Optimal. Leaf size=50 \[ \frac {x \sqrt {b x^2+c x^4}}{3 c}-\frac {2 b \sqrt {b x^2+c x^4}}{3 c^2 x} \]
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Rubi [A] time = 0.08, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 1588} \begin {gather*} \frac {x \sqrt {b x^2+c x^4}}{3 c}-\frac {2 b \sqrt {b x^2+c x^4}}{3 c^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 1588
Rule 2016
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {b x^2+c x^4}} \, dx &=\frac {x \sqrt {b x^2+c x^4}}{3 c}-\frac {(2 b) \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx}{3 c}\\ &=-\frac {2 b \sqrt {b x^2+c x^4}}{3 c^2 x}+\frac {x \sqrt {b x^2+c x^4}}{3 c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 0.68 \begin {gather*} \frac {\left (c x^2-2 b\right ) \sqrt {x^2 \left (b+c x^2\right )}}{3 c^2 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 34, normalized size = 0.68 \begin {gather*} \frac {\left (c x^2-2 b\right ) \sqrt {b x^2+c x^4}}{3 c^2 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.59, size = 30, normalized size = 0.60 \begin {gather*} \frac {\sqrt {c x^{4} + b x^{2}} {\left (c x^{2} - 2 \, b\right )}}{3 \, c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \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 = 0.74 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (-c \,x^{2}+2 b \right ) x}{3 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.50, size = 34, normalized size = 0.68 \begin {gather*} \frac {c^{2} x^{4} - b c x^{2} - 2 \, b^{2}}{3 \, \sqrt {c x^{2} + b} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.25, size = 33, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {c\,x^4+b\,x^2}\,\left (\frac {2\,b}{3\,c^2}-\frac {x^2}{3\,c}\right )}{x} \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}}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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