivative size = 21, 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 = {1588} \begin {gather*} -\frac {x}{c \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin {align*} \int \frac {x^4}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {x}{c \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -\frac {x}{c \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 32, normalized size = 1.52 \begin {gather*} -\frac {\sqrt {b x^2+c x^4}}{c x \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 29, normalized size = 1.38 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}}}{c^{2} x^{3} + b c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{\sqrt {c + \frac {b}{x^{2}}} c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 1.38 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) x^{3}}{\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 14, normalized size = 0.67 \begin {gather*} -\frac {1}{\sqrt {c x^{2} + b} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.15, size = 30, normalized size = 1.43 \begin {gather*} -\frac {\sqrt {c\,x^4+b\,x^2}}{c\,x\,\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^{4}}{\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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