Optimal. Leaf size=77 \[ \frac {2 x}{b \sqrt [4]{a-b x^2}}-\frac {4 \sqrt {a} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a-b x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, 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 = {288, 229, 228} \[ \frac {2 x}{b \sqrt [4]{a-b x^2}}-\frac {4 \sqrt {a} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 228
Rule 229
Rule 288
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a-b x^2\right )^{5/4}} \, dx &=\frac {2 x}{b \sqrt [4]{a-b x^2}}-\frac {2 \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{b}\\ &=\frac {2 x}{b \sqrt [4]{a-b x^2}}-\frac {\left (2 \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{b \sqrt [4]{a-b x^2}}\\ &=\frac {2 x}{b \sqrt [4]{a-b x^2}}-\frac {4 \sqrt {a} \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 )}{b^{3/2} \sqrt [4]{a-b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 56, normalized size = 0.73 \[ \frac {2 x-2 x \sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {b x^2}{a}\right )}{b \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}} x^{2}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {x^{2}}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (-b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx \]
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 \[ \int \frac {x^{2}}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\left (a-b\,x^2\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.93, size = 29, normalized size = 0.38 \[ \frac {x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3 a^{\frac {5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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