Optimal. Leaf size=56 \[ \frac{2 \sqrt{a} \left (\frac{b x^2}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{\sqrt{b} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.0096891, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {233, 231} \[ \frac{2 \sqrt{a} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 233
Rule 231
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx &=\frac{\left (1+\frac{b x^2}{a}\right )^{3/4} \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx}{\left (a+b x^2\right )^{3/4}}\\ &=\frac{2 \sqrt{a} \left (1+\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.006091, size = 46, normalized size = 0.82 \[ \frac{x \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )}{\left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}\, 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{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{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{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.68075, size = 24, normalized size = 0.43 \begin{align*} \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{3}{4}}} \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{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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