Optimal. Leaf size=103 \[ \frac{b^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^2}}+\frac{b \left (a-b x^2\right )^{3/4}}{2 a x}-\frac{\left (a-b x^2\right )^{3/4}}{3 x^3} \]
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Rubi [A] time = 0.0317913, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {277, 325, 229, 228} \[ \frac{b^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a-b x^2}}+\frac{b \left (a-b x^2\right )^{3/4}}{2 a x}-\frac{\left (a-b x^2\right )^{3/4}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 277
Rule 325
Rule 229
Rule 228
Rubi steps
\begin{align*} \int \frac{\left (a-b x^2\right )^{3/4}}{x^4} \, dx &=-\frac{\left (a-b x^2\right )^{3/4}}{3 x^3}-\frac{1}{2} b \int \frac{1}{x^2 \sqrt [4]{a-b x^2}} \, dx\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{3 x^3}+\frac{b \left (a-b x^2\right )^{3/4}}{2 a x}+\frac{b^2 \int \frac{1}{\sqrt [4]{a-b x^2}} \, dx}{4 a}\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{3 x^3}+\frac{b \left (a-b x^2\right )^{3/4}}{2 a x}+\frac{\left (b^2 \sqrt [4]{1-\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1-\frac{b x^2}{a}}} \, dx}{4 a \sqrt [4]{a-b x^2}}\\ &=-\frac{\left (a-b x^2\right )^{3/4}}{3 x^3}+\frac{b \left (a-b x^2\right )^{3/4}}{2 a x}+\frac{b^{3/2} \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 )}{2 \sqrt{a} \sqrt [4]{a-b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0094421, size = 52, normalized size = 0.5 \[ -\frac{\left (a-b x^2\right )^{3/4} \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};-\frac{1}{2};\frac{b x^2}{a}\right )}{3 x^3 \left (1-\frac{b x^2}{a}\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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^{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}}}{x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.18824, size = 36, normalized size = 0.35 \begin{align*} - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{3 x^{3}} \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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