Optimal. Leaf size=65 \[ -\frac{2 \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (-\frac{1}{4},-2 p;\frac{3}{4};-\frac{b x^2}{a}\right )}{d \sqrt{d x}} \]
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Rubi [A] time = 0.0221593, antiderivative size = 65, 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} \[ -\frac{2 \left (\frac{b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (-\frac{1}{4},-2 p;\frac{3}{4};-\frac{b x^2}{a}\right )}{d \sqrt{d x}} \]
Antiderivative was successfully verified.
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Rule 1113
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, 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 \frac{\left (1+\frac{b x^2}{a}\right )^{2 p}}{(d x)^{3/2}} \, dx\\ &=-\frac{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{1}{4},-2 p;\frac{3}{4};-\frac{b x^2}{a}\right )}{d \sqrt{d x}}\\ \end{align*}
Mathematica [A] time = 0.0088025, size = 54, normalized size = 0.83 \[ -\frac{2 x \left (\left (a+b x^2\right )^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-2 p} \, _2F_1\left (-\frac{1}{4},-2 p;\frac{3}{4};-\frac{b x^2}{a}\right )}{(d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.198, size = 0, normalized size = 0. \begin{align*} \int{ \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{p} \left ( dx \right ) ^{-{\frac{3}{2}}}}\, 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 \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\left (d x\right )^{\frac{3}{2}}}\,{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 (\frac{\sqrt{d x}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{d^{2} x^{2}}, 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 \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{p}}{\left (d x\right )^{\frac{3}{2}}}\, 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 \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\left (d x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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