Optimal. Leaf size=46 \[ \frac {\sqrt {2} b \sin ^{-1}\left (\frac {a x-b \sqrt {\frac {a^2 x^2}{b^2}+\frac {a}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rubi [A] time = 1.17, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2131, 2130, 216} \[ \frac {\sqrt {2} b \sin ^{-1}\left (\frac {a x-b \sqrt {\frac {a^2 x^2}{b^2}+\frac {a}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 216
Rule 2130
Rule 2131
Rubi steps
\begin {align*} \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx &=\int \frac {\sqrt {-a x^2+b x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\\ &=-\frac {\left (\sqrt {2} b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}{a}\\ &=\frac {\sqrt {2} b \sin ^{-1}\left (\frac {a x-b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [B] time = 0.19, size = 161, normalized size = 3.50 \[ \frac {\sqrt {2} b^2 \sqrt {\frac {a \left (a x^2+1\right )}{b^2}} \sqrt {a x \left (a x-b \sqrt {\frac {a \left (a x^2+1\right )}{b^2}}\right )} \sqrt {x \left (b \sqrt {\frac {a \left (a x^2+1\right )}{b^2}}-a x\right )} \tanh ^{-1}\left (\frac {\sqrt {a x \left (a x-b \sqrt {\frac {a \left (a x^2+1\right )}{b^2}}\right )}}{\sqrt {2} a x}\right )}{a^2 \left (-b x^2 \sqrt {\frac {a \left (a x^2+1\right )}{b^2}}+a x^3+x\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 12.96, size = 161, normalized size = 3.50 \[ \left [\frac {1}{2} \, \sqrt {2} b \sqrt {-\frac {1}{a}} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}} + 2 \, \sqrt {-a x^{2} + b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}} {\left (\sqrt {2} a x \sqrt {-\frac {1}{a}} - \sqrt {2} b \sqrt {-\frac {1}{a}} \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}\right )} + 1\right ), -\frac {\sqrt {2} b \arctan \left (\frac {\sqrt {2} \sqrt {-a x^{2} + b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}}}{2 \, \sqrt {a} x}\right )}{\sqrt {a}}\right ] \]
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 \[ \int \frac {\sqrt {-{\left (a x - \sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} b\right )} x}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\left (-a x +\sqrt {\frac {a^{2} x^{2}}{b^{2}}+\frac {a}{b^{2}}}\, b \right ) x}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}}+\frac {a}{b^{2}}}\, x}\, dx \]
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 \[ \int \frac {\sqrt {-{\left (a x - \sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} b\right )} x}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {-x\,\left (a\,x-b\,\sqrt {\frac {a}{b^2}+\frac {a^2\,x^2}{b^2}}\right )}}{x\,\sqrt {\frac {a}{b^2}+\frac {a^2\,x^2}{b^2}}} \,d x \]
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 \[ \int \frac {\sqrt {- x \left (a x - b \sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}}\right )}}{x \sqrt {\frac {a \left (a x^{2} + 1\right )}{b^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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