Optimal. Leaf size=25 \[ \frac {\left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2000} \[ \frac {\left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \]
Antiderivative was successfully verified.
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Rule 2000
Rubi steps
\begin {align*} \int \sqrt {b x^2+c x^4} \, dx &=\frac {\left (b x^2+c x^4\right )^{3/2}}{3 c x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2}}{3 c x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 28, normalized size = 1.12 \[ \frac {\sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + b\right )}}{3 \, c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 27, normalized size = 1.08 \[ \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} \mathrm {sgn}\relax (x)}{3 \, c} - \frac {b^{\frac {3}{2}} \mathrm {sgn}\relax (x)}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 1.16 \[ \frac {\left (c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3 c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 14, normalized size = 0.56 \[ \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}}}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.14, size = 29, normalized size = 1.16 \[ \frac {\left (\frac {b}{3\,c}+\frac {x^2}{3}\right )\,\sqrt {c\,x^4+b\,x^2}}{x} \]
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 \[ \int \sqrt {b x^{2} + c x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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