Optimal. Leaf size=140 \[ 2 a \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^5 a^2-2 \text {$\#$1}^4 b+b^2\& ,\frac {\text {$\#$1} a^2 \log \left (\sqrt {a x-\sqrt {a^2 x^2+b}}-\text {$\#$1}\right )+b \log \left (\sqrt {a x-\sqrt {a^2 x^2+b}}-\text {$\#$1}\right )}{-2 \text {$\#$1}^4+5 \text {$\#$1} a^2+2 b}\& \right ]-a \log \left (\sqrt {a^2 x^2+b}-a x\right ) \]
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Rubi [F] time = 4.84, 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 {b+a^2 x^2}}{x^2-\sqrt {a 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 {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx &=\int \left (\frac {x^2 \left (b+a^2 x^2\right )}{b+2 a x^5-x^8}+\frac {x^3 \sqrt {b+a^2 x^2} \left (-a+x^3\right )}{-b-2 a x^5+x^8}+\frac {a x \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}-\frac {x^4 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}+\frac {\left (b+a^2 x^2\right ) \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}\right ) \, dx\\ &=a \int \frac {x \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\int \frac {x^2 \left (b+a^2 x^2\right )}{b+2 a x^5-x^8} \, dx+\int \frac {x^3 \sqrt {b+a^2 x^2} \left (-a+x^3\right )}{-b-2 a x^5+x^8} \, dx-\int \frac {x^4 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\int \frac {\left (b+a^2 x^2\right ) \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx\\ &=a \int \frac {x \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx-\int \frac {x^4 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\int \left (\frac {b x^2}{b+2 a x^5-x^8}+\frac {a^2 x^4}{b+2 a x^5-x^8}\right ) \, dx+\int \left (\frac {a x^3 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8}+\frac {x^6 \sqrt {b+a^2 x^2}}{-b-2 a x^5+x^8}\right ) \, dx+\int \left (\frac {b \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}+\frac {a^2 x^2 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}\right ) \, dx\\ &=a \int \frac {x^3 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8} \, dx+a \int \frac {x \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+a^2 \int \frac {x^4}{b+2 a x^5-x^8} \, dx+a^2 \int \frac {x^2 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+b \int \frac {x^2}{b+2 a x^5-x^8} \, dx+b \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\int \frac {x^6 \sqrt {b+a^2 x^2}}{-b-2 a x^5+x^8} \, dx-\int \frac {x^4 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx\\ \end {align*}
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Mathematica [F] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.61, size = 140, normalized size = 1.00 \begin {gather*} -a \log \left (-a x+\sqrt {b+a^2 x^2}\right )+2 a \text {RootSum}\left [b^2-2 b \text {$\#$1}^4-4 a^2 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {b \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right )+a^2 \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{2 b+5 a^2 \text {$\#$1}-2 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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^{2} x^{2} + b}}{x^{2} - \sqrt {a 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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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}+b}}{x^{2}-\sqrt {a x -\sqrt {a^{2} x^{2}+b}}}\, dx\]
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^{2} x^{2} + b}}{x^{2} - \sqrt {a 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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\sqrt {a^2\,x^2+b}}{\sqrt {a\,x-\sqrt {a^2\,x^2+b}}-x^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 {\sqrt {a^{2} x^{2} + b}}{x^{2} - \sqrt {a x - \sqrt {a^{2} x^{2} + b}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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