Optimal. Leaf size=99 \[ -\frac {3 \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a-b x^2}}-\frac {3 \left (a-b x^2\right )^{3/4}}{a^2 x}+\frac {2}{a x \sqrt [4]{a-b x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {290, 325, 229, 228} \[ -\frac {3 \left (a-b x^2\right )^{3/4}}{a^2 x}-\frac {3 \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a-b x^2}}+\frac {2}{a x \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 228
Rule 229
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a-b x^2\right )^{5/4}} \, dx &=\frac {2}{a x \sqrt [4]{a-b x^2}}+\frac {3 \int \frac {1}{x^2 \sqrt [4]{a-b x^2}} \, dx}{a}\\ &=\frac {2}{a x \sqrt [4]{a-b x^2}}-\frac {3 \left (a-b x^2\right )^{3/4}}{a^2 x}-\frac {(3 b) \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{2 a^2}\\ &=\frac {2}{a x \sqrt [4]{a-b x^2}}-\frac {3 \left (a-b x^2\right )^{3/4}}{a^2 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 a^2 \sqrt [4]{a-b x^2}}\\ &=\frac {2}{a x \sqrt [4]{a-b x^2}}-\frac {3 \left (a-b x^2\right )^{3/4}}{a^2 x}-\frac {3 \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 )}{a^{3/2} \sqrt [4]{a-b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 53, normalized size = 0.54 \[ -\frac {\sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (-\frac {1}{2},\frac {5}{4};\frac {1}{2};\frac {b x^2}{a}\right )}{a x \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b^{2} x^{6} - 2 \, a b x^{4} + a^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {5}{4}} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.10, size = 41, normalized size = 0.41 \[ -\frac {2\,{\left (1-\frac {a}{b\,x^2}\right )}^{5/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{4},\frac {7}{4};\ \frac {11}{4};\ \frac {a}{b\,x^2}\right )}{7\,x\,{\left (a-b\,x^2\right )}^{5/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.09, size = 29, normalized size = 0.29 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{a^{\frac {5}{4}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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