Optimal. Leaf size=431 \[ \frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {c}}\right )}{16 a c^{7/2}}+\frac {\sqrt {a^2 x^2-b} \left (\left (-2560 a^2 c^4 x^2-2048 a c^6 x-1575 b^2+640 b c^4\right ) \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}+\sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c} \left (2240 a^2 c^3 x^2+1536 a c^5 x-10640 b c^3+2048 c^7\right )\right )+\sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c} \left (2240 a^3 c^3 x^3+1536 a^2 c^5 x^2-11760 a b c^3 x+2048 a c^7 x+1050 b^2 c-768 b c^5\right )+\left (-2560 a^3 c^4 x^3-2048 a^2 c^6 x^2-1575 a b^2 x+1920 a b c^4 x-840 b^2 c^2+1024 b c^6\right ) \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+c}}{5040 a c^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}} \]
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Rubi [F] time = 1.09, 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} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\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} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}
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Mathematica [F] time = 10.90, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.10, size = 431, normalized size = 1.00 \begin {gather*} \frac {\left (-840 b^2 c^2+1024 b c^6-1575 a b^2 x+1920 a b c^4 x-2048 a^2 c^6 x^2-2560 a^3 c^4 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (1050 b^2 c-768 b c^5-11760 a b c^3 x+2048 a c^7 x+1536 a^2 c^5 x^2+2240 a^3 c^3 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-1575 b^2+640 b c^4-2048 a c^6 x-2560 a^2 c^4 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (-10640 b c^3+2048 c^7+1536 a c^5 x+2240 a^2 c^3 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{5040 a c^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}+\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{16 a c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 564, normalized size = 1.31 \begin {gather*} \left [\frac {1575 \, b^{2} \sqrt {c} \log \left (2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} - b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}} + 2 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} + b\right ) + 2 \, {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} - 10640 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} + 560 \, a c^{4} x - 525 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (1024 \, c^{7} + 1680 \, a^{2} c^{3} x^{2} - 840 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 315 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} - 336 \, a c^{3} x - 315 \, b c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{10080 \, a c^{4}}, -\frac {1575 \, b^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{c}\right ) - {\left (2048 \, c^{8} + 1120 \, a^{2} c^{4} x^{2} - 10640 \, b c^{4} + 6 \, {\left (128 \, a c^{6} + 175 \, a b c^{2}\right )} x + 2 \, {\left (384 \, c^{6} + 560 \, a c^{4} x - 525 \, b c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (1024 \, c^{7} + 1680 \, a^{2} c^{3} x^{2} - 840 \, b c^{3} + 5 \, {\left (128 \, a c^{5} + 315 \, a b c\right )} x + 5 \, {\left (128 \, c^{5} - 336 \, a c^{3} x - 315 \, b c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{5040 \, a c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}\, \sqrt {a x +\sqrt {a^{2} x^{2}-b}}}{\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}\, dx\]
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} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{\sqrt {c + \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 = -0.00 \begin {gather*} \int \frac {\sqrt {a\,x+\sqrt {a^2\,x^2-b}}\,\sqrt {a^2\,x^2-b}}{\sqrt {c+\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}{\sqrt {c + \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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