Optimal. Leaf size=101 \[ \frac {4 a^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a-b x^2}}-\frac {2 a x \left (a-b x^2\right )^{3/4}}{15 b}+\frac {2}{9} x^3 \left (a-b x^2\right )^{3/4} \]
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Rubi [A] time = 0.03, antiderivative size = 101, 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 = {279, 321, 229, 228} \[ \frac {4 a^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a-b x^2}}+\frac {2}{9} x^3 \left (a-b x^2\right )^{3/4}-\frac {2 a x \left (a-b x^2\right )^{3/4}}{15 b} \]
Antiderivative was successfully verified.
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Rule 228
Rule 229
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^2 \left (a-b x^2\right )^{3/4} \, dx &=\frac {2}{9} x^3 \left (a-b x^2\right )^{3/4}+\frac {1}{3} a \int \frac {x^2}{\sqrt [4]{a-b x^2}} \, dx\\ &=-\frac {2 a x \left (a-b x^2\right )^{3/4}}{15 b}+\frac {2}{9} x^3 \left (a-b x^2\right )^{3/4}+\frac {\left (2 a^2\right ) \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{15 b}\\ &=-\frac {2 a x \left (a-b x^2\right )^{3/4}}{15 b}+\frac {2}{9} x^3 \left (a-b x^2\right )^{3/4}+\frac {\left (2 a^2 \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{15 b \sqrt [4]{a-b x^2}}\\ &=-\frac {2 a x \left (a-b x^2\right )^{3/4}}{15 b}+\frac {2}{9} x^3 \left (a-b x^2\right )^{3/4}+\frac {4 a^{5/2} \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 )}{15 b^{3/2} \sqrt [4]{a-b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 64, normalized size = 0.63 \[ \frac {2 x \left (a-b x^2\right )^{3/4} \left (\frac {a \, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {3}{2};\frac {b x^2}{a}\right )}{\left (1-\frac {b x^2}{a}\right )^{3/4}}-a+b x^2\right )}{9 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-b x^{2} + a\right )}^{\frac {3}{4}} x^{2}, x\right ) \]
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 \[ \int {\left (-b x^{2} + a\right )}^{\frac {3}{4}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (-b \,x^{2}+a \right )^{\frac {3}{4}} x^{2}\, dx \]
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 \[ \int {\left (-b x^{2} + a\right )}^{\frac {3}{4}} x^{2}\,{d x} \]
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 \[ \int x^2\,{\left (a-b\,x^2\right )}^{3/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.08, size = 31, normalized size = 0.31 \[ \frac {a^{\frac {3}{4}} x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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