Optimal. Leaf size=74 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {453}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt {2} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt {2} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 453
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 1.98, size = 65, normalized size = 0.88 \begin {gather*} -\frac {-\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )+\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (-b \,x^{2}-2\right ) \left (-b \,x^{2}-1\right )^{\frac {3}{4}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs.
\(2 (57) = 114\).
time = 0.71, size = 274, normalized size = 3.70 \begin {gather*} \left [-\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (-b x^{2} - 1\right )}^{\frac {1}{4}}}{\sqrt {b} x}\right ) - \sqrt {2} \sqrt {b} \log \left (-\frac {b^{2} x^{4} + 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} + 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}, \frac {2 \, \sqrt {2} \sqrt {-b} \arctan \left (\frac {\sqrt {2} {\left (-b x^{2} - 1\right )}^{\frac {1}{4}} \sqrt {-b}}{b x}\right ) - \sqrt {2} \sqrt {-b} \log \left (-\frac {b^{2} x^{4} - 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} - 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {-b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}\right ] \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}}{b x^{2} \left (- b x^{2} - 1\right )^{\frac {3}{4}} + 2 \left (- b x^{2} - 1\right )^{\frac {3}{4}}}\, dx \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*} \text {could not integrate} \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^2-1\right )}^{3/4}\,\left (b\,x^2+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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