Optimal. Leaf size=46 \[ \frac {\sqrt {2} b \sinh ^{-1}\left (\frac {b \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x}{\sqrt {a}}\right )}{\sqrt {a}} \]
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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 = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {2131, 2130, 215} \[ \frac {\sqrt {2} b \sinh ^{-1}\left (\frac {b \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x}{\sqrt {a}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 215
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 \sinh ^{-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 = 0.14, size = 148, normalized size = 3.22 \[ \frac {\sqrt {2} x \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \tanh ^{-1}\left (\frac {\sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}}{\sqrt {2} a x}\right )}{\sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \left (x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 14.40, size = 161, normalized size = 3.50 \[ \left [\frac {\sqrt {2} b \log \left (-4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (\sqrt {2} \sqrt {a} x + \frac {\sqrt {2} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}{\sqrt {a}}\right )} + 1\right )}{2 \, \sqrt {a}}, -\sqrt {2} b \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} \sqrt {-\frac {1}{a}}}{2 \, x}\right )\right ] \]
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 \[ \int \frac {\sqrt {x\,\left (a\,x+b\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}\right )}}{x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}} \,d x \]
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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