Optimal. Leaf size=44 \[ \frac {(d x)^{1+m} \, _2F_1\left (4,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a^4 d (1+m)} \]
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Rubi [A]
time = 0.01, 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, 371}
\begin {gather*} \frac {(d x)^{m+1} \, _2F_1\left (4,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a^4 d (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 371
Rubi steps
\begin {align*} \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {(d x)^m}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {(d x)^{1+m} \, _2F_1\left (4,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a^4 d (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 42, normalized size = 0.95 \begin {gather*} \frac {x (d x)^m \, _2F_1\left (4,\frac {1+m}{2};1+\frac {1+m}{2};-\frac {b x^2}{a}\right )}{a^4 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (d x \right )^{m}}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\left (d x\right )^{m}}{\left (a + b x^{2}\right )^{4}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d\,x\right )}^m}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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