Optimal. Leaf size=73 \[ \frac{\left (a+b x^2\right ) (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a d (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.0308932, antiderivative size = 73, 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 = {1112, 364} \[ \frac{\left (a+b x^2\right ) (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a d (m+1) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 364
Rubi steps
\begin{align*} \int \frac{(d x)^m}{\sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{\left (a b+b^2 x^2\right ) \int \frac{(d x)^m}{a b+b^2 x^2} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{(d x)^{1+m} \left (a+b x^2\right ) \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{b x^2}{a}\right )}{a d (1+m) \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0194491, size = 62, normalized size = 0.85 \[ \frac{x \left (a+b x^2\right ) (d x)^m \, _2F_1\left (1,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right )}{a (m+1) \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.266, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx \right ) ^{m}{\frac{1}{\sqrt{{b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{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 (d x\right )^{m}}{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{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{\left (d x\right )^{m}}{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{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 (d x\right )^{m}}{\sqrt{\left (a + b x^{2}\right )^{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 (d x\right )^{m}}{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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