Optimal. Leaf size=530 \[ \frac {4 b x}{3 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 x \sqrt {a x^2+b^2}}{3 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 \sqrt {\sqrt {2}-1} b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 i \left (\sqrt {2 \left (\sqrt {2}-1\right )} b^{3/2}+\sqrt {\sqrt {2}-1} b^{3/2}\right ) \tan ^{-1}\left (\frac {\frac {i a x}{\sqrt {\sqrt {a x^2+b^2}+b}}-i \sqrt {a} \sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {1+\sqrt {2}} b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 i \left (\sqrt {2 \left (1+\sqrt {2}\right )} b^{3/2}-\sqrt {1+\sqrt {2}} b^{3/2}\right ) \tanh ^{-1}\left (\frac {\frac {i a x}{\sqrt {\sqrt {a x^2+b^2}+b}}-i \sqrt {a} \sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}} \]
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Rubi [F] time = 0.92, 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 {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx &=\int \left (\sqrt {b+\sqrt {b^2+a x^2}}+\frac {2 b^2 \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2}\right ) \, dx\\ &=\left (2 b^2\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx+\int \sqrt {b+\sqrt {b^2+a x^2}} \, dx\\ &=\frac {2 a x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {2 b x}{\sqrt {b+\sqrt {b^2+a x^2}}}+\left (2 b^2\right ) \int \left (-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 b \left (b-\sqrt {a} x\right )}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 b \left (b+\sqrt {a} x\right )}\right ) \, dx\\ &=\frac {2 a x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {2 b x}{\sqrt {b+\sqrt {b^2+a x^2}}}-b \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {a} x} \, dx-b \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {a} x} \, dx\\ \end {align*}
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Mathematica [F] time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.53, size = 195, normalized size = 0.37 \begin {gather*} \frac {4 b x}{3 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 x \sqrt {a x^2+b^2}}{3 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 \sqrt {\sqrt {2}-1} b^{3/2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {2}-1} \sqrt {a} x}{\sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{\sqrt {a}}-\frac {2 \sqrt {1+\sqrt {2}} b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{\sqrt {a}} \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 {{\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{a x^{2} - b^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{2}+b^{2}\right ) \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{a \,x^{2}-b^{2}}\, dx \end {gather*}
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 {{\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{a x^{2} - b^{2}}\,{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 = -0.00 \begin {gather*} \int \frac {\left (b^2+a\,x^2\right )\,\sqrt {b+\sqrt {b^2+a\,x^2}}}{a\,x^2-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 {\sqrt {b + \sqrt {a x^{2} + b^{2}}} \left (a x^{2} + b^{2}\right )}{a x^{2} - b^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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