Optimal. Leaf size=76 \[ -\frac{\left (a-b x^2\right )^{3/4}}{x}-\frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}} \]
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Rubi [A] time = 0.0204751, antiderivative size = 76, 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 = {277, 229, 228} \[ -\frac{\left (a-b x^2\right )^{3/4}}{x}-\frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 277
Rule 229
Rule 228
Rubi steps
\begin{align*} \int \frac{\left (a-b x^2\right )^{3/4}}{x^2} \, dx &=-\frac{\left (a-b x^2\right )^{3/4}}{x}-\frac{1}{2} (3 b) \int \frac{1}{\sqrt [4]{a-b x^2}} \, dx\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{x}-\frac{\left (3 b \sqrt [4]{1-\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1-\frac{b x^2}{a}}} \, dx}{2 \sqrt [4]{a-b x^2}}\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{x}-\frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0087876, size = 50, normalized size = 0.66 \[ -\frac{\left (a-b x^2\right )^{3/4} \, _2F_1\left (-\frac{3}{4},-\frac{1}{2};\frac{1}{2};\frac{b x^2}{a}\right )}{x \left (1-\frac{b x^2}{a}\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{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{{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}{x^{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 (-b x^{2} + a\right )}^{\frac{3}{4}}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.03792, size = 31, normalized size = 0.41 \begin{align*} - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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