Optimal. Leaf size=43 \[ \frac {2 b \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{a x} \]
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Rubi [F] time = 1.77, 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 {1}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx &=\int \frac {1}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.65, size = 72, normalized size = 1.67 \begin {gather*} \frac {2 \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right )}{\sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.89, size = 43, normalized size = 1.00 \begin {gather*} \frac {2 b \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{a x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 37, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} b}{a x} \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 {1}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\, \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}\, dx\]
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 {1}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} 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 = -0.02 \begin {gather*} \int \frac {1}{x\,\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}} \,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 {1}{x \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )} \sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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