Optimal. Leaf size=59 \[ \frac {B x \sqrt {b x^2+c x^4}}{3 c}-\frac {\sqrt {b x^2+c x^4} (2 b B-3 A c)}{3 c^2 x} \]
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Rubi [A] time = 0.14, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2039, 1588} \[ \frac {B x \sqrt {b x^2+c x^4}}{3 c}-\frac {\sqrt {b x^2+c x^4} (2 b B-3 A c)}{3 c^2 x} \]
Antiderivative was successfully verified.
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Rule 1588
Rule 2039
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx &=\frac {B x \sqrt {b x^2+c x^4}}{3 c}-\frac {(2 b B-3 A c) \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx}{3 c}\\ &=-\frac {(2 b B-3 A c) \sqrt {b x^2+c x^4}}{3 c^2 x}+\frac {B x \sqrt {b x^2+c x^4}}{3 c}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 40, normalized size = 0.68 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (3 A c-2 b B+B c x^2\right )}{3 c^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 36, normalized size = 0.61 \[ \frac {\sqrt {c x^{4} + b x^{2}} {\left (B c x^{2} - 2 \, B b + 3 \, A c\right )}}{3 \, c^{2} x} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{2}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 0.71 \[ \frac {\left (c \,x^{2}+b \right ) \left (B c \,x^{2}+3 A c -2 b B \right ) x}{3 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 50, normalized size = 0.85 \[ \frac {\sqrt {c x^{2} + b} A}{c} + \frac {{\left (c^{2} x^{4} - b c x^{2} - 2 \, b^{2}\right )} B}{3 \, \sqrt {c x^{2} + b} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 41, normalized size = 0.69 \[ \frac {\left (\frac {3\,A\,c-2\,B\,b}{3\,c^2}+\frac {B\,x^2}{3\,c}\right )\,\sqrt {c\,x^4+b\,x^2}}{x} \]
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 \[ \int \frac {x^{2} \left (A + B x^{2}\right )}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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