Optimal. Leaf size=119 \[ \frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (\frac {\sqrt {a-b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {441} \[ \frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (\frac {\sqrt {a-b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 441
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a-b x^2\right )^{3/4} \left (2 a-b x^2\right )} \, dx &=\frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 68, normalized size = 0.57 \[ \frac {x^3 \left (\frac {a-b x^2}{a}\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};\frac {b x^2}{a},\frac {b x^2}{2 a}\right )}{6 a \left (a-b x^2\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.06, size = 211, normalized size = 1.77 \[ -2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \arctan \left (\frac {4 \, {\left (\sqrt {\frac {1}{2}} \left (\frac {1}{4}\right )^{\frac {3}{4}} a b^{4} x \sqrt {\frac {b^{4} x^{2} \sqrt {-\frac {1}{a b^{6}}} + 2 \, \sqrt {-b x^{2} + a}}{x^{2}}} \left (-\frac {1}{a b^{6}}\right )^{\frac {3}{4}} - \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a b^{4} \left (-\frac {1}{a b^{6}}\right )^{\frac {3}{4}}\right )}}{x}\right ) - \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} - {\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) \]
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 \[ \int -\frac {x^{2}}{{\left (b x^{2} - 2 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (-b \,x^{2}+a \right )^{\frac {3}{4}} \left (-b \,x^{2}+2 a \right )}\, dx \]
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 \[ -\int \frac {x^{2}}{{\left (b x^{2} - 2 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {x^2}{{\left (a-b\,x^2\right )}^{3/4}\,\left (2\,a-b\,x^2\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{2}}{- 2 a \left (a - b x^{2}\right )^{\frac {3}{4}} + b x^{2} \left (a - b x^{2}\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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