Optimal. Leaf size=69 \[ \frac {\sqrt {b x^2+c x^4} (2 b B-A c)}{b c^2 x}-\frac {x^3 (b B-A c)}{b c \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.15, antiderivative size = 69, 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 = {2037, 1588} \begin {gather*} \frac {\sqrt {b x^2+c x^4} (2 b B-A c)}{b c^2 x}-\frac {x^3 (b B-A c)}{b c \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1588
Rule 2037
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {(b B-A c) x^3}{b c \sqrt {b x^2+c x^4}}+\frac {(2 b B-A c) \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx}{b c}\\ &=-\frac {(b B-A c) x^3}{b c \sqrt {b x^2+c x^4}}+\frac {(2 b B-A c) \sqrt {b x^2+c x^4}}{b c^2 x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 35, normalized size = 0.51 \begin {gather*} \frac {x \left (-A c+2 b B+B c x^2\right )}{c^2 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.73, size = 46, normalized size = 0.67 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-A c+2 b B+B c x^2\right )}{c^2 x \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 45, normalized size = 0.65 \begin {gather*} \frac {\sqrt {c x^{4} + b x^{2}} {\left (B c x^{2} + 2 \, B b - A c\right )}}{c^{3} x^{3} + b c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 60, normalized size = 0.87 \begin {gather*} -\frac {2 \, B \sqrt {b}}{{\left ({\left (\sqrt {c + \frac {b}{x^{2}}} - \frac {\sqrt {b}}{x}\right )}^{2} - c\right )} c} + \frac {B b - A c}{\sqrt {c + \frac {b}{x^{2}}} c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.64 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (-B c \,x^{2}+A c -2 b B \right ) x^{3}}{\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.57, size = 39, normalized size = 0.57 \begin {gather*} \frac {{\left (c x^{2} + 2 \, b\right )} B}{\sqrt {c x^{2} + b} c^{2}} - \frac {A}{\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 = 0.21, size = 44, normalized size = 0.64 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}\,\left (B\,c\,x^2-A\,c+2\,B\,b\right )}{c^2\,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 (A + B x^{2}\right )}{\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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