Optimal. Leaf size=61 \[ -\frac {\sqrt {b x^2+c x^4} (3 b B-2 A c)}{3 b^2 x^2}-\frac {A \sqrt {b x^2+c x^4}}{3 b x^4} \]
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Rubi [A] time = 0.17, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2034, 792, 650} \begin {gather*} -\frac {\sqrt {b x^2+c x^4} (3 b B-2 A c)}{3 b^2 x^2}-\frac {A \sqrt {b x^2+c x^4}}{3 b x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 792
Rule 2034
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^3 \sqrt {b x^2+c x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^2 \sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {A \sqrt {b x^2+c x^4}}{3 b x^4}+\frac {\left (-2 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b x+c x^2}} \, dx,x,x^2\right )}{3 b}\\ &=-\frac {A \sqrt {b x^2+c x^4}}{3 b x^4}-\frac {(3 b B-2 A c) \sqrt {b x^2+c x^4}}{3 b^2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 43, normalized size = 0.70 \begin {gather*} -\frac {\sqrt {x^2 \left (b+c x^2\right )} \left (A \left (b-2 c x^2\right )+3 b B x^2\right )}{3 b^2 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 44, normalized size = 0.72 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-A b+2 A c x^2-3 b B x^2\right )}{3 b^2 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 38, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}} {\left ({\left (3 \, B b - 2 \, A c\right )} x^{2} + A 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 = 0.20, size = 88, normalized size = 1.44 \begin {gather*} \frac {3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )}^{2} B + 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )} A \sqrt {c} + A 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 = 0.05, size = 47, normalized size = 0.77 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (-2 A c \,x^{2}+3 B b \,x^{2}+A 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 = 1.47, size = 70, normalized size = 1.15 \begin {gather*} \frac {1}{3} \, A {\left (\frac {2 \, \sqrt {c x^{4} + b x^{2}} c}{b^{2} x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}}}{b x^{4}}\right )} - \frac {\sqrt {c x^{4} + b x^{2}} B}{b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 39, normalized size = 0.64 \begin {gather*} -\frac {\sqrt {c\,x^4+b\,x^2}\,\left (A\,b-2\,A\,c\,x^2+3\,B\,b\,x^2\right )}{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 {A + B x^{2}}{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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