Optimal. Leaf size=44 \[ \frac {(d x)^{m+1} \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a^2 d (m+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 44, 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 = {28, 364} \[ \frac {(d x)^{m+1} \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a^2 d (m+1)} \]
Antiderivative was successfully verified.
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Rule 28
Rule 364
Rubi steps
\begin {align*} \int \frac {(d x)^m}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac {(d x)^m}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {(d x)^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a^2 d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.95 \[ \frac {x (d x)^m \, _2F_1\left (2,\frac {m+1}{2};\frac {m+1}{2}+1;-\frac {b x^2}{a}\right )}{a^2 (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d x\right )^{m}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {\left (d x\right )^{m}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{b^{2} x^{4}+2 a b \,x^{2}+a^{2}}\, dx \]
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 \[ \int \frac {\left (d x\right )^{m}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d\,x\right )}^m}{a^2+2\,a\,b\,x^2+b^2\,x^4} \,d x \]
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 \[ \int \frac {\left (d x\right )^{m}}{\left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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