Optimal. Leaf size=52 \[ \frac {2 x^{3/2}}{3}+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1802, 827, 1162, 617, 204} \[ \frac {2 x^{3/2}}{3}+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 827
Rule 1162
Rule 1802
Rubi steps
\begin {align*} \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx &=\int \left (\sqrt {x}-\frac {1+x}{\sqrt {x} \left (1+x^2\right )}\right ) \, dx\\ &=\frac {2 x^{3/2}}{3}-\int \frac {1+x}{\sqrt {x} \left (1+x^2\right )} \, dx\\ &=\frac {2 x^{3/2}}{3}-2 \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 x^{3/2}}{3}-\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )-\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 x^{3/2}}{3}-\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )+\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right )\\ &=\frac {2 x^{3/2}}{3}+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )-\sqrt {2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 1.00 \[ \frac {2 x^{3/2}}{3}+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )-\sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 23, normalized size = 0.44 \[ \frac {2}{3} \, x^{\frac {3}{2}} - \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 1\right )}}{2 \, \sqrt {x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 46, normalized size = 0.88 \[ \frac {2}{3} \, x^{\frac {3}{2}} - \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 97, normalized size = 1.87 \[ \frac {2 x^{\frac {3}{2}}}{3}-\sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )-\sqrt {2}\, \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )-\frac {\sqrt {2}\, \ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )}{4}-\frac {\sqrt {2}\, \ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.81, size = 46, normalized size = 0.88 \[ \frac {2}{3} \, x^{\frac {3}{2}} - \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.17, size = 43, normalized size = 0.83 \[ \frac {2\,x^{3/2}}{3}-\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x}}{2}+\frac {\sqrt {2}\,x^{3/2}}{2}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x}}{2}\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 44, normalized size = 0.85 \[ \frac {2 x^{\frac {3}{2}}}{3} - \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )} - \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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