Optimal. Leaf size=81 \[ \frac{4 a^{3/2} \left (1-\frac{b x^2}{a}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{3 b^{3/2} \left (a-b x^2\right )^{3/4}}-\frac{2 x \sqrt [4]{a-b x^2}}{3 b} \]
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Rubi [A] time = 0.0216295, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {321, 233, 232} \[ \frac{4 a^{3/2} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 b^{3/2} \left (a-b x^2\right )^{3/4}}-\frac{2 x \sqrt [4]{a-b x^2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 233
Rule 232
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} \sin ^{-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 = 0.0195983, size = 64, normalized size = 0.79 \[ \frac{2 x \left (a \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right )-a+b x^2\right )}{3 b \left (a-b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.013, 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{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} x^{2}}{b x^{2} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.725392, size = 29, normalized size = 0.36 \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^{2 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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