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 = 0.03, 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 364
Rule 1113
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*}
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Mathematica [A] time = 0.02, 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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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \left (d x\right )^{m}, x\right ) \]
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 \[ \int {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}\, dx \]
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 \[ \int {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,x\right )}^m\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p \,d x \]
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 \[ \int \left (d x\right )^{m} \left (\left (a + b x^{2}\right )^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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