Optimal. Leaf size=71 \[ -\frac {\sqrt {2} \log \left (\sqrt {a^2 x^2-b}-\sqrt {2} \sqrt {a} \sqrt {x \sqrt {a^2 x^2-b}+a x^2}+a x\right )}{\sqrt {a}} \]
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Rubi [F] time = 1.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}{x \sqrt {-b+a^2 x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}{x \sqrt {-b+a^2 x^2}} \, dx &=\int \frac {\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}{x \sqrt {-b+a^2 x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.14, size = 104, normalized size = 1.46 \begin {gather*} \frac {\sqrt {2} \sqrt {x \left (\sqrt {a^2 x^2-b}+a x\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}}\right )}{\sqrt {a} \sqrt {x} \sqrt {\sqrt {a^2 x^2-b}+a x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.56, size = 71, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \log \left (a x+\sqrt {-b+a^2 x^2}-\sqrt {2} \sqrt {a} \sqrt {a x^2+x \sqrt {-b+a^2 x^2}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 18.60, size = 142, normalized size = 2.00 \begin {gather*} \left [\frac {\sqrt {2} \log \left (-4 \, a^{2} x^{2} - 4 \, \sqrt {a^{2} x^{2} - b} a x - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a^{2} x^{2} - b} \sqrt {a}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b} x} + b\right )}{2 \, \sqrt {a}}, -\sqrt {2} \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b} x} \sqrt {-\frac {1}{a}}}{2 \, x}\right )\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 {\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b} x}}{\sqrt {a^{2} x^{2} - b} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a \,x^{2}+x \sqrt {a^{2} x^{2}-b}}}{x \sqrt {a^{2} x^{2}-b}}\, dx\]
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*} \int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b} x}}{\sqrt {a^{2} x^{2} - b} x}\,{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 {\sqrt {x\,\sqrt {a^2\,x^2-b}+a\,x^2}}{x\,\sqrt {a^2\,x^2-b}} \,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 {\sqrt {x \left (a x + \sqrt {a^{2} x^{2} - b}\right )}}{x \sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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