Optimal. Leaf size=78 \[ \frac{2 x \sqrt [4]{a+b x^2}}{3 b}-\frac{4 a^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{3 b^{3/2} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.0195595, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {321, 233, 231} \[ \frac{2 x \sqrt [4]{a+b x^2}}{3 b}-\frac{4 a^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 b^{3/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 233
Rule 231
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x^2\right )^{3/4}} \, dx &=\frac{2 x \sqrt [4]{a+b x^2}}{3 b}-\frac{(2 a) \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx}{3 b}\\ &=\frac{2 x \sqrt [4]{a+b x^2}}{3 b}-\frac{\left (2 a \left (1+\frac{b x^2}{a}\right )^{3/4}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx}{3 b \left (a+b x^2\right )^{3/4}}\\ &=\frac{2 x \sqrt [4]{a+b x^2}}{3 b}-\frac{4 a^{3/2} \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 )}{3 b^{3/2} \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0168684, size = 62, normalized size = 0.79 \[ \frac{2 x \left (-a \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )+a+b x^2\right )}{3 b \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \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{x^{2}}{{\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{x^{2}}{{\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 = 0.71829, size = 27, normalized size = 0.35 \begin{align*} \frac{x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 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{x^{2}}{{\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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