Optimal. Leaf size=294 \[ \frac {2 \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{a}-\frac {\sqrt {\sqrt {2} \sqrt {b}-2 \sqrt {a} c} \left (-\sqrt {b}+\sqrt {2} \sqrt {a} c\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {\sqrt {2} \sqrt {b}-2 \sqrt {a} c}}\right )}{a^{5/4} \left (-\sqrt {2} \sqrt {b}+2 \sqrt {a} c\right )}-\frac {\left (\sqrt {b}+\sqrt {2} \sqrt {a} c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {\sqrt {2} \sqrt {b}+2 \sqrt {a} c}}\right )}{a^{5/4} \sqrt {\sqrt {2} \sqrt {b}+2 \sqrt {a} c}} \]
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Rubi [F]
time = 0.29, 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 {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\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 {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx &=\int \frac {\sqrt {c+\sqrt {a x^2+x \sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 5.03, size = 275, normalized size = 0.94 \begin {gather*} \frac {4 \sqrt {a} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}-\sqrt {2 \sqrt {2} \sqrt {a} \sqrt {b}-4 a c} \text {ArcTan}\left (\frac {\sqrt {2 \sqrt {2} \sqrt {a} \sqrt {b}-4 a c} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {2} \sqrt {b}-2 \sqrt {a} c}\right )+\sqrt {2} \sqrt {-\sqrt {2} \sqrt {a} \sqrt {b}-2 a c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {-\sqrt {2} \sqrt {a} \sqrt {b}-2 a c} \sqrt {c+\sqrt {x \left (a x+\sqrt {-b+a^2 x^2}\right )}}}{\sqrt {2} \sqrt {b}+2 \sqrt {a} c}\right )}{2 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c +\sqrt {a \,x^{2}+x \sqrt {a^{2} x^{2}-b}}}}{\sqrt {a^{2} x^{2}-b}}\, dx\]
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*} \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 {c + \sqrt {a x^{2} + x \sqrt {a^{2} x^{2} - b}}}}{\sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\sqrt {c+\sqrt {x\,\sqrt {a^2\,x^2-b}+a\,x^2}}}{\sqrt {a^2\,x^2-b}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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