Optimal. Leaf size=107 \[ \frac {4 b x}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \sqrt {b^2+a x^2}}{3 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {4 b^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]
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Rubi [F]
time = 0.61, 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 (\left (2 b^2\right ) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b^2+a x^2} \, dx\right )+\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 [A]
time = 0.23, size = 86, normalized size = 0.80 \begin {gather*} \frac {2 x \left (2 b+\sqrt {b^2+a x^2}\right )}{3 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {4 b^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}-b^{2}\right ) \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{a \,x^{2}+b^{2}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\left (a\,x^2-b^2\right )\,\sqrt {b+\sqrt {b^2+a\,x^2}}}{b^2+a\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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