Optimal. Leaf size=78 \[ \frac{6 \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{2 x}{5 a \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.0173645, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {199, 197, 196} \[ \frac{6 \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{2 x}{5 a \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 197
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{9/4}} \, dx &=\frac{2 x}{5 a \left (a+b x^2\right )^{5/4}}+\frac{3 \int \frac{1}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a}\\ &=\frac{2 x}{5 a \left (a+b x^2\right )^{5/4}}+\frac{\left (3 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{5 a^2 \sqrt [4]{a+b x^2}}\\ &=\frac{2 x}{5 a \left (a+b x^2\right )^{5/4}}+\frac{6 \sqrt [4]{1+\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt{b} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0313522, size = 72, normalized size = 0.92 \[ \frac{-3 x \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+8 a x+6 b x^3}{5 a^2 \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.035, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{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{9}{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{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}{b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.77887, size = 24, normalized size = 0.31 \begin{align*} \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{9}{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{9}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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