Optimal. Leaf size=98 \[ \frac{x^{m-3} (b B (3-m)-A c (5-m)) \, _2F_1\left (1,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right )}{2 b^2 c (3-m)}-\frac{x^{m-3} (b B-A c)}{2 b c \left (b+c x^2\right )} \]
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Rubi [A] time = 0.0595799, antiderivative size = 92, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1584, 457, 364} \[ \frac{x^{m-3} \left (\frac{b B}{c}-\frac{A (5-m)}{3-m}\right ) \, _2F_1\left (1,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right )}{2 b^2}-\frac{x^{m-3} (b B-A c)}{2 b c \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 457
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^{-4+m} \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac{(b B-A c) x^{-3+m}}{2 b c \left (b+c x^2\right )}+\frac{(-A c (-5+m)+b B (-3+m)) \int \frac{x^{-4+m}}{b+c x^2} \, dx}{2 b c}\\ &=-\frac{(b B-A c) x^{-3+m}}{2 b c \left (b+c x^2\right )}+\frac{\left (\frac{b B}{c}-\frac{A (5-m)}{3-m}\right ) x^{-3+m} \, _2F_1\left (1,\frac{1}{2} (-3+m);\frac{1}{2} (-1+m);-\frac{c x^2}{b}\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0749948, size = 80, normalized size = 0.82 \[ \frac{x^{m-3} \left ((A c-b B) \, _2F_1\left (2,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right )+b B \, _2F_1\left (1,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right )\right )}{b^2 c (m-3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.235, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( B{x}^{2}+A \right ){x}^{m}}{ \left ( c{x}^{4}+b{x}^{2} \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )} x^{m}}{{\left (c x^{4} + b x^{2}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} x^{m}}{c^{2} x^{8} + 2 \, b c x^{6} + b^{2} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \left (A + B x^{2}\right )}{x^{4} \left (b + c x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )} x^{m}}{{\left (c x^{4} + b x^{2}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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