Optimal. Leaf size=55 \[ x \left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac {1}{2},-2 p;\frac {3}{2};-\frac {b x^2}{a}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1103, 251}
\begin {gather*} x \left (\frac {b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac {1}{2},-2 p;\frac {3}{2};-\frac {b x^2}{a}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 1103
Rubi steps
\begin {align*} \int \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx &=\left (\left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \int \left (1+\frac {b x^2}{a}\right )^{2 p} \, dx\\ &=x \left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac {1}{2},-2 p;\frac {3}{2};-\frac {b x^2}{a}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 46, normalized size = 0.84 \begin {gather*} x \left (\left (a+b x^2\right )^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-2 p} \, _2F_1\left (\frac {1}{2},-2 p;\frac {3}{2};-\frac {b x^2}{a}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}\, dx\]
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 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}\, 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 {\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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