Optimal. Leaf size=98 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {442} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 442
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-2 a-b x^2\right ) \left (-a-b x^2\right )^{3/4}} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 70, normalized size = 0.71 \begin {gather*} -\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}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 2.32, size = 100, normalized size = 1.02 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}{\sqrt {b} x}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 211, normalized size = 2.15 \begin {gather*} 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 ) \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 (b x^{2} + 2 \, a\right )} {\left (-b x^{2} - a\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (-b \,x^{2}-2 a \right ) \left (-b \,x^{2}-a \right )^{\frac {3}{4}}}\, dx \end {gather*}
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 (b x^{2} + 2 \, a\right )} {\left (-b x^{2} - a\right )}^{\frac {3}{4}}}\,{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 (-b\,x^2-a\right )}^{3/4}\,\left (b\,x^2+2\,a\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}}{2 a \left (- a - b x^{2}\right )^{\frac {3}{4}} + b x^{2} \left (- a - b x^{2}\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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