Optimal. Leaf size=52 \[ \frac {2 c \sqrt {b x^2+c x^4}}{3 b^2 x^2}-\frac {\sqrt {b x^2+c x^4}}{3 b x^4} \]
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Rubi [A] time = 0.08, antiderivative size = 52, 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, 2014} \begin {gather*} \frac {2 c \sqrt {b x^2+c x^4}}{3 b^2 x^2}-\frac {\sqrt {b x^2+c x^4}}{3 b x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {b x^2+c x^4}} \, dx &=-\frac {\sqrt {b x^2+c x^4}}{3 b x^4}-\frac {(2 c) \int \frac {1}{x \sqrt {b x^2+c x^4}} \, dx}{3 b}\\ &=-\frac {\sqrt {b x^2+c x^4}}{3 b x^4}+\frac {2 c \sqrt {b x^2+c x^4}}{3 b^2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.67 \begin {gather*} \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (2 c x^2-b\right )}{3 b^2 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 35, normalized size = 0.67 \begin {gather*} \frac {\left (2 c x^2-b\right ) \sqrt {b x^2+c x^4}}{3 b^2 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 31, normalized size = 0.60 \begin {gather*} \frac {\sqrt {c x^{4} + b x^{2}} {\left (2 \, c x^{2} - b\right )}}{3 \, b^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 57, normalized size = 1.10 \begin {gather*} \frac {3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )} \sqrt {c} + b}{3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )}^{3}} \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 = 0.71 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (-2 c \,x^{2}+b \right )}{3 \sqrt {c \,x^{4}+b \,x^{2}}\, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 44, normalized size = 0.85 \begin {gather*} \frac {2 \, \sqrt {c x^{4} + b x^{2}} c}{3 \, b^{2} x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}}}{3 \, b x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.26, size = 29, normalized size = 0.56 \begin {gather*} -\frac {\left (b-2\,c\,x^2\right )\,\sqrt {c\,x^4+b\,x^2}}{3\,b^2\,x^4} \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 {1}{x^{3} \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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