Optimal. Leaf size=98 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}{\sqrt {a} x}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \]
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Rubi [A] time = 0.04, antiderivative size = 96, normalized size of antiderivative = 0.98, 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 = {442} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 442
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 68, normalized size = 0.69 \begin {gather*} -\frac {x^3 \left (1-\frac {a x^2}{b}\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};\frac {a x^2}{b},\frac {a x^2}{2 b}\right )}{6 b \left (a x^2-b\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 2.47, size = 98, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 207, normalized size = 2.11 \begin {gather*} 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \arctan \left (\frac {4 \, {\left (\sqrt {\frac {1}{2}} \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{4} b x \sqrt {\frac {a^{4} x^{2} \sqrt {\frac {1}{a^{6} b}} + 2 \, \sqrt {a x^{2} - b}}{x^{2}}} \left (\frac {1}{a^{6} b}\right )^{\frac {3}{4}} - \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a x^{2} - b\right )}^{\frac {1}{4}} a^{4} b \left (\frac {1}{a^{6} b}\right )^{\frac {3}{4}}\right )}}{x}\right ) - \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} - {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{{\left (a x^{2} - b\right )}^{\frac {3}{4}} {\left (a x^{2} - 2 \, b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a \,x^{2}-2 b \right ) \left (a \,x^{2}-b \right )^{\frac {3}{4}}}\, dx\]
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 \begin {gather*} \int \frac {x^{2}}{{\left (a x^{2} - b\right )}^{\frac {3}{4}} {\left (a x^{2} - 2 \, b\right )}}\,{d x} \end {gather*}
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 \begin {gather*} -\int \frac {x^2}{{\left (a\,x^2-b\right )}^{3/4}\,\left (2\,b-a\,x^2\right )} \,d x \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\left (a x^{2} - 2 b\right ) \left (a x^{2} - b\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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