Optimal. Leaf size=67 \[ \frac{2 (d x)^{3/2} \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac{3}{4},-2 p;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 d} \]
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Rubi [A] time = 0.0204767, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1113, 364} \[ \frac{2 (d x)^{3/2} \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (\frac{3}{4},-2 p;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 1113
Rule 364
Rubi steps
\begin{align*} \int \sqrt{d x} \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 \sqrt{d x} \left (1+\frac{b x^2}{a}\right )^{2 p} \, dx\\ &=\frac{2 (d x)^{3/2} \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{3}{4},-2 p;\frac{7}{4};-\frac{b x^2}{a}\right )}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0085003, size = 56, normalized size = 0.84 \[ \frac{2}{3} x \sqrt{d x} \left (\left (a+b x^2\right )^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-2 p} \, _2F_1\left (\frac{3}{4},-2 p;\frac{7}{4};-\frac{b x^2}{a}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.171, size = 0, normalized size = 0. \begin{align*} \int \sqrt{dx} \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 \sqrt{d x}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}\,{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 (\sqrt{d x}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}, 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 \sqrt{d x} \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 \sqrt{d x}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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