Optimal. Leaf size=74 \[ \frac{\left (a+b x^2\right ) (d x)^{m+1} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (1,\frac{1}{2} (m+4 p+3);\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a d (m+1)} \]
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Rubi [A] time = 0.0263096, antiderivative size = 77, normalized size of antiderivative = 1.04, 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 = {1113, 364} \[ \frac{(d x)^{m+1} \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac{m+1}{2},-2 p;\frac{m+3}{2};-\frac{b x^2}{a}\right )}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 1113
Rule 364
Rubi steps
\begin{align*} \int (d x)^m \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 (d x)^m \left (1+\frac{b x^2}{a}\right )^{2 p} \, dx\\ &=\frac{(d x)^{1+m} \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+m}{2},-2 p;\frac{3+m}{2};-\frac{b x^2}{a}\right )}{d (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0192336, size = 66, normalized size = 0.89 \[ \frac{x (d x)^m \left (\left (a+b x^2\right )^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-2 p} \, _2F_1\left (\frac{m+1}{2},-2 p;\frac{m+1}{2}+1;-\frac{b x^2}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.331, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p}\, 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{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{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 ({\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \left (d x\right )^{m}, 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 \left (d x\right )^{m} \left (\left (a + b x^{2}\right )^{2}\right )^{p}\, 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{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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