Optimal. Leaf size=71 \[ \frac {x^{m-1} (b B-A c) \, _2F_1\left (1,\frac {m-1}{2};\frac {m+1}{2};-\frac {c x^2}{b}\right )}{b c (1-m)}-\frac {B x^{m-1}}{c (1-m)} \]
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Rubi [A] time = 0.05, antiderivative size = 71, normalized size of antiderivative = 1.00, 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, 459, 364} \[ \frac {x^{m-1} (b B-A c) \, _2F_1\left (1,\frac {m-1}{2};\frac {m+1}{2};-\frac {c x^2}{b}\right )}{b c (1-m)}-\frac {B x^{m-1}}{c (1-m)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 459
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^m \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac {x^{-2+m} \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=-\frac {B x^{-1+m}}{c (1-m)}-\frac {(b B (-1+m)-A c (-1+m)) \int \frac {x^{-2+m}}{b+c x^2} \, dx}{c (-1+m)}\\ &=-\frac {B x^{-1+m}}{c (1-m)}+\frac {(b B-A c) x^{-1+m} \, _2F_1\left (1,\frac {1}{2} (-1+m);\frac {1+m}{2};-\frac {c x^2}{b}\right )}{b c (1-m)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 55, normalized size = 0.77 \[ \frac {x^{m-1} \left ((A c-b B) \, _2F_1\left (1,\frac {m-1}{2};\frac {m+1}{2};-\frac {c x^2}{b}\right )+b B\right )}{b c (m-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{2} + A\right )} x^{m}}{c x^{4} + b x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{m}}{c x^{4} + b x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{2}+A \right ) x^{m}}{c \,x^{4}+b \,x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{2} + A\right )} x^{m}}{c x^{4} + b x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,\left (B\,x^2+A\right )}{c\,x^4+b\,x^2} \,d x \]
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 \[ \int \frac {x^{m} \left (A + B x^{2}\right )}{x^{2} \left (b + c x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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